Intelligence Games

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INTELLIGENCEGAMES

Franco Agostini - Nicola Alberto De Carlo

INTELLIGENCEGAMES

Color plates by Lino Simeoniby Chiara Molinaroli and Vittorio Salarolo

PublishedA Fireside Bookby Simon & Schuster, Inc.

New York

Drawings

PhotographsMondadori Archnves, Milan, Italy

Copyrights 1985 byArno do Mondadori Ed tore S p.A r MilanEngl sh language translat on copyright A) 1987 byArnoldo Mondadori Editore S.p.A, MilanAll rights reserved ncludng the right of reproduction in whole or inpart in any form

A Fireside BookPublished by Simon & Schuster, Inc.Simon & Schuster Bu Id ngRockefeller Center1230 Avenue of the AmericasNew York, New York 30020

Orgina iy publ shed as G OCH DELLA INTELLIGENZA byArnoldo Mondadori Ed tore S.p A., in lta y

FIRESIDE and colophon are reg stered trademarks ofSimon & Schuster, Inc.Pr nted and bound n Italy by Off c ne Graf ohe,Arnoldo Moncdador. Editore, Verona1 2 3 4 5 6 7 8 9 10

Library of Congress Cataloging-in-Publication Data

Agostini, Franco.Inte0I gence games

Translation of G ochi della inte I genza."PA Fireside book" --Verso tLp1. Mathematical recreations. I. De Car o, N cola.

II TitleQA95 A3313 1987 793 7'4 86-33908ISBN 0-671-63201-9 (pbk.)

Contents

Foreword 7

A general surveyIntelligence is ..Two facets of intelligenceThis bookConventional wisdom and intelligenceHorny hands and intelligenceA history of intelligenceThe dwarfLet's exercise our intelligenceSenerThe royal game of UrLanguage and intelligencePlaying with wordsNatural "tools" for gamesFirst games with wordsThe game of question and answerDefinitionsWord chainsPalindromesNumbers and intelligenceGames with numbersVisual intelligenceGames with shapesAlquerqueBagh-bandiCity miragesFigures in motionTopological gamesGames with numbers or figures?Memory and intelligenceHow's your memory 7

The spirit of adaptabilityThe tale of AlathielEnigmas, riddles, games of logicThe tools of the trade

Defining intelligenceA curious thing: I.Q.Let's gauge our intelligenceFrom the laboratory to everyday experienceHistorical digressionsOedipus and the Sphinx: a tragic precedent!EnigmaSolitaireFox andgeeseThe intelligence behind riddlesWhat is it that . . ? (Some sample riddles)How many hares have the hunters bagged?How is it possible?How many are we in our family?How old is Peter?A logical riddleThe eyes of the mindWeights and scalesPing-Pong ballsIn the world of oppositesThe two roadsA variantSh rook !Two buffoonsThe meeting with the prime ministerAtthe Assembly of the WiseHow many members?The game never endsA serious gameChinese checkers - AlmaNine men's MorrisA liar from antiquityThe antimony of the liarAn invitation to logicTrue/false: an old dichotomyGames of logicWhat colour are their clothes?

A mixed bunchWho is the guard?Only one sort of logic?A tough case for Inspector BillA problem of logical deductionReasoning with figuresThe game of true and falseCorrect but not true, true but incorrect!Areas without boundsAgainst the mechanicalMind journeysImagination and creativityTwo different ways of thinking, two different ways of

playingMaking up a storyThe story of the exclamation markA square, a circle, and .. . a childThe long history of matchesMatches: to spark the imagination!How did the little match girl die?Four balls, five coins, six matchesAnimals and matchesArt and creativityA first exerciseRe-creative gamesSuccessThe game of successWorkCapacity for synthesisTechnical aptitudeHow is your mechanical skill?GoLoveA sad storyJealousyGames and friendshipWho is it?Guess the personWhat is it?Yes, noThe hidden trickThe analogy gameWho said . . ?Blind associationsA love storyA macabre gameThe same-letter gameA meal in companyAn ancient remedyAt the "Fuli Moon" innSeega - DerrahWar/The point: numbers and imaginationMathematics and realityA synthesisThe intelligent crow

848586888788889293939394

9495959696979798989999

100100101102102105105107113113114116117118118120120121121122122123123124125125126129131131131132

Counting: a human faculty!The tools of countingNumerical bases and systemsHow many hands?The planet of the one-handedThree times one-five = five-oneThe "Black Cat" SocietyThe origins of indigitationThe oldest mode of calculationOne, two, three ... ten: on your fingers"Mathematics gives a V sign"Hand calculationsWhat is XLVlll by CCLXXXVII1?Roman numerals with matchesA system for the human brainThe "farmer's system of multiplication"Only two symbolsIs there a reason?The hidden binary principleA system for a computer "brain"How to count in binary on your fingersA game of strategy in binaryAnother problemLogic and mathematics: true/false - naught/oneAppendix of games with numerical systemsPick your own gamesA brilliant solutionGeometrical figures with matchesBackgammonPuff- alea or tabu/aAdding three matches ...Removing three matchesLet's play with squaresWhat are the coins in Peter's pocket?How many horses has the farmer got?A square and a triangleThe ocean linerHow old was Livy?Tony's socks... and Prudence's glovesThe millThe bridge to the islandA wheel with paddlesClaustrophobiaThe hidden squareGames with clocksMagic squaresThe 14-17 puzzleHow do you make out in mathematics?The prince's legacyThe trapezoid1985: rendezvous with Halley's cometGiotto's cometThe craterSelect bibliographyIndex of games

132133133133134134135136136137141141143143144144144145145145146146148148149153153153'55159160161161163163164164165166165166166166167167168169773175176179179179180181183

Foreword

Al/ of us more or less know what intelligence is: we have probably classed some ofour friends as more intelligent than others. We might say that someone has reached ahigh position because of their intelligence, or we might-perhaps wrongly-thinkthat another has only a humdrum job because he is not very intelligent. Again, weclaim that humans are more intelligent than apes, and apes in turn more intelligentthan cats, and so on. The man in the street will have a rough and ready concept ofwhat intelligence is, which helps him to assess and orientate himself in the society inwhich he lives. But on what is this concept based? What is intelligence, really?

It is not easy to anwer this. A book entitled I ntelligence Ga mes will naturally beexpected to offer some precise definition. And we shall try to meet such expectations.Yet not even the experts (for example, the psychologists) can give a conclusivedefinition. The subject is in the end so vast that it seems impossible to wrap upneatly: whenever one discusses intelligence, one is always left with a feeling ofincompleteness-that something important has been left out. Intelligence is part ofwhat makes a human being. And it is no easy business sorting out what makes ahuman being! However, it is possible to describe certain facets and behaviourpatterns of humankind, starting for instance with the discoveries and formulations ofpsychology. In particular, skill with words, a facility with numbers, and the ability toargue clearly are all accepted as characteristics of intelligence.

Using games, puzzles, and stories, this book deals with verbal, visual,mathematical, and logical forms of intelligence. It must be stressed, though, thatwhile psychology tends to concentrate on those aspects of intelligence that are mosteasily accessible to objective analysis, it also acknowledges that intelligence is asingle faculty, at once a unified whole and an immensely complex entity, embracingthe individual's entire psyche, and is determined by genetic, environmental, andcultural factors.

7

This book is an opportunity for you to reflect about yourself Some of the exercisesare versions of material used in intelligence tests, adapted here in the form of games.Yet there is always the danger of feeling somehow "judged. " Often, newspapers andmagazines promise an objective "measurement" of intelligence that in fact createsmuch doubt and disappointment, because the methods adopted are not set in properproportion. Rather than helping people to know themselves better, they seemdesigned to instill a certain unease. The tests in this book are simply games. Throughthem, each individual will be able to express his or her intelligence and personalityfreely and entertainingly.

Intelligence has been understood differently over the years, and this book alsotraces the evolution of our primitive faculty for solving problems of survival toabstract notions of intelligence such as were held by the Greeks; then on to Romancall id itas-a down-to-earth, practical quality; thence to the quick, lively, dynamicability of the emergent mercantile bourgeoisie in the late Middle Ages to attain acertain goal or resolve unforeseen crises; and finally to the developments of modernpsychology, which sets the problem of intelligence in a systematic, organic overallview. If our intelligence is expressed not merely in the traditionally understood ways,but involves our total being, then we are entitled to ask how it manifests itself inrelation to otherpeople-to friends especially, or in our choice of a partner, our work,or our desire for a successful career and financial position. We attempt to answerthese questions, always inviting the reader to step outside his or her own self bymeans of games, tests, and exercises. Intelligence is to some degree the ability to seeoneself from the outside, with that irony and spirit of freedom through which we areable to feel both mastery of and solidarity with our own selves. And this is not al/:every moment of life can be lived with intelligence Good humour, a sense of thecomic, the ability to see problems and difficulties for what they are-real savoi r-vivre,in fact-a/I help dispel that sense of boredom, ennui, and emptiness that some dayscan bring. Life becomes a true joy when wine, good food, company, wit, and humourappropriate to the time and place combine with a basically balanced life-style,dictated by good sense. Intelligence under these conditions can be seen as the abilityto spend one's days happily, rather than fixed on a distant, abstract happinessprojected into the future. The sixteen colour plates show "mental games'' from a/Iover the world and from every age. Cards, checkers, chess, dominoes, and other verycommon games have not been included. Many books already exist on such games(some are included in the bibliography at the end of this book).

a

A general surveyAl/intel/eftual improvementarises from leisure. Samuel Johnsor

Intelligence is ...

Intelligence is a credit card. Anyone possessing it isthought to be able to face the most tangled prob-lems and solve them. In everyday use, the adjective"intelligent" implies a number of qualities: the abil-ity to identify objectives quickly and to achievethem; sensitivity in dealings with others; skill inassessing people's characters; balanced judgment;and readiness to alter one's own ways.

A child behaves intelligently if it abandons itstantrums, once it is clear they lead nowhere. Inbusiness a sign of intelligence is the ability to ignorelesser problems in order to concentrate on themajor ones in the fields of accounting, production,or management. Parents who recognize and canhelp their children to see the most vital elements ofthe educational process are similarly "intelligent."

Intelligence is thus a virtue with many practicalfeatures, so highly prized and so useful that it issomething we can admire even in our enemies.

Intelligence is a safety door. Much is forgiven"intelligent" people, both male and female: lack ofpracticality, inconstancy, laziness, irritability, andinattentiveness. Some people rarely seem able todo a job on time. Yet if they are held to be intelligent,they are judged much more tolerantly. Tribute al-ways seems to be paid to those with intellectual

potential, even if it is never properly used and ismostly hypothetical.

A pun, a witty remark, or a clever riposte can turnthe most awkward situation to one's advantage."My friend," said the highly revered professor ofanatomy Riccardo Anzalotti to Francesco Lalli, athird-year student, "your work has not been what itshould have been. For your efforts in this exam, Ishall offer you a seventeen and a cigarette." "Thankyou," Lalli replied, fresh from a week's wild livingand a successful amorous encounter. "Give mesixteen, will you, and a light?"

Accused by her husband, Sir Andrew, after beingsurprised kissing the young gardener Pettygreen,Lady Miligham contemptuously denies the evi-dence: "How can you possibly say you love me,Andrew, if you prefer to believe your own eyesrather than my words?"

Intelligence is a proof of breeding. It is associatedwith important things such as good taste, success,agreeable feelings, and hopes, wealth, and power.

Margaret, flattered by Faust's compliments, la-ments over the untold numbers of women moreintelligent than herself on whom he has exercisedhis powers of seduction. Here intelligence goes withculture, ancestry, and personal magnetism.

Intelligent people enjoy brilliant careers, earn for-tunes, and have an intense emotional life. "Sir Fran-cis Drake is an intelligent man," Queen Elizabeth

9

A general survey

observed, "and we owe a great deal to intelligence."And with that she boarded the admiral's ship andspent the entire night in conversation with him.

Like an official title or honour, a reputation forintelligence can compensate for many defects.Quirks, oddities, and negative personality traits thatwould be considered serious in an "ordinary" per-son are looked on more kindly. Wit can even makemeanness seem entertaining: "Ah, virtue is price-less. Alas! Were it not,wecould sell or mortgage it!"

Intelligence is knowing how to live well. In theoffice, at school, in the factory, the theater, or hotel,on a cruise, walking about a city center, out in thecountry, alone, in company, with a date or withsomebody one loathes, with children or with an oldfriend-in any of those countless everyday situa-tions that make up our lives, intelligence representsthe ability to achieve the greatest possible satisfac-tion, the best results, the most experience, and thetruest pleasure. Intelligence is knowing how to eatwell without putting on weight or suffering fromindigestion. It is, however, also knowing when toignore the rules of "healthy" eating and enjoy thepleasures of the table to the full (and take the con-sequences) without suffering from guilt at doing so.

Thus intelligent behaviourgoes hand in handwithawareness. It entails an ability to approach prob-lems, people, facts, and events in a constructiveway: anticipating possible developments, bal-ancing positive and negative factors, and makingdecisions accordingly. Such considerations will de-termine whether one copes with any givenpredicament with a touch of humour, say, or with adecisive attack.

Intelligence is a game. Surely a characteristic ofintelligent people is also that they are able to seethemselves, events, other people, and the worldabout them in all its beauty and all its awfulness,with humour. This is wise. Reality is not whollywithin our powers: old age and the whims offortunestill loom over us. While remaining totally com-mitted to all we hold most dear (family, profession,science, art, ideas, the ethical life), it is useful, too, tomaintain a certain detachment-to be able to smile,to take things with a pinch of salt. Setbacks andfrustrations can then be turned to good account. Atthe sametime, any momentof the day can becomesource of unexpected pleasure, affording some-thing comic, curious, grotesque, stylish, or new andoriginal. A card game is more interesting when oneknows its origins, its ancient esoteric symbolism, itshistory as it evolved into a pastime, and the import-ance it has for those who regularly spend theirevenings playing it, over a liter or two of wine. Withdue detachment, a "nonevent"-a love affair thatnever got off the ground, for example-can give

cause for laughter rather than misery. There will beothers. The English novelist J. R. R. Tolkien tells atouching but entertaining story of a young mangreatly in love with a rather haughty young lady.The man went to a ladies' outfitters, accompaniedby his sister, to buy his beloved a pair of fine gloves.It being a typical English winter, the sister took theopportunity of buying herself a pair of woolly draw-ers. Sadly, of course, the shopgirl made the inevit-able mistake of sending the drawers, instead of thegloves, to the lady in Belgrave Square. The errormight have been rectified had not the young manleft a letter to accompany them.

Dear Velma,This little gift is to let you know I have not forgottenyour birthday. I did not choose them because Ithought you needed them or were unaccustomed towearing them, nor because we go out together inthe evenings. Had it not been for my sister, I shouldhave bought long ones, but she tells me you wearthem short, with just one button. They are a delicatecolour, I know, but the shopgirl showed me a pairshe had worn for three weeks, and there was not theslightest stain on them. Iow I would love to putthem on you for the first time myself.

Doubtless many another man's hand will havetouched them before lam able to see you again, but thope you will think of me every time you put themon. I had the shopgirl try them, and on her theylooked marvellous. I do not know your exact size,but I feel I am in a position to make a better guessthan anyone else. When you wear them for the firsttime, put a bit of talc in them, which will make themslide on more smoothly; and when you removethem, blow into them before putting them away;obviously they will be a little damp inside. Hopingthat you will accept them in the same spirit in whichthey are offered, and that you will wear them to theball on Friday evening, I sign myself.

Your very affectionateJohn

P.S. Keep count of the number of times I kiss themover the next year.

Two facets of intelligence

It will be clear by nowthat"intelligent" behaviour ascommonly defined takes many different forms. Andthe connotations of such "intelligence" are equallymany and varied: success, charm, originality, inde-pendence of judgment and action, and so on.

10

A general survey

An unusual, to say the least, physics. Yet at the right time andpicture of Einstein that perhaps place, joking high spirits, andbelies the normal image of the extrovert good humour are veryscientific genius who much part of humanrevolutionized traditional intelligence.

But alongside this notion of what, broadly speak-ing, is meant by intelligence, is a more preciseunderstanding. A more academic approach tries todefine the specific features that distinguish intelli-gence from other psychological traits and to createcategories into which those features can be ordered(skill in problem solving, abilitywith language, mas-tery of figures, speed and efficiency of response tophysical stimuli, and so forth). And then there is thefurther question as to whether intelligence is aninnate quality or more a matter of something ac-quired through teaching.

Each of these two aspects of intelligence-the"practical," everyday side, and the theoretical,analytical side that is the particular preserve ofpsychologists and researchers-are worth study-ing. The diversity of human behaviour and condi-

tion in which they appear give us a chance to investi-gate our own "self' as well as that of others. No areaof human life, individual or social, is outside the fieldof the study of intelligence.

This book

Together with the reader, we should like to considerthe different forms that "intelligent" behaviourtakes. A tale by Boccaccio, the ability to rememberseries of numbers, the social use of some particulartalent, the solutionstospecific problems, all provideways of exploring "how the mind works." We shalladopt two levels of approach. First, various data willbe given (stimuli, problems, "unusual" cases) anddifferent ways of understanding them suggested.The first chapter will thus be a sort of introduction.

In the second chapter we shall give a brief surveyof present scientific knowledge about intelligence-about its development within the individual, theways it is expressed, its connection with creativityand with personality traits.

The succeeding chapters will follow up thesethemes and, most important, provide "stimulusopportunities" for the reader to exercise his or herown faculties.

The key feature is the games-a pointer in every-day life for developing and maintaining good spiritsin difficult circumstances. In a way it is a sort oftraining for one's ability to "see the funny side,"both in others' lives and our own.

Conventional wisdom and intelligence

From the Greekswe inherited an essentially abstractnotion of intelligence, which expressed itself mostlyin cultural forms. It was thus above all intellectual,centered on words, distinct from practical mattersand other aspects of human behaviour. The institu-tion of schools is based on-and in a sense alsocontinues-this idea. Intelligence is commonly con-sidered in terms of performance in study and schoolexams, although in fact the equation of intelligencewith academic ability is less popular nowthan itwasnot so long ago, when fewer people went to school.Schoolwork certainly still remains an objective fac-tor in assessing intelligence, but it is not all. Indeed,it has been seen that the institutionalization ofschool (its structures, the categorization of subjects,the relationships between teachers and pupils)often actually blunts children's liveliness andcuriosity for learning. Gradually, a less strictlyacademic concept of intelligence has thus evolved.Different kinds of ability are recognized as constitut-

11

A general survey

ing intelligence, other than brilliance at solvingmathematical problems or translating Latin tags.

We would like here to concentrate on aspects ofintelligence that are perhaps not ordinarily thoughtof as having a bearing on "intelligence." It wasmodern psychology that broke the traditionalmould of ideas about intelligence, seeing it as basi-cally a capacity for wholeness-just as the indi-vidual is one whole being. And while differentaspects are identified (verbal intelligence, perform-ance, and concrete, synthetic, and analytic intelli-gence), it is only because human beings need toanalyze, make distinctions, and classify.

As a faculty for wholeness of being, intelligenceinvolves all of human life. Thus it can be foundwhere one would least expect-in joking, in the waysomeone faces up to difficulties, in irony, inventive-ness, artistic taste, feelings, or just the humdrumbusiness of getting on with others.

Horny hands and intelligence

The ancient Romans took their idea of intelligencefrom the Greeks as something basically intellectualand cultural. They had not always thought thus,however. Study of the Latin language reveals that itwas at one time viewed as something more practicaland concrete. "Intelligence" is a Latin word, so it isworth finding out what the Romans meant by it.

Etymologically, "intelligence" means "to readinto" (intus legere) and refers to the ability to com-prehend; in a more limited use, it meant sensitivityand good sense. It was in the classical period (firstcentury B.C.), however, that intellegentia becamewidely employed by educated people, when Hellen-ism had penetrated deeply into Roman society. Andthe more aware and sensitive intellectuals such asCicero sought to adapt Latin to the new culturaldemands. The word "intellegentia" thus came todenote essentially intellectual characteristics. Yetthe Romans are noted historically for their practicalbent of mind and their administrative, organiz-ational, and juridical genius. When they used wordslike "ingenious," "dexterous," or "sagacious" (intheir Latin forms, of course), they were referringmainly to practical talents of such a kind. An intelli-gent or wise person was also called ca/lidus (onewho possessed calliditas). This understanding ofintelligence is most clearly evident in the plays ofPlautus, who lived in Rome in the third and secondcenturies B.C. We know that his comedies wereaimed at popular audiences. It may be supposed,therefore, that cafliditas was used mostly by thecommon people and those who were least Hellen-ized. The word is commonly translated as "skill,"

Scenes from an old edition ofBoccaccio's Decameron,illustrating the tale ofAndreuccio of Perugia, in which

native wit, good fortune, and theelement of the unexpected givea lively and realistic picture ofreal life.

"aptitude," or "ability," but a meaning closer to theoriginal would be "knowledge of that which arisesfrom experience, from practice," and the qualitiesstemming therefrom. Often rendered as"expert" or"skillful," ca/lidus is clearly related to practical in-telligence by its own etymology: for callidus, callidi-tas, and the associated verb called, all derive fromca//um, meaning a callus-the areas of thick, hardskin that develop on the soles of the feet after muchwalking, or on the hands of manual labourers.

Now we can get an idea of the true meaning ofcalliditas: intelligence is wisdom acquired by ex-perience, the ability to understand things becausethey are the materials of experience. Roman intelli-gencewasthusfirstandforemosta practical quality-wisdom drawn from experience.

A history of intelligence?

For a long time the notion of intelligence as intellec-tual or cultural activity, to do with study and contem-plation, reigned supreme over the more practicalmeaning of the term.

In the eleventh century, with the revitalization ofeconomic and urban life, a new concept of intelli-gence emerged-thoroughly down to earth, con-cerned with concrete, contingent problems. Theintelligent person was nowthe person who was ableto achieve aims (be they economic, amorous, orpolitical) without illusion and with fixed determina-tion, using every tool available in the real world. Wefind such a view of intelligence in the tales of Boc-caccio (1313-1375). Forhim it is a faculty raising the

12

A general survey

individual abovethe common run of human beings,driving him to pursue his ends, letting nothing di-vert his energies. It might almost be said that intelli-gence is one of the main characters in his writing-adynamic, ever-active force. Abstract contemplationceases to be a sign of intelligence, in favour of thatfaculty of self-control that is able to profit from thepassions of others in order to obtain a set goal. (See"The Tale of Alathiel," pages 45-50.)

It was this concept of intelligence that under-girded the activities of the new merchant class thatdominated the European economic and cultural re-vival of the Middle Ages. Boccaccio reflected theworld and the outlook of this mercantile class. Forthem a man could only be fulfilled through hisintelligence, which was seen as an ability to get thebest out of any situation however complex or dif-ficult. The same utilitarian urge extended also torelations between men and women: Boccaccio wasfascinated by the way lovers managed to overcomeobstacles placed in their way by parents who hadforgotten the delights they themselves had onceenjoyed, or by jealous spouses outmaneuvered bysharper intelligence. This supremacy of the intelli-gence is manifested in an individual's quickness ofresponse to unforeseen hurdles or in the ability toturn the tables in an embarrassing situation andgain the upper hand. Such intelligence was thus nota prerogative of any one social class but could occurin anyone, plebeian, merchant class, or nobility.Clearly, this accorded well with the ethos of thenascent bourgeoisie, for whom individual meritcounted more than birthright.

The dwarf

Here is quite a well-known trick question, to whichpeople normally give the first answer that happensto come to mind. Being creative also involves beingable to direct our imagination to seeking the mostprobable answers and not allowing spontaneousfancy to have its head. The story goes as follows.

On the twentieth floor of a skyscraper there lived adwarf. Methodical in his habits, and dedicated to hiswork, he would rise early, get himself ready, makebreakfast, take the lift down, and go off for the day.Every evening at the same hour he would return andrelax. All so far seems perfectly normal. However,one feature of the dwarf's day was odd: on his wayback in the lift in the evenings, he would stop at thetenth floor and walk the remaining ten. A fair hike!

Why did he not take the lift all the way to thetwentieth floor? Was there some reason?

This question evokes, interestingly enough, a vastrange of answers, all quite original in their own way,

all in some sense creative. The most common are:- the dwarf wants some exercise, so he uses the tenflights of stairs every day for this purpose;- he is a rather overweight dwarf who has beenadvised by his doctor to take some exercise;- he has a friend, who may be ill, living between thetenth and twentieth floors.

Less frequent replies (though equally, in a sense,original and quite entertaining) are:-the dwarf comes home drunk every night, late,and climbs the last ten flights on tiptoe for fear hiswife should hear him and be waiting for him with arolling pin;-the lift once got stuck between the tenth andtwentieth floors, giving him such a fright that nowhe prefers to climb the last ten flights of stairs;- between the tenth and twentieth floors the liftshakes slightly and makes a noise, which is ratherworrying, and the dwarf climbs this section on footto avoid the unpleasant sensation.

All these answers are clearly related to the subjec-tive experience of those who give them, in somemanner or another. The simplest solution, and alsothe likeliest, is that the dwarf is unable to reach anybutton higher than the tenth. Thus, while he is ableto press the bottom button and descend all the wayin the lift, on the upward journey he cannot gohigher than the tenth floor and is thus obliged to dothe last ten on foot.

Let's exercise our intelligence

At an intuitive level, we all know what intelligence is.The problems start when we try to define it. Someview it as "the ability to adapt to new circum-stances" or "the ability to learn from experience";others more often define it as "the ability to findsolutions to the problems raised in everyday life."All, however, agree that it is an ability, a potential,that is manifested in many and varied ways. Thereare in fact all kinds of modes of behaviour that wewould unhesitatingly call "intelligent."

On the basis of such a concept, experts havedevised intelligence tests characterized by theirsheer variety. In line with the historical develop-ments we have traced, it is worth stressing thatthese tests are confined to the conditions of a certainculture. The figurative, numerical, and verbalmaterial of which most are composed all relate tocontemporary Western culture. Their value is thusrelative. Together they test a number of facultiesthat all, to varying degrees, make up "intelligence."Those presented on the following pages are verbal,numerical, spatio-visual, and evolved from certainpsychological experiments.

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A general survey

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(continued on page 2015

A general survey

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SENETJudging from the evidence thathas come down to us, theancient Egyptians invented andplayed numerous board gamesas pure mental diversions. Themost widespread of thesegames would appearto havebeen senet, the rules of whichhave been worked out by thearchaeologists C. Jdquier(Swiss) and E. B. Pusch(German). It was a game playedby all, from the common people,to the rich nobility and even thepharaoh. The photograph belowis of a fresco in the tomb ofNepheronpet (nineteenthdynasty), showing the deceasedand his wife playing senet. Thegame is in fact a contest ofspeed, played by two, on a boardconsisting of three parallel linesdivided into ten compartmentseach (see diagram above-the

order of the numbers indicatesthe direction to be followed),with 5 counters of differentcolours (normally 5 black and 5white) to each player. To startwith, these counters are placedalternately in the first row, asshown on the opposite page.I To move, the Egyptians wouldappear to have had four specialdice, with only two faces suchas one black and one white),thus allowing only twopossibilities each throw. Thescoring for each throw could

therefore be as follows:- 1 white and 3 black 1 point- 2 white and 2 black 2 points- 3 white and 1 black 3 points- white 4points- 4 black 6 points* To start, the players throw thedice in turn until one of themthrows a 1. That player thentakes black and moves the firstcounter from square 1ltosquare 11. Then hethrowsthedice again. If he throws either a1-, a 4-, or a 6-point throw, hecan move any of his counters

1 2 3 4 5 617 8 9 10

20 19 18 17 16 15 14 13 12 11

21 22 23 24 25 26 27 28 29 30

16

according to his score and throwthe dice again' if his throw earnshim 2 or 3 points, however, hecan move whichever counter hewishes two or three places butthen has to pass the dice to hisopponent. At his first throw, theother player must move thecounter on square 9. After thathe can move any piece he maywish to. He, too, throws the diceuntil he scores a 2 or a 3, atwhich it once again becomes theother player's turn. (Their turnsalways change with a 2 or 3.)* The counters can be movedeither forwards or backwards.However, certain rules have tobe respected. If your countermoves to a square alreadyoccupied by your opponent, thelatter has to move to where youhave just come from: there cannever be more than one counteron any one square. If there arecounters of the same colour ontwo consecutive squares, theycannot be attacked, and theopponent must make a differentmove. Where there are threeconsecutive counters of thesame colour, not only can theynot be attacked, they cannoteven be passed. The opponentmust make other moves, untilthe throw of the dice forces thestronger one to move on.Counters of the same colour areof course permitted to crosssuch a "castle."* Every throw of the dice entailsa move, but if the move isbackwards, the counter cannotland on a square alreadyoccupied by the opposition-forthat would merely advance therelevant counter to your owndisadvantage. If a player cannotmove, he has to forgo his turn.I As appears on the oppositepage, the square correspondingto number 27 is marked X. Anycounter that lands here mustreturn to number 1, or to the firstunoccupied square after 1, andbegin again. Squares 26, 28, 29,and 30, however, are "freezones," and no counter can beforced backwards from them.I When all the counters of onecolour are in the last row (from21 to 301, they can come out, oneat a time, as each lands exactlyon the final square. If, however,during this operation onecounter lands on 27 and has togo back to the beginning, theother counters of that colourmust all wait until it has againreached the end row beforebeing able to leave the board.The winner is the first player toclear the board of all his pieces.The board, counters, and dice forsenet are not availablecommercially, but any readerwho wants to will easily be ableto improvise well enough.

I - i

1 .y -. t .I

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3 -i. .* f .. .4 -C

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1 I . .-1 ~ 1 1'1

Is 4 II I

l i

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�0150�

THE ROYAL GAMEOF URIn the Sumerian section of theBritish Museum is a finely inlaidgame board from the royaltombs at Ur (the ancient citymentioned in the Bible as Ur ofthe Chaldeans). This board(illustrated below) was used fora game played some four and ahalf thousand years ago in thepalace of the Sumerian kings.Models of the board areavailable in some gift shops,together with the rules of thegarne-or at least such rules ashave been supposed by scholarsto be the probable originalgame.The board on display in themuseum is valuable not only asan historical curiosity, but alsofor its exquisite inlay work ofstone, mother-of-pearl, and lapislazuli (a deep blue mineral usedas a gem or pigment). Alongwith other objects of similarlyskilled craftsmanship, the boardtestifies to the sophistication ofSumerian culture and the luxuryand lively refinement of courtlife. It was discovered togetherwith other game boards by anarchaeological expedition led bySir Leonard Woolley, which wasmounted by the British Museumand the University ofPennsylvania in the 1920s and1930s to work on sites insouthern Iraq. Similar boardshave been found in Egypt andCyprus, This suggests that thevarious wealthy courts of thoseancient civilizations shared agame (or variants of one basicgame) in common. Whatever thelinks between these differentpastimes, and whatever the

rules that can be inferred forthem at this distance in time, wesh all here set forth those for theroyal game of Ur, which Woolleydescribed as "the most strikingexample found" during hisexcavations.As the illustration below and thereconstruction (opposite page)show, the board is made up of 20squares, arranged in threesections: at the bottom, having12 squares 14 rows of 3), at thetop, having 6 (2 rows of 3); and 2squares connecting them. Fiveare specially marked with arosettelike star with eight points;5 have little circles; 5 havedesigns that resemble eyes; andthe remaining 5 are variouslypatterned. As we shall see,however, only those squareswith stars have any particularsignificance.To playthe game you need 14counters (7 white and 7 black)and six special dice (three foreach player) shaped liketriangular pyramids, ofwhichtwo apexes are coloured. Thepoint of the game is for each ofthe two contestants to get hispieces around one of the twotracks (shown arrowed in thediagram at the right). Thecounters are moved according tothe throw of the dice, thepossible scoring for each throwbeing as follows:

- three coloured apexes: 5points (Ms probability);- three plain apexes: 4 points VAprobability);- two coloured and one plain: 1point (¾ probability);- one coloured andtwo plain: 0points (¾ probability).

A throw of the dice decides who

starts. Then the first piece canonly be put on the board after ascore of 1 or 5; for example, ithas to enter on the first or fifthstarred square. Subsequently,the other pieces can be broughtinto the game on square 1whenever a counter of the samecolour lands on a starred square.Once started, no piece can movebackwards. Once a piece hasmoved 14 squares (so hascrossed back over the 2-sq uare"bridge") it is turned upsidedown. Several pieces can bepiled on top of each other, Whena piece is on the bridge or in thetop section, it can attack itsopponents, whether single orpiled up, by landing on the samesquare. When this happens, theopponent's pieces are removedfrom the board and have to startagain. The starred squares andthe final square are "free zones,"where tokens of either colour areimmune from attack.Upside-down pieces can only beattacked by other upside-downpieces, The same principleapplies to pieces the right wayup. To end the course, eachpiece has to land exactly on thefinal square (where any numbercan accumulate}, and then a 4has to be scored-at which pointall the tokens on the end squarefinish together. The player whogets all his pieces around thecourse first is the winner.As one might imagine, althoughthe game is largely determinedby the throw of the dice, the funconsists in choosing how tomove once each player has morethan one piece on the board.The royal game of Ur is to someextent a game of strategy, inwhich reason and intuition canhelp win the day.

19

A general survey

oHow does the series continue?

4 . 3 ,

.9 .89 k . 8 ' I

I1�1 , I

_, 1 ., .. 3�,� 1. " 2 ) I ( 2 ). I, , " I

0


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goes bwith -

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then c goesI with

1 -. -

e f;. - - ,I

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20

A general survey

(a1913

L Here are five words with jumbled-up letters.Four of them are anagrams of birds. Which isthe odd one out?

a)b)c)d)e)

CCNIHEKCAANYRCCOUOKCEIALNPCISFTAH

These words can be rearranged to read as asentence. If the sentence is true, tick T; if false,tick F.

FLOATS ALL NOT WOOD

mD

R21

0

:21LHI 1 -L 1 1-2

If a goeswith

Two of these words are similar in meaning.Which ones?

a) VIVACIOUSb) SUPERFICIALc) INCAPABLEd) SPIRITED


14 6 ,

9 18

21

b then c d e f

11

A general survey

©Figure a can be constructed with four of thesix fragments designated b, c, d, e, f, g. Whichare the unnecessary fragments?

©Fill in the gaps so as to make each line intotwo complete words.

a) MISTJ..RORb) LEM(..)LYc) LO(..)XED

a

08Which of sections a, b, and c are contained inline d?

b

d

U- -

Figure a can be constructed with three of thefive fragments designated b, c, d, e, f. Whichare the two unnecessary fragments?

Figure a can be constructed with three of thefive fragments designated b, c, d, e, f.Which are the two unnecessary fragments?

a

a b

EF0

22

0a ©

C

A general survey

Complete these three series of numbers.

a) <i> <::> <+> <8> <> <2> <a> <a>

[ 117 | I11 II1] 9 1 LIII] 111?W

c) 31 28 /Ej\ lN z& Za

Answers

(Of course, there are other solutions besides the ones given bythe author-for example, "token" and "money" both have fiveletters. A coin is called money, and money can be referred to ascoin or coins.)

1) a, d

2)9:6 (+1), 7 (+2), 91+31, 12(+4), 16

3) e (2:00, 5:00, 8:00, 11:00)

41 d, e

5) 13 or 17 depending on whetherthe number begins or endsthe series: 13 (-11).12 (+21,14 (-1,13 (+2l, 15 (-11,14(+2),16 (-1, 15 (+2), 17

6) d

71 c (a fox has a den; a bird a nest)

8) 4: 4 (-4), 0 (+51, 5 (-4), 1 (+5), 6 (-4), 2 (+5), 7 (-4), 3 (+5),8 (-41, 4

91 e

10) c (a wheel has the same function as a hoof for a horse)

11) 44: 2 (+21, 4 (XZ), 8 (+2), 10 (x21, 20 (+2), 22 (x2),44

12) e

13) Except for e, which conceals the word ATLANTA, which isnot a country.a) FRANCE, b) GERMANY, c) AMERICA, d) FINLAND

14) The sentence reads: DOLPHINS BELONG TO THE FAMILYOF MAMMALS, and is true

15) The series continues with 16: 1 (x2), 2(-11, 1 (x3), 3 (-1).2 1x2> 4 (-1), 3 ix3), 9 (-1), 8 (x2), 16

16) The series continues with 1: 13 (-61, 7 (+31, 101-5), 5 (+2),7(-4),3(+1),41-31,1

17) e or f

18) e (Figs. a and b are visually symmetrical, and the only figuresimilarly symmetrical to c among d, e and f is el

19) e (CATFISH is the odd man out. The others are a) CHICKEN,b) CANARY, c) CUCKOO, d) PELICAN

20) The sentence reads "NOT ALL WOOD FLOATS." This is true:the specific weight of some types of wood is greaterthanthat of water, and will therefore make them sink.

21) 60: 5 (-1, 4 {x21,8 1-21, 6 (x31,18 (13),15 {x4),60

22) e is visually symmetrical with c

23) a, d

24) 15: 9 1+5), 141+4), 18 (3),61+5), 11 (+4), 15

25) The unnecessary fragments are b and c

26) The unnecessary fragments are e and f

27) a) MIST(ERJRORbI LEMION)LYc) LO(VE)XED

28) a, c

29) The unnecessary fragments are b and d

30) a) 5: 25 (+2), 27 (+3), 30 (÷2), 151+ 31, 5 1+2), 7 (+3),10 .2),5

bI 8: 17 1-6),11 (+3),14 (-5), 9 (+2),11 1-4p, 7 (+11, 8c) 20: 3 (x4), 12 (-41,81(.4),2 (+4),6 (x4),24(-41,20

23

A general suTvey

Language and intelligence

Can a faculty such as intelligence be independent ofverbal language? It is true that not all "language" isverbal: we communicate also through gesture andwith figures and other symbols. These are oftenspoken of as codes, artificial languages, or animalcommunication. It is nevertheless hard to imagineanother language capable of encompassing such ahuge variety of objects and expressing such a vastrange of experiences and feelings as our everydayverbal language.

Hence it is natural to suppose that what we call"intelligence" finds particularly good expression inordinary language. One need only observe how, inbabies learning to speak (which can start at the ageof one), their relationship with others (parents andstrangers) is extended, and they begin to be able tomaster many of the mysteries of their immediateenvironment. Yet it must be remembered that whilefor animals the development of crude communica-tion is a natural thing and does not evolve, in hu-mans it is the fruit of learning and does evolve, bothin the lives of individuals and in history, changing inform and content. This learning process is in turndetermined by the family, the environment, thecultural atmosphere, and the stimuli present inthese. Thus it is true that thought finds a naturalvehicle in language; yet it is also true that this verylanguage has resulted from a slow, continuous pro-cess of cultural evolution. It is necessary to bearthisin mind when one looks at intelligence as man-ifested in language. We also know that the richness,variety, and distinguishing characteristics of lan-guage, and the accessibility of abstract concepts,vary among social groups. Someone brought up ina narrow cultural environment, speaking mostly alocal language or dialect, will find it hard to shine inverbal tests relating to the language as it is spokennationwide and to a culture very remote from hisown experience. The risk in tests of verbal ability isthat rather than showing merely lack of expertisewith words, they can suggest low intelligence, sincethey are moulded according to certain cultural con-texts. It is, however, difficult to devise verbal tests ofuniversal applicability in a society in which greatlinguistic and cultural variety still exists.

Finally, it must also be remembered that verbalintelligence is normally deemed to consist of beingable to speak clearly and expressively, with readycomprehension of information read or heard, andwith the facility for conversation.

However, we shall return to such detailed consid-erations of the phenomenon of intelligence in thesecond chapter. For now we can confine ourselvesto introductory observations.

Playing with words

Here is a series of tests in which the reader canexercise his or her ability with words. They go fromeasy to difficult, from simple to complex: the firstconcrete, the later ones more abstract.

These little tests should be entered upon in thespirit of a game. There is no scoring system, and anyresults are entirely for the player's own interest andbenefit.

0Two of these words refer to objects with basiccharacteristics in common.Which are they?

a)b)c)d)e)f)g)

BELFRYSPIRETERRACEEMBRASURESBATTLEMENTSKEEPBELL TOWER

Two of these words refer to objects with basiccharacteristics in common.Which are they?

a)b)c)d)e}

WINECASKBOTTLEVINTAGEBARREL


a) PYLONb) BRIDGEc) SCAFFOLDd) PARAPETe) VIADUCT

24

A general 4urvey

0Two in this series refer to objects with basiccharacteristics in common.Which are they?

a)b)c)d)e)f)

HAYSICKLEGRAINEARBILLHOOKRAKE


a)b)c)d)e)

Two of these words are similar in meaning.Which are they?

a)b)c)d)e)

SADSERIOUSMALEVOLENTGRAVESPIRITED

0Two of these words are similar in meaning.Which are they?

a)b}e)d)e)

MUSICGUITARFLUTEPIANOPIPE

REFUSALLISTLESSNESSINTROVERSIONAPATHYREPROOF

: 0I Two of these words are similar in meaning.

Which are they?

Here are five words with their letters jumbledup. Four are names of planets.Which is the odd one out?

a)b)c)d)e)

a)b)c}d)e)ETHAR

TEENPUNASIRUNASTRUNIPURJET

Fi l in the empty spaces to create two fullwords:

a) NOG. )AL b) CAI,.AR

CONVENIENCECONVENTIONCONVECTIONCONCESSIONCONVOCATION

Answers 6) c (is not URANUS)

1) a,g 7) a) NO{SE)ALb) CA(GE)AR

2) b, e

3)be 8)b,d

4) b,e 9i bd

5)ce 1O)b,e

25

41

A general survey

Natural "tools" for games

The history of human civilization reveals the pres-ence of games of one kind and another among allpeoples of every society. Games seem to satisfy abasic need in human nature.

They have always stimulated the imagination,and mankind has been able to make use of the mostunlikely objects for play. Even when there wasnothing to provide entertainment, human beingsmanaged to play games with their own most vitalmeans of communication-words!

One needs only listen to a child beginning toarticulate sounds: it is as though new horizons wereopening up to it. It derives visible pleasure frombeing able to control the sequence of its own sounds("ta-ta," "pa-pa, ma-ma," "ga-ga"), until finally itcan utter words with meanings (relating to parents,useful objects, other people in the home, and soforth). Its verbal ability then improves steadily,along with understanding. Then the first sentencesare spoken, bringing with them satisfaction andpleasure. As the child develops further, this samebasic pleasure will act as a powerful stimulus, en-couraging it to use verbal language to extend andmaster its own world of relationships.

First games with words

The acquisition and mastery of verbal languageby children is manifested by increased confidenceand self-awareness, and is a feature of the generalgrowing-up process. Another clear sign of this istheir delight in constructing odd sequences ofwords, as soon as they begin to have a vocabularyand to know how to construct sentences. They seemto have a faci lity for making games with words, bothfor adults' entertainment and their own Words andthe different meanings they acquire in conjunctionwith other words are a constant source of surpriseand discovery for young children, bringing themcloser to a world of adult control. They use words asif they had an independent existence and were allparts of a huge puzzle-bits to be played aroundwith in ever new ways, to create ever new results.Thus it is quite normal to hear children playing withrhyming words:

mouse

ground

-> house

-+ drowned

jelly-cake belly-ache

trees breeze

dollar

nose

scholar

-* toes

Words that are associated with each other in thisway often sound strange and comical because of theunusual or non-sensical meaning they assume.Many nursery rhymes are based on this principleand can often provide parents and teachers with anamusing and lighthearted way of teaching childrenhowto count, or recite the alphabet, as the followingtwo examples illustrate:

was an ArcherA who shot at a frog

was a ButcherB who kept a bull-dog

was a CaptainC all covered with lace

was a DrummerD who played with much grace

was an EsquireE with pride on his browwas a FarmerF who followed the plough

GH

was a Gamesterwho had but ill luckwas a Hunterand hunted a buckwas an Italian

I who had a white mousewas a JoinerJ and built up a housewas a KingK so mighty and grand

LM

was a Ladywho had a white handwas a Miserwho hoarded up goldwas a NoblemanN gallant and boldwas an Organ boy

0 who played about townwas a Parson

P who wore a black gownwas a Queen

Q who was fond of her people

26

A general survey

Rwas a Robinwho perched on a steeplewas a Sailor

S who spent all he gotwas a TinkerT who mended a potwas an Usher

U who loved little boyswas a Veteran

V who sold pretty toyswas a Watchman

W who guarded the doorwas expensive

X and so became poorwas a Youth

Y who did not love schoolwas a Zany

Z who looked a great foolANON

One, two, buckle my shoe,Three, four, knock at the door,Five, six, pick up sticks,Seven, eight, lay them straight,Nine, ten, a big fat hen,Eleven, twelve, dig and delve,Thirteen, fourteen, maids a-courting,Fifteen, sixteen, maids in the kitchen,Seventeen, eighteen, maids in waiting,Nineteen, twenty, my plate's empty.

ANON

The game of question and answer

This is played with friends, in a circle. The first playerasks his right-hand neighbour a question; heanswers, then puts a question to the neighbour onhis right-and so on, until all have replied to theirleft-hand neighbours and put questions to those ontheir right. Everyone has heard all that has beensaid. Then each player has to repeat the question hewas asked and the answer he received to his ownquestion. In other words, he does not repeat his ownquestion, but does repeat what his neighbours oneither side of him said-the question asked on hisright and the answer given on his left!

A variant of the game of question and answer isoften played by mixed groups of teenagers, who sitalternately boy-girl-boy-girl. On a sufficiently long

piece of paper, they then one after another writesentences on a predetermined theme (in block cap-itals, to conceal identity as far as possible).

It is interesting, for instance, to suggest they writea compliment or a love message to one of the girlsthere (chosen at random}. The first person writessomething, then passes the sheet to his neighbourafter carefully folding over the top of the paper tohide what he has written. The next' player writes asentence, again folds over the top of the sheet (thatthus gradually becomes a thicker and thickerwedge), and passes it on. Finally the sheet of paperis unrolled, and everything read out in one go, asthough written as a single piece. The results arealways good fun.

Definitions

Defining a word, either concrete or abstract, meansdescribing what it means by using other words andrelated meanings. The game of definitions consistsof guessing a word from its definition. A number ofpeople can play. Either at random or turn by turn,the players one at a time take a dictionary and(without showing which page they are readingfrom) read out a definition. Before the start of thegame, it should be decided how the answers are tobe given-whether by individual players in turn orin a free-for-all. The winner is the player who gues-ses most words. The fun of this game is that it is notalways the simplest or most common words that areeasiest to guess. Here are a few examples.

possible: that which comes within the bounds ofabstract or concrete supposition; that which comeswithin the bounds of an objective or subjectivefaculty

gramophone: an instrument for reproducingsounds by means of the vibration of a needle follow-ing the spiral groove of a revolving disk

emulation: praiseworthy effort to imitate, equal,or surpass others

corbel: architectural element jutting out from theface of a wall serving to support a structure above

cerulean: the blue of the sky, used for example todescribe a small patch of water

rnisoneist: one who through fear, hatred, or in-tolerance is opposed to novelty or change

27

A general survey

humour: the vision of the ridiculous in things, notnecessarily hostile or purely entertaining, butreflecting an acute wit and frequently a kindlyhuman sympathy

beautiful: capable of arousing physical or spir-itual attraction; worthy of admiration and contem-plation

This game can be played the other way round:each player in turn chooses a word to be defined.The winner is the one who gives the most precisedefinition. Words that are too difficult, too abstractor too idiomatic are naturally excluded. Nor is itnecessary to give a definition strictly according tothe one in the dictionary. An approximation will do.After all, dictionaries do not always exactly agreeeven among themselves!

The players can agree on a judge who will be thefinal arbitrator,to maintain good sense and fairness.

Word chains

The society we live in, with its frenetic pace of lifeand timetabled mode of working, seems to leavefew idle moments in the day. However, there areoccasionally times when we find ourselves withnothing to do: waiting to leave somewhere, waitingfora delayed train, ora long car journey all can seemto be wasting more time than they in fact are takingup. Often on such occasions we cannot think howto"kill" time. Games are an ideal way not only ofmaking the most of these tedious hours, but aboveall of relaxing the tension during a period of waiting

or overcoming the boredom of a long journey.One game, which even children can join in with, is

to go from one word to another by changing onlyone letter at a time. For example i . .

LOVE LOSE -- LOST -- LUST

Every word must mean something, and all con-jugations of verbs and declensions of nouns arepermitted.

Two boys waiting for a plane at an airport adoptedthe following sequences: one had to go from ROMEto BONN, the other from YUMA to RENO. Bothmanaged their respective word chains, the first onewent as follows:

ROME, TOME, TONE, BONE, BONN,

and the second:

YUMA, PUMA, PUMP, RUMP, RUMS, RUNS,RUNT, RENT, REND.

It can be agreed before the start of the game thatthe winner will be the player who reaches thesecond word within a certain time limit. And then ifthat results in a tie, the winner can be the one whowent through the fewest word permutations.

In the example of the boys at the airport, the firstone was thus the winner.

This game can also be played as a sort of patience,alone. Here are some word chains to try out:

I)VI)

Ill)IV)V)

GAME-BALLMORE-CARDFEEL-GOODTHICK-BRIDEREAR-BACK

Palindromes

Another game to play with words is to think whichwords are palindromes, or symmetrical in theirspelling. "Palindrome" itself is a word of Greekorigin, meaning "running back again." It therefore

Answers1) GAME-GALE-GALL-BALL

1fl MORE-MARE-CARE-CARD111) FEEL-FELL-TELL-TOLL-TOLD-GOLD-GOODIV) THICK-TRICK-TRICE-TRINE-BRINE-BRIDEVU REAR-BEAR-BEAK-BECK-BACK

28

A general survey

refers to words that are the same when read for-wards or backwards.

In our language we write and read from left toright, at the end of each line returning to the leftagain to start the next. However, some languages(such as Arabic and Hebrew) adopt the oppositesystem and read from right to left.

Words read in a single direction, either leftto rightor right to left, are mostly asymmetrical. To take asimple example, "apple" in our language has aprecise meaning when read from left to right, yet ifread from right to Jeft is a meaningless muddle ofletters. The word "elppa" conjures no object in ourmind and is merely a haphazard group of letterswithout reference to any objective reality. The gameof palindromes consists of hunting through the lan-guage for words that have the same meaning re-gardless of which way they are read. The word"radar," for instance, means the same whether it isread from left to right or from right to left.

Other examples are "level" and "rotor."Try to think which is the longest palindrome in the

English language. (Here is a long one: "Malayalam"-the language spoken in Kerala, southern India.But perhaps even better ones exist.)

Numbers and intelligence

When we talk about language we tend to think ofthat which we pick up naturally, and which is themain means of communication between humans.But there is another language, not commonlythought of as such, though in constant use in dailylife. That is the language of mathematics.

Often, unfortunate mental habits picked up atschool, where different mental disciplines are keptvery separate, mean that we view mathematics as aworld of its own, abstract and quite apart fromnormal everyday language. Thus we ignore theessentially linguistic nature of mathematics, whichuses signs and symbols (numbers) to conveythought. Though basically a language, the mainpurpose of mathematics is not communication be-tween individuals, but systematization of humanpowers of knowledge and ability to present it as aunified, objective experience. Hence it is above allthe language of the sciences. Intelligence musttherefore necessarily include the capacity for usingnumbers, for establishing the relationships be-tween them and handling them effectively.

Intelligence is by no means purely mathematical,but equally it is commonly accepted that mathema-tical ability is an expression thereof. It is for goodreason that intelligence tests contain a significantproportion of numerical exercises.

Games with numbers

Let us now test our ability with numbers, using thefollowing exercises. Each is a numerical series inwhich the idea is to discover the general law gov-erning it. The first ones are simple, the later onesmore difficult. Having discovered a law of the series,it is then possible to guess the next number.

0Insert the missing number.


29

A general survey

How does the series continue?Jo-- - I-

2 I --I 5 10

L 2 L~ L- -i Lo i A---.

How does the serie

3-

t --. 3

s continue?

6 5

"--- O f >

10

10


( 10 12 )


3 6

/ *6

:2 -* 15545-,*--

18

,./i

0How does the series continue?

15

6 -5 105 >

30

1 0 22

9

/'

18

/1

-A

?"II

3C

A-

/

'N

4

5

A general survey


5

7


4\> "I7X


Answers and explanations

1) 11:8(+1I),9+2),11 (+3),141+4),181+51,23

2) 6: 2 1+2), 4-I), 3 (+2), 5 -11,4 (+2),6 (-1),5±+2, 7 (-1),6

3) 23: 2 (X2), 4 (+1I, 5 X 2 , 10 (+11,11 (X2), 22 1+1), 23

4) 17: 3 (x2X, 6 (-1), 5 (x2), 10 (-1), 9 x21, 18 (-1), 17

5) 26: 31+21, 5 (x2), 10 +2), 12 (X21, 24(+2), 26

6) 42: 2 (x3),6 (-3),3 (x3,1 9 1-3), 6 (x3), 18 (-3 15 (x3),45 1-3X, 42

7 14: 6 (+5), 11 (+4),15 ÷.3),5 (+)), 10 (+4),14

817:2 (+21,41+4),81(2), 4(+21,6(+4), 10 (±2),5(+2)1,7

9) 8and 12: 6 (x41, 24 -4).20 4), 5 (+41, 9 (x4), 361-4),32 + .4X, 8 (+41, 12

10) 10: 19 (-6),131+3),16(-5),11 (+2.,131-4),9 (+1),10

31

A general survey

Visual intelligence

The attitudes of an individual or a community areexpressed above all through verbal language. Thissupremacy of language in turn affects the develop-ment of the individual's other faculties (especiallythe logical and cognitive faculties). A rich, well-articulated language influencesthe whole culture ofa social group, feeding it with terms and concepts.There are, however, other faculties and modes ofhuman activity that are also indisputably "intelli-gent." The architect's ability to order space in ahouse, is without doubt a clear form of intelligence.Perhaps, too, a painter, in the treatment of shapesand colours within the limitations of his materials,displays a similar kind of "intelligence." Intellectual

ability is exhibited in many activities in which simi-lar or dissimilar geometrical figures have to bediscerned.

Games with shapes

Let us now exercise our visual powers in the follow-ing tests. The object is to recognize similarities indissimilar figures and vice versa, to discern theindividual shapes that all together make up a singlegeometric form. Some of these exercises are pre-sented as a progression: the development of thevisual features of one unit into those of the next-amental step that can be taken without the aid of ver-bal language. Intelligence is a unifying faculty.

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A general survey

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If goes then goesa with b C with d e f



(continued on page 38)33

ALQUERQUEIn the thirteenth century, whenSpain was largely ruled by theMoors, Alfonso X of Swabia,known as the "the Learned,"was king of Castile and Lein forthirty-two years (1 252-1284). Heearned his nicknamethrough histransformation of the court ofSeville into an internationalcultural center. A poet himself,he wrote numerous secular andreligious songs. Oneundertaking he encouraged wasthe translation of Greek andArabic philosophy into Spanish.He also endeavoured to compilea collection of all the knowledgeof his age, and the scholarsworking on this projectproduced a remarkable body oflegal works (Las site partidas),historical works (Grande egenera/ Estoria; Crdnicageneral), and artistic, literary,and scientific documents (theAlphonsine astronomical charts,for instance). Among these therewas a book of games, theso-called Libro de juegos. This isa large illustrated collection ofevery game known and played atthat time, some of which arementioned as being "playedwith the mind." One of the mostinteresting is called a/Querquede doce ("for twelve pieces"),the name being a Spanishapproximation to the Arabicel-qirkat. In fact, although thisgame may have been introducedinto Spain by the Moors, it had

fig. a

for a long time been familiar tothe ancient Egyptians and thepeoples of the Near East and isessentially an early variant ofcheckers. It is still widely playedtoday-not only in the housesand bodegas of Spain, but alsoin certain areas of Asia, Africa,and Central America. Likecheckers, it is a board game fortwo players. The board isdivided up with 5 horizontallines, 5 verticals, and 6diagonals, creating 25 points ofintersection (see opposite page,below). There are 24 pieces: 12per player, of two coloursusually 12 black and 12 white),

and at the start of the game theyare positioned as indicated infigure a. The players move oneat a time, turn by turn. Unlikecheckers, the pieces can bemoved in any direction along thelines of the board, and the playerwho has the advantage at thebeginning is not the starter (whois constrained by having only theone empty space) but thesecond. Really, though, theoutcome depends on eachplayers ability to gauge thepossible moves-using bothreason and intuition-and tochoose the best, predicting hisopponent's moves. Any piececan be moved lone at a time)from where it is to any emptyneighbouring place. However,when a piece-let us say,white-moves to a position nextto a black piece, leaving anempty space behind it, the blackcan jump the white and "gobbleit up"-for example, take it andeliminate it from the game (seefigure b i. If the position of thepieces allows, it is possible totake more than one of anopponent's pieces at a single go,even when changes of directionare involved. At the same time, ifit is possible to take a piece, it isnot permitted notto take it. If aplayer fails to notice that he cantake a piece, and makes anothermove, his opponent can removethe piece that should have takena "prisoner" from the board.Once a player has moved andtaken his hand away from thepiece, the move isirreversible-unless he haspreviously said, "Testing," toindicate that he is not making aproper move but is just tryingthe ground to see what hewishes to do. The winner is theone who takes all the otherplayer's pieces orforces him intoa position where he cannotmove. It is quite common to endin a draw, when the players areeve n ly matched. M any va ria ntsof alquerque are to be foundthroughout the world, thedifferences depending primarilyon differences in board design.

fig. d

Some are simplerthan theSpanish-Moorish version, othersmore complex. Among theformer is the version played bythe Zuni Indians in New Mexico,who learnt it from theConquistadors, naming itKolowisAwithlaknannai ("snakefight"). This version still has 24pieces, but the board consists of8 lozenges intersected by a line.At the start of the game, thepieces are arranged as in figurec. A more complex variant ofalquerque is played on the islandof Madagascar, where the gameis known as fanorana: the boardis as illustrated in figure d, andeach player starts with 22 pieces(thus 44 pieces in all).

Another complex variant is to befound in Sri Lanka, where it isknown as peralikatuma. Theboard for this is the one shownon the opposite page (above).Each player has 23 pieces,arranged as in figure e. At theoutset there are three emptyspaces in the middle. Thisversion takes somewhat longerto play, but all the differentvariants share the same rules.Like the better-known games ofcheckers and chess, alquerqueand its variants belong in thecategory of "games ofintelligence," for they alldemand a combination ofintuition, strategy, andreasoning power.

34

fig. e i. .

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12 0 12 01-�LI! lu 0 11 rld IN,-12 Li 12 OR,

A'

BAGH-BANDISagh-bandi-the game of tigersand goats-is an entertainingIndian game for two players,played throughout the vastexpanse of Asia under differentnames and with rules of varyingdegrees of strictness. Theexample given here is one of themore easily graspable forms,with a simple board and rules.Ragh-bendiis the Bengali name(Bengal being famous for itstigers, andthe board (see figurea) is in fact the simplest of all. Ascan be seen, it is square, with 5horizontal lines and 6 diagonals,intersecting to form 32 equaltriangles, exactly as foralquerque (see pp. 34- 351.Anyone can make this boardwith squared paper, (thehandiest size beingapproximately 8 x 8 in. 120 x 20cmE. To start, one player takes 2pieces of the same shape andcolour, while the other(generally the more experiencedplayer) has 20 pieces, which areall the same but different fromthose of the other player. It ispossible to use coloured buttonsor painted coins-one colour forthe tigers and another for thegoats, which should also besmaller and stackable). There arethus 2 tigers and 20 goats. Thetigers start off from intersectionsa3 and c3, while the goats, inpiles of 5, start on b2, b4, d2, andd4 isee figure a: tigers pink,goats red).The aim of the game is of courseto win: the tigers win if theymanage to eat all the goats or ifthe goat player capitulates whenit is impossible for him to win);the goats win if they managetohedge the tigers in so that theyare unable to move.Each player moves in turn, asfollows.First, the goats move. They canonly travel one at a time, to anext-door intersectionconnected to where they are by astraight line, corresponding tothe side of a triangle, both on thearea of the board and around theperimeter. The top goat of thepile moves first, with the othersfollowing in successive moves.Actual piles of goats cannot bemoved. It takes some skill tomaneuver the goatswell-which is why, as wasindicated above, the moreexperienced player should takethem. The best way to defeat thetigers is to dismantle the pilesand move the goats around asmuch as possible.As well as being able to makethe same moves as the goats,the tigers can jump the latter,both individually and when inpiles, as long as there is a free

5

4

3

2

1

fig. a eats only the toi one. If the oaats

a

intersection to the other side.(The intersections thuscorrespond to the squares on acheckerboard.) Every time a tiger

fig. b

fig. c

b c

jumps a goat, it "eats" it, and the"dead" goat is removed fromthe board. When it jumps awhole pile of goats, however, it

have been so foolish as to permitit, a tiger can devour several atonce, in a series of leaps(including the top goat of anypile). Of course, the tiger cannotjump back and forth, over thesame pile of goats eating themall in a single turn!The following rules must beobserved, however:11 goats cannot eattigers;2) when a tiger is able to eat agoat, it must do so wheneverthegoat player insists. If, however,there are two "eatable" goats indifferent directions, the tiger canchoose which one to eat;3) when a tiger becomes hedgedin and cannot move, it isremoved from the board.Expert players will play severalisay, four) games in a row,swapping sides each time. if theresult is 2-2, they then play onuntil one of them wins twogames in a row.

Rimau rimauFarther east, in Malaysia, adifferent tiger-hunting game isplayed. Known as rimau-rimau,it is played on a board similar tothe Sinhaleseperalikatumaboard (see pp. 34-351, but in asimplified form. (See figure b.LAnother game fortwo players, itconsists of I tiger (white; and 24hunters Iblack). To start, 9hunters are positioned on theboard as shown in figure b. Thetiger then moves, devouring 3 ofthem (whichever 3 it likes) andending up on any of the 3 freedpositions. At each successivemove, black introduces anotherpiece onto the board, placing iton any empty intersection, untilall the hunters have entered thegame. They can then move, onestep at a time, in any directionbut are not able to jump thetiger. Meanwhile, the tiger alsomoves one space at a time,though it then has the advantageof being able to jump anddevour hunters, as long as thereare an odd number of hunters onthe hn--4

The hunters win if they manageto +r-o the ti-e 50 that it i.unable to move. The tiger wins ifit is able to eat at least 14hunters.

Mogol Pult'hanUsing the same board as forrimau-rirmau, but with 2 armiesof 16 pieces each, the game ofMogolPutt'han is played inIndia. To start, the pieces arepositioned as shown in figure c.The moves are the same as inalquerque. Whoever captures allthe enemy soldiers, or managesto prevent their moving farther,is the winner.

37

A general survey

Figure a is composed of three of the five fragments indicated b, c, d, e, f.Which are the unnecessary fragments?

a

0 Answers and explanations

Figure a is composed of three of the fivefragments indicated b, c, d, e, f.Which are the unnecessary fragments?

1) f

2) e

3) a

4) e (If the half-moon a in its position is visually symmetricalwith b,then c is similarly symmetrical with e.)

5) f (Here, too, a and b are related by visual symmetry; hence cmust go with f.)

6) e (for the same reason as above)

0Figure a is comDosed of four of thesix fragments indicatedb,cld,e,fg.Which are the unnecessary

bAa

7) e lb is clearly visually symmetrical with a, thus c goes withe.)

Tests 4-7 work on visual perception of symmetry. This is acommon phenomenon found almost everywhere, from the artsto the natural sciences, occurring in objects of widely differingnature. In these problems, basic data in a given set of visualstimuli are arranged in a certain pattern (described as"symmetrical") in relation to the other elements of the whole.The distinctive feature of such data is that they are seen asequivalent, in their spatial arrangement, with respect to a termof reference (a point, a plane, a straight line, or some moreabstract factor). The concept of symmetry is of particularinterest in science.

8) The unnecessary fragments are c and d.

9) The unnecessary fragments are c and e.

10) The unnecessary fragments areg andf.

38

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A general survey

City mirages

Our car has stopped at a red light. The lights thenchange and turn green, and we proceed. What wehave experienced is a sort of movement from red togreen. The same kind of visual impression is evenclearer when we watch the strings of lights on aChristmas tree flashing on and off, or fairgroundlights and neon signs lighting up and switching off.In fact, of course, nothing moves; the individuallights stay where they are, and electricity as such is aphysical phenomen, which cannot be seen. And yetthat illusion of movement remains, even for thosewho are unaware that electric current is generatedby the movement of electrons. The movements wesee are pure illusion.

C0

0--S0 We have turned over the figure

on the left and then spun itaround. Which of the patterns onthe right shows the result?

How is this possible?We know that such visual tricks occur because

images remain on the retina of our eye, thus fusingwith the next, so producing an impression of move-ment. It is not hard to see here the basic principlebehind the projection of images in movie films.

Figures in motion

Let us exercise our powers of perceiving motion.These are problems in which we need to "follow"mentally-in our imagination-a moving figure orgroup of figures. In some the correct answerwill beimmediately evident but others will require greaterattention.

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We have turned over the figureon the left and then spun itaround. Which of the patterns onthe right shows the result?

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39

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A general survey

0-"

We have turned over the figureon the left and then spun itaround. Which of the patterns onthe right shows the result?

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a

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'<' %. We have turned over the figureon the left and then spun itaround. Which of the patterns onthe right shows the result?

C

a

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We have turned over the fig u reon the left upside down and thenspun it around. Which of thepatterns on the right shows theresult?

a

C

Answers:

1) d (To prove this, turn the page over, twist the book 180

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degrees, then hold the page up to a strong light and lookthrough it at the original figure. You will see the design shown isanswer d.) 21 d 3) b 41 a 51 c

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I

A general survey

Topological games

Has it ever occurred to you that an ordinary coffeecup with a handle might have a significance beyondits normal function?

A coffee cup is a particular geometrical form, withcharacteristics that are of great importance fortopology, a curious branch of modern geometry.Topology deals with shapes that can be altered(enlarged, shrunk, curved, crushed, twisted, and soon) without their losing certain mathematical andgeometrical properties.

In the following exercises, the idea is to identifythe "positional" features that remain constant de-spite the alterations. They test our motor-perceptivefaculties, but in a slightly unusual way: concentrat-ing on topology.

02

* The figure above has undergone a simpledistortion. Can you recognize it in either a, b,

' c,ord?

a

0D

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07

The figure above has undergone a simpledistortion. Can you recognize it in either a, b,c, or d? a,

J The figure above has undergone a simpledistortion. Can you recognize it in either a, b,

. cord?

a b

d

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41

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A general survey

0D

The figure above has undergone a simpledistortion. Can you recognize it in either a, b,c, or d?


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Guide to the answers of the topological tests

1) Figure a' consists of an almost regular intersection of twoellipses and thus has unique spatial characteristics thatdistinguish it from otherfigures: there is an inner area,defined by four curved lines exactly adjacent to four outerareas. Of figures a, b, c, and d, the only one to share the samepositional (topological) features is b, in which a central area isbordered by four peripheral areas. Figure a is not possiblebecause in the top right there is a side thattouches twoperipheral areas; and it is easy to see that c and d do notperpetuate the same geometrical form.

21 This figure is slightly more complex than the previous one.Let us consider what the unique features of figure a'are.Clearly it has two internal points at which three lines intersect.

Of figures a, b, c, and d, which shares this characteristic?Obviously the only one to do so is a.

3) Figure a'has four distinct areas and two internal points atwhich three lines intersect: of a, b, c, and d, the only one toshare these topographical features is c.

4) It is always best to start by analyzing the essential features offigure a'. Here there is a clear internal point at which four linesintersect. Of a, b, c, and d, the only one to share this feature isd.

6) Figure a'has two internal points where two lines intersect. Ofthe four figures a, b, c. and d, only a retains this feature.

42

0

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JIII

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J.:~

A general survey

Games with numbers or figures?

The variety of ways in which intelligence is mani-fested is a symptom of the human mind's need todivide into components whatever is under examina-tion or analysis. Here is an exercise in which theboundary between mathematical reasoning andvisual thought is highly tenuous.

In each column, identify the two displaced boxes.

a

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Memory and intelligence

In preparing for an exam, the exactness, the com-pleteness, or the partiality of certain answers canrest on the degree of visual memory of the booksstudied. It is probable that successful results willcorrespond to the degree of clarity of memory ofrelevant explanations and illustrations. The layoutof a page, underlining of certain passages, plans,and so forth, all can aid memory.

Memory plays a vital part in the learning process.And the capacity to retain what has been learned byexperience is a powerful contributory factor in ourability to adapt-thus also to our intelligence.

A good memory is a "must" for many activities:for study and intellectual work of any sort, but also,too, for manual work (a mechanic, for instance, hasto remember many tiny parts of an engine).

How's your memory?

"Memory shrinks unless exercised," according toCicero (writing some two thousand years ago). Andan important truth is grasped here: memory is afaculty that can be improved.

Psychology offers various methods: associationsof stimuli, the use of rhymes in verbal memory, andthe use of codes, symbols, and metaphor. The waywe learn-our attentiveness, motivation, and ap-plication-all affect the working of our memory.

Let us try some concrete exercises for testing ourcapacity for visual mnemonics. Although memoryembraces all the sense experiences, (we have audi-tory and tactile memory, as well as memories oftastes and smells), the visual is the easiesttotry out.

Look carefully at the figures in the orange boxbelow for two minutes, then reproduce them on aseparate piece of paper. The order in which theycome out does not matter.

43

Answers: a) IV and VI b) IV and ViII c V and VI d) III and V

, .

I

A general survey

Carefully study the words printed above, for notless than two minutes; then close the book and writedown as many of them as you can remember.

The spirit of adaptability

One of the most commonly identified characteris-tics of intelligence is adaptability-an individual'sability to change according to developments in thephysical and social environment.

The events of individual lives are largely unfore-seeable. Society, on the other hand, now evolves ata rapid rate. Never before have profound changes inthe organization of work and of leisure time-involved whole generations. Industrial methods ofproduction, which were at the source of thesechanges, proved a considerable test of people'sspirit of adaptability. Millions of people left thecountry for the cities or emigrated to places verydifferent both socially and culturally from wherethey had originated.

An industrialized society is by nature in a state ofconstant change. One need only observe currentdevelopments: we are at present experiencing arapid and vast reorganization of production thanksto the computer "revolution"-for example, theintroduction of microtechnology into industry.Naturally this creates demands for adaptability.

Let us at this point examine what we mean by"adapt." When our environment (social, economic,physical) changes, then we, too, must change.Knowing how to adapt means being open to thepossibility of having to change our habits, our wayof life, and even in the long run our whole way ofthinking. Thus, intelligence manifests itself in thespirit of adaptability as a willingness to accept forourselves new values and new standards of be-haviour, and to establish new relations with thesociety around us. It is essentially the ability tochange along with the world in which we live.

We may also take an example of the spirit ofadaptabilityfrom the sphere of individual emotionallife: when one falls in love-that is, brings anotherperson into one's life-a new situation is created, inwhich one has oneself to change. True, in suchcases the power of love helps the spirit of adaptabil-ity, and often two people who are in love adapt veryquickly to each other, without problems. An indi-vidual is a complex entity, however, and when oneelement of a complex system undergoes change,thewholesystem hastoadjustitsbalanceandfind anew mode of being. It may be that a change like lovecan bring a new strength to the whole system; or itmay be only a partial adjustment, in which caseimbalance can occur. In such cases, intelligence isrevealed in the quest for a new equilibrium, a newway of relating to the other.

44

P7;1-1=

A general survey

The tale of Alathiel

In the summer of 1348 the serene city ofFlorence was struck by plague. The nor-mal scenes of bustling, colourful lifewere replaced by scenes of death andgrief. From this disaster ten youths re-solved to flee to the haven of the coun-tryside, seeing life as a mode of reality inwhich, through intricate interaction ofevents of all kinds, intelligence, nature,and fortune held sway. This briefly is theunifying background to the Decameron,written by Giovanni Boccaccio (1313-1375), a collection of a hundred novelle(short stories) evoking the burgeoning.dynamic world of industry and enter-prise created by the mercantile middleclasses of the late Middle Ages. Uponthis broad, wide-open stage was playedout the comedy of human life: a contestbetween two forces, one external(circumstances and fortune), the otherinternal (natural instinct, manifestedpredominantly in the experience oflove). These forces are both at play inthe tale of Alathiel, the unhappy daugh-ter of the king of Egypt, who, althoughcaught up in events beyond her control,still managed to follow her own nature,intelligently enjoying what pleasuresoffered themselves to her despite themisfortunes of her existence. In the end,Alathiel's ability constantly to adapt toaltered circumstances is rewarded.

It is now a long time since there lived asultan of Babylon, called Beminedab,

who was fortunate in all his affairs.Amongst other children, both male andfemale, he had a daughter namedAlathiel, who, in the opinion of all thatsaw her, was the fairest lady in the wholeworld. And because the King of Algarvehad afforded him great assistance in adefeat occasioned to a most numerousarmy of Arabians that had assailed him,and had demanded her afterwards inmarriage, he consented as a most specialfavour: and providing a ship, well equip-ped for the purpose, with all necessaryprovisions, and sending an honourabletrain both of lords and ladies to bear hercompany, he commended her to the pro-tection of Heaven, and took his leave.The sailors, as soon as a fit opportunityoffered, hoisted their sails, and leavingthe port of Alexandria, sailed prosper-ously many days; when, having passedthe island of Sardinia, and now seemingto be near the end of their voyage, on asudden, contrary winds arose, whichwere so boisterous, and bore so hardupon the ship, that they often gave them-selves over for lost. Nevertheless, fortwo days together, they tried all themeans they could devise. amidst an in-finite number of tempests, to weather itout; but all to no purpose. And not beingable to comprehend by marinal judg-ment where they were, or to see to anydistance on account of the clouds anddark night, being now not far fromMajorca, they felt the ship split; andperceiving no hopes of escaping, everyone caring for himself only, they threw alittle boat into the sea, reposing moreconfidence of safety that way than byabiding any longer in the broken ship.The men therefore that were in the shipwent into it, one after another; althoughthose who were first down made strongresistance with their drawn weaponsagainst other followers; and thinking toavoid death by this means, they randirectly into it; for the boat, not beingable to bear them all, sunk straight to thebottom, and the people therein all per-ished. The ship being driven furiouslyby the winds, though it was burst andhalf full of water, was at last strandednear the Island of Majorca, no otherperson remaining on board but the ladyand her women, all lying as it were life-less, through the terror occasioned bythe tempest. It struck with such violence,that it was fixed upon the sand about a

stone's throw from the shore; where itcontinued all that night, the winds notbeing able to move it. When day-lightappeared, and the storm was somethingabated, the lady, almost dead, lifted upher head, and began to call her servants;but all to no purpose, for such as shecalled for were far enough from her:wherefore, receiving no answer, andseeing no one, she was greatly aston-ished; and raising herself up as well asshe could, she beheld the ladies thatwere of her company, and some other ofher women, lying all about her; andtrying first to rouse one, and thenanother of them, she scarcely found anythat had the least understanding left; somuch had sickness and fear togetheraffected them, which added greatly toher consternation. Nevertheless, nec-essity constraining her, seeing that shewas alone, she knew not where, sheshook those that were living till she madethem get up, and perceiving that theywere utterly ignorant of what was be-come of all the men, and seeing the shipdriven upon the sands, and full of water,she began with them to lament mostgrievously. It was noon-day before theycould descry any person from on shore,or elsewhere, to afford them the leastassistance. At length, about that time, agentleman, whose name was Pericon daVisalgo, passing that way, with many ofhis servants, on horseback, upon seeingthe ship, imagined what had happened,and immediately sent one of them onboard, to see what was remaining in her.The servant got into the ship with somedifficulty, and found the lady with thelittle company that was left her, who hadall hidden themselves, through fear,under the deck of the ship. As soon asthey saw him, they begged for mercy;but not understanding each other, theyendeavoured, by signs, to inform him oftheir misfortune. The servant carried thebest account he could to his master ofwhat he had seen; who ordered theladies, and every thing that was in theship of any value, to be brought onshore, conducting them to one of hiscastles, where he endeavoured to com-fort them under their misfortunes by thisgenerous entertainment. By the richnessof her dress, he supposed her to be someperson of great consequence, whichappeared more plainly by the great re-spect that was paid to her by all the

45

A general survey

women: and although she was pale andin disorder, through the great fatigue shehad sustained yet was he much takenwith her beauty; and he resolved, if shehad no husband, to make her his wife;or, if he could not have her as such, stillnot to lose her entirely. Pericon was aman of stern looks, and rough in hisperson; and having treated the lady wellfor some time, by which means she hadrecovered her beauty, he was grievedthat they could not understand eachother, and that he was unable to learnwho she was; yet, being passionately inlove, he used all the engaging arts hecould devise to bring her to a com-pliance, but all to no purpose; she re-fused all familiarities with him, whichinflamed him the more. This the ladyperceived, and finding, after some staythere, by the customs of the place, thatshe was among Christians, and where, ifshe came to be known, it would be of nogreat service to her; supposing also, that,at last, Pericon would gain his will, if notby fair means, yet by force; she resolved,with a true greatness of spirit, to tread allmisfortune under foot, commanding herwomen, of whom she had but three nowalive, never to disclose her quality, un-less there should be hopes of regainingtheir liberty; recommending it farther tothem to maintain their chastity, and de-claring her fixed resolution never tocomply with any one besides her hus-band; for which they all commendedher, promising to preserve their honour,as she had commanded them. Every daydid his passion increase so much themore as the thing desired was more near,and yet more difficult to be obtained:wherefore, perceiving that entreaty wasto no purpose, he resolved to try what artand contrivance could do, reservingforce to the last. And having onceobserved that wine was pleasing to her,not having been accustomed to it, asbeing forbidden by her country's law, hedetermined to surprise her by means ofthis minister of Venus. And seemingnow to have given over his amorouspursuit, which she had used her bestendeavours to withstand, he providedone night an elegant entertainment, atwhich she was present, when he gave it incharge to the servant that waited uponher, to serve her with several winesmingled together, which he accordinglydid; whilst she, suspecting no such

treachery, and pleased with the richflavour of the wine, drank more thansuited with her modesty, and forgettingall her past troubles, became gay andmerry; so that, seeing some womendance after the custom of Majorca, shealso began to dance after the manner ofthe Alexandrians; which when Periconobserved, he supposed himself in a fairway of success, and plying her still withmore wine, continued this revelling thegreatest part of the night. At length,when the guests departed, he went withthe lady into her chamber, who having atthat time more wine than modesty, un-dressed herself before him, as if he hadbeen one of her women, and got intobed. He instantly followed, and accom-plished his purpose. They afterwardscohabited together without any reserve,till at length, fortune, unwilling that shewho was to have been the wife of a king,should become the mistress of a noble-man, prepared for her a more barbarousand cruel alliance.

Pericon had a brother, Twenty-fiveyears of age, of a most complete person,called Marato; who having seen her, andflattering himself, from her behaviourtowards him, that he was not displeasingto her: supposing also that nothing ob-structed his happiness, except the guardwhich his brother had over her; he con-sequently contrived a most cruel design,which was not long without its wickedeffect. There was by chance a ship in thehaven at that time, laden with merchan-dize bound for Chiarenza in Romania, ofwhich two young Genoese were the mas-ters, who only waited for the first fairwind to go out: with them Marato madea contract, to receive him with the ladythe following night. When night came,having ordered how the thing should bemanaged, he went openly to the house,nobody having the least mistrust of him,taking with him some trusty friends,whom he had secured for that service,and concealed them near the house: inthe middle of the night. therefore, heopened the door to them, and they slewPericon as he was asleep in bed with thelady; seizing upon her, whom they foundawake and in tears, and threatening tokill her if she made the least noise. Theytook also everything of value that be-longed to Pericon, with which Maratoand the lady went instantly on board,whilst his companions returned about

their business. The wind proving fair,they soon set sail, whilst the lady reflect-ing on both her misfortunes, seemed tolay them much to heart for a time; tillbeing over persuaded by Marato, shebegan to have the same affection for himthat she had entertained for his brother;when fortune, as if not content with whatshe had already suffered, preparedanother change of life for her. Her per-son and behaviour were such, as to en-amour the two masters of the ship, whoneglected all other business to serve andplease her; taking care all the while thatMarato should have no cause to suspectit. And being apprised of each other'slove, they had a consultation togetherabout it, when it was agreed to have herin common between them, as if love, likemerchandise, admitted of partnership;and observing that she was narrowlywatched by Marato, and their designthereby frustrated, they took the oppor-tunity one day, as the ship was under fullsail, and he standing upon the stern look-ing towards the ship, to go behind andthrow him over-board; whilst the shiphad sailed on a full mile before it wasknown that he had fallen in: as soon asthe lady heard of it, and saw no likelymeans of recovering him again, she fellinto fresh troubles, when the two loverscame quickly to comfort her, using manykind and tender expressions, which shedid not understand; though indeed shedid not then so much lament Marato asher own private misfortunes. After somelittle time, imagining that she was suf-ficiently comforted, they fell into a dis-pute together which should have the firstenjoyment of her; and from words theydrew their swords, and came to blows,the ship's crew not being able to partthem, when one soon fell down dead, theother being desperately wounded, whichoccasioned fresh uneasiness to the lady.who now saw herself left alone, withoutany one to advise and help her: she wasfearful also of the resentment of the twomasters' relations and friends: but theentreaties of the wounded survivor, andtheir speedy arrival at Chiareoza, savedher from the danger of death. She wenton shore with him there, and they con-tinued together at an inn; whilst the fameof her beauty was spread all over thecity, till it reached the ears of the Princeof Morea, who was then by chance atChiarenza. He was impatient to get a

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sight of her; and after he had seen her,was so charmed, that he could think ofnothing else: and being told in whatmanner she came hither, he began tocontrive means how to obtain her; whichwhen the man's relations understood,they immediately sent her to him, to hergreat joy, no less than the prince's, nowthinking herself freed from all danger.The prince perceiving her rare accom-plishments, joined to a matchless per-son, though he could have no informa-tion concerning her, yet concluded thatshe must be nobly descended; and suchwas his fondness for her, that he treatedher not as a mistress but a wife. She nowrecollecting what she had already suf-fered, and being pretty well satisfiedwith her present situation, began to beeasy and cheerful, whilst her charms in-creased to that degree, that she was thechief subject of discourse throughoutRomania. Hereupon the Duke ofAthens, a young and gay person, a rela-tion also to the prince, had a mind to seeher; and came one day thither underpretence of a visit to him, as usual, with anoble retinue, when he was handsomelyentertained. Talking together, aftersome time, concerning the lady's greatbeauty; the duke asked whether she wassuch as fame had reported; to which theprince replied, 'she far exceeds it; butlet your own eyes convince you, and notmy bare assertion." The duke solicitingthe prince very earnestly to gratify hiscuriosity, they went into her apartmenttogether, when she received them withgreat good manners and cheerfulness,being apprised of their coming; andthough they could not have the pleasureof conversing together, as she under-stood little or nothing of their language,yet they looked upon her, the duke moreespecially, as a prodigy of nature, scarce-ly believing her to he a mortal creature;and, without perceiving how much of theamorous poison he had taken in by in-tently gazing upon her, and meaningonly to gratify himself with the sight ofher, he soon became over head and earsin love. After they had parted from her,and he had time to reflect, he began tothink the prince the happiest person inthe universe, in being possessed of such abeauty; and, after much musing upon it,having more regard to his lust than to hishonour, he resolved at all adventures todeprive him of that bliss, and to secure it

for himself: and having a heart to putwhat he had resolved into execution,setting all reason and justice aside, hismind was wholly taken up in devising afit stratagem for his purpose. One day,therefore, according to a most wickedagreement, which he had made with avalet de chambre belonging to theprince, whose name was Ciuriaci, hegave secret orders to have his horses andthings got ready for a sudden departure;and the following night. taking a friendwith him, and being both completelyarmed, they were introduced by thatservant into the prince's chamber, whomthev found in his shirt, looking out of awindow towards the sea, to take the coolair, the weather being very hot, whilstthe lady was fast asleep. Having theninstructed his friend what he would havedone; he went softly up to the window,and stabbed him with a dagger throughthe small of his back, and threw him out.Now the palace was seated upon thesea-shore, and very lofty; and the win-dow at which the prince stood lookingfrom, was directly over some houses,which the force of the waves had beatendown, and which were but little fre-quented; on which account, as the dukehad before contrived it, there was nogreat likelihood of its being discovered.The duke's companion when he saw thatit was over, took a cord which he carried

with him for that purpose, and seemingas if he was going to caress Ciuriaci.threw it about his neck, and drew it sotight, that he prevented his crying out,whilst the duke came to his assistance,and they soon dispatched, and threw himdown after the prince. This being done,and plainly perceiving that they were notheard or seen by the lady, or any oneelse, the duke took a light in his hand,and went on softly to bed, where she layin a sound sleep, and he stood beholdingher for some time with the utmostadmiration; and if she appeared socharming before in her clothes, what wasshe not without them? Not at all dis-mayed with his late-committed sin, hishands yet reeking with blood, he creptinto bed to her, she taking him all thewhile for the prince.

After he had been with her for sometime. he ordered his people to seize herin such a manner, that she could make nooutcry; and going through the same backdoor at which he had been introduced,he set her on horseback, and carried heraway towards Athens. But, as he wasmarried, he did not choose to bring herthither, but left her at one of his countryseats, a little way out of town, where hesecretly kept her, to her great grief;allowing her, in a most genteel manner,everything that was necessary. Theprince's servants waited till nine o'clockthat morning, expecting his rising; buthearing nothing of him, and thrustingopen the chamber doors, which wereonly closed, and finding nobody within,they concluded that he and the lady weregone privately to some other place todivert themselves for a few days, andtherefore thought no more about thematter. The next day it happened, bygreat chance, that a fool going amongstthose ruinous houses where the deadbodies were lying, took hold of the cordthat was about Ciuriaci's neck, and drag-ged him along after him: which surprisedmany people to whom he was known;who, by fair words and much persuasion,prevailed upon the fellow to shew themwhere he had found him; and there, tothe great grief of the whole city, they sawthe prince's body also, which theycaused to be interred with all due pompand reverence. Inquiring afterwardswho should commit so horrid a deed,and perceiving that the Duke of Athenswas not to be found, but was gone pri-

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vately away, they judged (as it reallywas) that he had done it, and taken thelady with him. Immediately they electedthe prince's brother to be theirsovereign, inciting him to revenge sohorrid a fact, and promising to assist himto the utmost of their power. He beingafterwards fully assured of the truth ofwhat they had but before surmised, col-lected together all his relations, friends,and vassals, and mustering a powerfularmy, directed his course against theduke: who had no sooner heard of thesepreparations, but he also levied a greatarmy, and many princes came to hisrelief. Amongst the rest, Constantius,son to the Emperor of Constantinople,and Emanuel the nephew, attended by agoodly body of troops, who were kindlyreceived by the duke, and the duchessmore especially, being their sister-in-law. Things tending every day more andmore to a war, the duchess had themboth one day into her chamber, when,with abundance of tears, she recountedto them the whole history and occasionof the war, and the ill-usage she hadreceived from the duke on account ofthis woman, whom she imagined he keptprivately; and complaining very earnest-ly to them, she conjured them, for hishonour, and her own ease and comfort,to give her their best assistance. The twoyoung lords knew all this matter before,and therefore, without asking manyquestions, they comforted her as well asthey could, and informing themselveswhere the lady was kept, they took theirleave. Hearing much talk of her beauty,they became very desirous of seeing her,and entreated the duke to shew her tothem; who, never remembering whathad happened to the prince, promised todo so; and ordering a magnificent enter-tainment to be prepared in a pleasantgarden belonging to the palace wherethe lady was kept, the next day he tookthem, and some more friends, to dinewith her. Constantius, being seated atthe table, began, full of admiration, togaze upon her, declaring to himself thathe had never seen anything like her, andthat the duke, or any other person, wasexcusable, who, to possess so rare abeauty, should commit any act of base-ness or treason: and looking still moreand more upon her, and evermore com-mending her, it happened just to him asit had done to the duke; for, going away

quite enamoured of her, he had givenover all thoughts of the war, contrivingonly how to steal her away from theduke, at the same time that he concealedhis love from every one. Whilst he was inthis agitation, the time came when theywere to march against the prince, whowas now advancing near the duke's terri-tories: upon which the duke, with Con-stantius and the rest, according to theresolution that was taken, marched outof Athens to secure the frontiers, and toprevent the prince's passing any further.Continuing there for some days, andConstantius having still the lady at heart,and concluding, now the duke was ab-sent, that he might more easily compasshis intent, he, that there might be apretence for his return, feigned himselfextremely sick: and, with the duke's con-sent, leaving the command of his troopsto Emanuel, he returned to Athens to hissister's, where, after some days, havingencouraged her to talk of her husband'sbaseness in keeping a mistress, he at lastsaid, that if she would give her consent,he would rid her of that trouble, byremoving the lady out of the way. Theduchess, supposing that this was spokenout of pure regard to her, and not to thelady, replied, that she should be veryglad if it could be done in such a manneras the duke should never know that shewas in any way accessory; which Con-stantius fully promised, and she accor-dingly agreed that he should do it as hethought most advisable. He provided,therefore, with all secrecy, a light vessel,and sent it one evening near to the gar-den where the lady was kept, having firstinformed some of his people that were init, what he would have them do; andtaking others with him to the house, hewas kindly received by the servants inwaiting there, and by the lady also her-self, who took a walk with him at hisrequest, attended by the servants be-longing to them both, into the garden;when, drawing her aside towards a doorwhich opened to the sea, as if he hadbusiness to communicate from the duke,on a signal given, the bark was broughtclose to the shore, and she seized uponand carried into it, whilst he, turningback to the people that were with her,said- "let no one stir or speak a word atthe peril of their lives; for my design isnot to rob the duke of his lady, but totake away the reproach of my sister." To

this none being hardy enough to returnan answer, Constantius, boarding thevessel, bid the men ply their oars, andmake the best of their way, which theyaccordingly did, so that they reachedEgina by the next morning. There theylanded, and reposed himself awhile withher, who had great reason to curse herbeauty. From thence they went to Chios,where, for fear of his father, and toprevent her being taken away from him,he chose to abide as a place of security:and though she seemed uneasy for atime, yet she soon recovered, as she haddone before, and became better recon-ciled to the state of life wherein badfortune had thrown her.

In the mean time Osbech, king of theTurks, who was constantly at war withthe emperor, came by chance to Smyrna,and hearing how Constantius lived alascivious life at Chios, with a mistressthat he had stolen, and no provisionmade for his safety, he went privatelyone night with some armed vessels, andmade a descent, surprising many peoplein their beds before they knew of hiscoming upon them, and killing all thatstood upon their defence; and after hehad burnt and destroyed the whole coun-try, he put the prisoners and booty whichhe had taken on board, and returned toSmyrna. Upon taking a view of the pris-oners, Osbech, who was a young man,saw this lady, and knowing that she wasConstantius's mistress, because she wasfound asleep in his bed, he was muchpleased at it, and took her for his ownwife, and they lived together very happi-ly for several months. Before this thinghappened, the emperor had been mak-ing a treaty with Bassano, king of Cappa-docia, who was to fall on Osbech on oneside, whilst he attacked him on theother; but they could not come to a fullagreement, because Bassano made a de-mand of some things which he was un-willing to grant; yet now, hearing of whathad befallen his son, and being in theutmost concern, he immediately closedwith the King of Cappadocia, requestinghim to march with all expedition againstOsbech, whilst he was preparing to in-vade him on his part. When Osbechheard of this, he assembled his armybefore he should be surrounded by twosuch mighty princes, and marched on tomeet the king of Cappadocia, leaving hislady behind, with a faithful servant of

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his, at Smyrna: they soon came to abattle, wherein his army was entirelyrouted, and himself slain. Bassano re-maining victorious, he proceeded on toSmyrna, the people making their sub-mission to him all the way as he went.But now Osbech's servant, whose namewas Antiochus, who had the lady incharge, although he was in years, yetseeing her so beautiful, and forgettingthe regard which was due to his lord,soon became in love with her himself;and, as he understood her language, itwas a great comfort to her, because shehad been forced to live for some yearslike a deaf and dumb person, for want ofunderstanding other people, or beingunderstood by them. This gave him greatadvantages, and whilst his master waswarring abroad, he spared no pains togain her consent, in which he succeeded:and when they understood that Osbechwas slain, and that Bassano carried allbefore him, without waiting for his com-ing upon them, they fled away privately,taking with them what belonged toOsbech of any value, and came toRhodes. They had not been there longbefore he was taken extremely ill, andhaving a merchant of Cyprus along withhim, who was his great friend, andfinding himself at the point of death, heresolved to bequeath to him the care ofhis lady and wealth also; and callingthem both to him, he spoke as follows:-"I find myself declining apace, whichgrieves me much, because I had nevermore pleasure in living than at present;yet one thing is a great comfort to me,viz., that I shall die in the arms of thosetwo persons whom I love and valuebeyond all the rest of the world; namely,in yours, my dearest friend, and in thatlady's, whom I have loved, ever since Ihave known her, more than my own life.I am uneasy, indeed, when I considerthat I leave her here a stranger, anddestitute both of help and advice, andshould be infinitely more so if you werenot with us, who, I know, will take thesame care of her, on my account, as youwould of myself; therefore I entreat you,in case I should die, to take my affairsand her together, under your protection,and to act, with regard to both, as youthink will be most for the comfort of mydeparted soul.-And you, my dearestlove, let me beg of you never to forgetme, that I may boast, in the next world.

that I have been beloved by the fairestlady that ever nature formed; assureme of these two things, and I shall diesatisfied." The merchant and lady wereboth much concerned, and promised tofulfil his desires, if he should chance todie; and soon afterwards he departedthis life, when they took care to have himdecently interred; which being done,and the merchant having dispatched allhis affairs, and wanting to return homein a Catalan ship that was there. ques-tioned the lady, to know what she in-tended to do, because it became necess-ary for him to go back to Cyprus: she.made answer, that she was willing to gowith him, hoping that, for the love hebore towards his friend, he would regardher as his own sister. He replied, that hewas ready to oblige her in everything;and, that he might the better defend herfrom all injuries whatever, till they cameto Cyprus, she should rather call herselfhis wife. Being on board the ship, theyhad a cabin and one little bed allottedthem, agreeable to the account they hadgiven of themselves, by which meansthat thing was brought about, whichneither of them intended when theycame from Rhodes: for they forgot allthe fine promises they had made to Anti-ochus, and before they reached Baffa,where the Cyprian merchant dwelt, theybegan to consider themselves as man andwife. Now a certain gentleman hap-pened to arrive at Baffa about that time,on his own private affairs, whose namewas Antigonus, one advanced in years.and of more understanding than wealth:for by meddling much in the affairs of theKing of Cyprus, he had found fortunevery unkind to him. Passing one day bythe house where she lodged, the mer-chant being gone about his business toArmenia, and seeing her by chance atthe window, he took more than ordinarynotice of her, on account of her beauty;till at length he began to recollect he hadseen her somewhere before, but couldby no means remember where. She,also, who had long been the sport offortune, the time now drawing nearwhen her sorrows were to have an end,as soon as she saw Antigonus, remem-bered that she had seen him in no meanstation in her father's service at Alexan-dria. And having now great hopes ofregaining her former dignity by hisadvice and assistance, she took the

opportunity of the merchant's absenceto send for him. Being come to her, shemodestly asked him whether he was notAntigonus of Famagosta, as she reallybelieved. He answered, that he was, andadded-"Madam, I am convinced that Iknow you, but I cannot call to mindwhere it is that I have seen you; there-fore, if it be no offence, let me entreatyou to tell me who you are." The lady,perceiving him to be the same person,wept very much, and throwing her armsabout his neck, asked him, at last, as oneconfounded with surprise, if he hadnever seen her at Alexandria? When heimmediately knew her to be Alathiel,the sultan's daughter, whom they sup-posed to have been drowned; and beingabout to pay homage to her, she wouldnot suffer him to do it, but made him sitdown. He, then, in a most humble man-ner, asked her where she had been, andfrom whence she now came; because forsome years it was believed, through allEgypt, that she was drowned. She re-plied, "I had much rather it had so hap-pened than to have led such a life as Ihave done; and I believe my father, if heknew it, would wish the same." Withthese words the tears ran down hercheeks in great abundance: and he re-plied, "Madam, do not afflict yourselfbefore it is necessary to do so; tell meonly what has happened to you; perhapsit may he of such a nature, that, by thehelp of God, we may find a remedy."-'Antigonus!" replied the fair lady, "Ithink when I see you that I behold myfather: moved therefore with the likeduty and tenderness that I owe to him, Ishall reveal to you what I might havekept secret: there are few persons that Ishould desire to meet with sooner thanyourself to advise me; if, therefore,when you have heard my whole story,you think there is any probability ofrestoring me to my former dignity, Imust beg your assistance: if you thinkthere is none, then I conjure you to tellno person living that you have eitherseen or heard anything about me." Afterwhich, shedding abundance of tears dur-ing the whole relation, she gave a fullaccount of what had befallen her, fromthe time of her shipwreck to that veryhour. Antigonus shewed himself trulyconcerned at what he had heard, and(thinking some little time about it) hesaid to her- "Madam, since it has never

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been known, in all your misfortunes,who you were, I will restore you to yourfather, to whom you shall be more dearthan ever, and afterwards you shall bemarried to the King of Algarve." Sheinquiring how that could be broughtabout, he let her know in what mannerhe intended to do it. Therefore, that nodelay might intervene to prevent it, hereturned directly to Famagosta. andwaiting upon the king, he thus addressedhim:-"My liege, you may, if youplease, do great honour to yourself, andservice to me, who am impoverished onyour account, and without any ex-pense." The king desiring to know bywhat means, Antigonus thus answered:

-"A young lady is just come to Baffa,daughter to the sultan, who was gener-ally thought to have been drowned, andwho, to preserve her honour, hathundergone great calamities, and is nowreduced, and desirous of returning to herfather: if, therefore, you will be so goodas to send her home under my conduct, itwill redound greatly to your honour, andprove much to my advantage, nor canthe sultan ever forget the favour." Theking, moved by a truly royal spirit, re-plied, that he was well pleased with theproposal, and immediately sent in greatstate for her to Famagosta, where shewas received with all honour and re-spect, both by him and the queen; andbeing questioned by them concerningher misfortunes, she made such answersas she had been before taught by Anti-gonus.

In a few days afterwards, at her ownrequest, she was sent with a great retinueboth of lords and ladies, and conductedall the way by Antigonus, to the sultan'scourt; where, with what joy they were allreceived, it is needless here to mention.When they had rested awhile after theirjourney, the sultan became desirous toknow how it happened that she was nowliving, and where she had been all thistime, without his being ever able to heara word about her. When she, who had allAntigonus's lectures perfectly by heart,gave her father the following narration:"Sir, about twenty days after my depar-ture from you, our ship was split in thenight by a violent tempest, and driven onthe western coasts; nor did I ever learnwhat befel the men that were in it: I onlyremember this, that when day-lightappeared, and I seemed recovered, as it

were, from death to life, certain peasantsof the country spying the ship's wreck,came to plunder it; whilst I was carriedfirst on shore, with two of my women,who were immediately borne away bysome young fellows, and taken differentways, so that I could never learn whatbecame of either of them. I also wasseized by two of them, making the bestdefence I could; and as they were drag-ging me towards the wood by the hair ofmy head, four persons on horsebackcame riding by, when they immediatelyleft me and fled. But the gentlemen onhorseback, who appeared to possesssome authority, came to me, and wespoke to each other, without knowingwhat either of us said. At last, afterconferring together, they set me uponone of their horses, and carried me to amonastery of religious women, accord-ing to their laws, where I was receivedwith great honour and respect. And afterI had been there for some time, andlearnt a little of their language, theybegan to inquire of me who I was, andfrom whence I came; whilst I (fearful oftelling the truth, lest they should haveturned me out as an enemy to their reli-gion) made them believe that I wasdaughter to a gentleman of Cyprus, whosending me to be married to one ofCrete, we happened to be driven thitherby ill weather, and shipwrecked. Con-forming to their customs in many things,for fear of the worst, I was asked, atlength, by the chief among them, whomthey call Lady Abbess, whether I desiredto return to Cyprus; and I answered, thatI desired nothing more. But she, tenderof my honour, would never trust me withany persons that were going to Cyprus,till about two months ago, certainFrench gentlemen with their ladies camethis way, one of whom was related to theabbess; who, understanding that theywere going to visit the holy sepulchre atJerusalem, where he, whom they believeto be God, was buried, after he had beenput to death by the Jews, recommendedme to them, and desired that they woulddeliver me to my father at Cyprus. Whatrespect and civilities I received bothfrom the gentlemen and their ladies,would be needless to mention. Accor-dingly we went on ship-board, and camein a few days to Baffa, where, when I sawmyself arrived, a stranger to every per-son, nor knowing what to say to these

gentlemen, who were to present me tomy father; behold (by the great provi-dence of God), whom should I meet withupon the shore, but Antigonus, the verymoment we were landed. I called to himin our language (that none of them mightunderstand us) and desired him to ownme as his daughter. He easily under-stood my meaning, and shewing greattokens of joy, entertained them as wellas his narrow circumstances wouldallow, and brought me to the King ofCyprus, who received and sent mehither, with such marks of respect as Iam no way able to relate; if there be anything omitted in this relation, Anti-gonus, who has often heard the wholefrom me, will report it." Antigonus thenturning to the sultan, said, "My lord,according both to her own account, andthe information of the gentlemen andtheir wives, she has said nothing buttruth. One part only she has omitted, asnot suiting with her great modesty toreport, namely, what the gentlemen andtheir ladies told me, of the most virtuouslife that she had led amongst those reli-gious women, and their great concern atparting; which, if I were fully to recountto you, would take up both this day andnight too. Let it suffice then that I havesaid enough (according to what I couldboth hear and see) to convince you thatyou have the fairest, as well as the mostvirtuous daughter of any prince in theworld." The sultan was overjoyed withthis relation; begging over and over, thatGod would pour down his blessings onall who had shewed favour to his daugh-ter; and particularly the King of Cyprus,who had sent her home so respectfully:and having bestowed great gifts uponAntigonus, he gave him leave to returnto Cyprus; sending letters, as also a spe-cial ambassador to the king, to thankhim on her account. And now, desiringthat what he had formerly proposedshould take effect; namely, that sheshould be married to the King ofAlgarve; he wrote to give him a fullrelation of the whole matter, adding,that he should send for her, if he desiredthe match to proceed. The king wasmuch pleased with the news, and sent ingreat state, and received her as hisqueen: whilst she, who had passedthrough the hands of eight men, nowcame to him as a pure virgin, and livedhappily with him the rest of their lives.

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Enigmas, riddles,games of logic

"Contrariwfse, "contmued Tweediedee, 'i it was so, it might be, andif awere so, t would be. but as it isn't, iat That's logic."

Lewis Carro I

The tools of the trade

Whatever job one is undertaking, it is necessary tobe properly equipped. A carpenter has to have hisplane, chisels, gimlets, and so forth constantly athand, in the same way as a language student needsto have up-to-date dictionaries and texts.

But, it may be objected, such "paraphernalia" isreally only of use to those who know how it ought tobe used. It is all a matter of "know-how."

In fact we all have personal traits that enable us todo well in some activities and not so well, or down-right badly, in others. And it is these characteristicsthat need to be concentrated on: expertise is ac-quired and refined all the better according to one'sintellectual powers, one's flexibility, and one's im-agination. Of prime importance are intelligence andcreativity since these form the basis of our more orless habitual, everyday activities, the ways in whichwe cope with problems, and our work performancein the factory, at school, or wherever it may be.

Defining intelligence

Specialiststend to equate intellective levels with theability to resolve specific problems. They thus adoptan "operational" mode to identify a certain charac-teristic-intelligence-through its manifestations.

The tests commonly used by psychologists are onthe one hand stimuli and on the other a gauge ofability in those who do well at them. By these meansan objective conclusion can be reached, howeverreductive it may be by comparison with the broadersignificance of "intelligent" as the word is normallyused.

In scientific investigations there is a need forprecise definitions. "Intelligent" behaviour reflectsan interplay of many different personality traits,such as the ability to control worry, to see problemsin context or in their individual details, and skill atcertain operations. Things must be ordered. Thencome operational definitions.

Mostly the tests are conceived and developed totest ability in the following areas:

- comprehension and use of words and phrases;- arithmetical calculations;- perception of relationships, similarities, and

differences between given geometrical forms;- powers of memorization;- perception of general rules governing certain

phenomena.

Hencethe "dimension" of intelligencecan be splitintovarious categories: verbal comprehension, ver-bal fluency, numerical ability, spatial ability, mem-ory, perceptual ability, and power of reasoning.

51

Enigmas, riddles, games of logic

A curious thing: I.Q.

It is difficult to ignore the desire to know oneselfbetter. The serpent that tempted our first parentsknew this as well as anyone. I.Q., used for the firsttime in the Stanford-Binet test at the beginning ofthis century, provides much information in con-densed form. It is arrived at through a variety ofexercises and represents an overall score.

The name is associated with the French psycho-logistAlfred Binet (1857-191 1), inventorof the scalefor measuring children's intelligence, in collabora-tion with Theodore Simon. The Binet-Simon scalebecame widely used in the decades following thefirst intelligence test (based on a series of exerciseslinked to the everyday experience appropriate todifferent ages) evolved by Binet in 1905, in responseto government requirements. Of the many revisionsof the scale, the so-called Stanford revision (thework of Lewis Terman of Stanford University) is thebest known: hence the "Stanford-Binet" method.

The phrase "intelligence quotient" derives fromthe original way of assessing the final score: estab-lishing the mental age of a child, on the basis of itsperformance in the tests it is able to do, in relation toits actual age, expressed in years and months. Itsperformance in these tests is based on the averagefor any given age. Thus 1.Q. was developed as ameans of measuring general intellectual ability inrelation to the average achievement of others of thesame age. It may therefore appear that a given childis either ahead of or behind his or her peers, in termsof mental age.

Today I.Q. is also used for adults, and the conceptof mental age has been dropped. The "crude" re-sults of a test are transformed by statistical cal-culations onto a scale on which 90-110 represents"normal." Above or below these results indicatesabove or below average intelligence.

A competitive spirit is aroused through all this: ameans of gauging the intellectual development ofchildren in relation to the norm, it came graduallytoacquire emotive significance and value way beyondmere scores. High performance could be taken toreveal an exceptional individual with unusual qual-ities. And who would not wish to be that?

What truly, though, is our level of intelligence?What are the most important "tools of the trade,"

the skills we need to do well at work, in study, and inour relationships with others?

Let us satisfy our curiosity. The following tests willallow you to assess your general 1.0. Note, however,that the results we supply will only help to gaugehigher levels of performance {from 115 to 145-thelatter representing a truly outstanding score). It isjust a game-so good luck!

Let's gauge our intelligence

InstructionsThirty minutes are allowed for the entire test. Onceyou have covered all the questions, it is worth goingback once again (if there is still time) and thinkingover the questions you were unable to answer.

05 7 . 15 23 36

What is the missing number?

0V.0 M H

What is the missing letter?

We have turned this figure over and then spunit around. Which of the patterns below is theresult-a, b, c, or d?

a 0 c d

AUTOMATICIMAGINEDSURPRISEDINVOLUNTARYEXPRESSED

Two of these five words are similar inmeaning. Which are they?

52

7

I I - N a,

The figure above has undergone a sindistortion. Can you recognize it in eithc, or d?


I<0

npleer a, b,

I

a

C

h

d

.. - "I - ---- --- 1 - - I III - -'--- .. '.1

0G P isto E R

as J M is to .

r - - .- - -... - -I.. ..I .. -_ - . .. .- .-_I -- . .I . . .-_ - -- _-.. -.

b

Eliminate one spoke, to leave a completeword on each circle.

I.....-...-.. ...-. ~. - .-..-.. .I...--.. -. . - -..... . . ......... .- .. .- -.

[ ... . .... - .. -...... ~. . . . . . . . . . . . A.. . . .. . . . . -. . . . .. .

18 12

286 568

What number should the triangle on thebottom row contain?

...|Which of these figures is the odcl man out? I

I It. . , . . .-.. -,- ., -. .. . .. ... = . . - ....... .. .. .. . - . . . .

GREEDYFERVENTSAVIOURAVARICIOUSZEALOUS

* of these five words are similar in

....... .... ^ -.... .. - L .- O-.- . II - ...... .- A-- . . . .. . . -.-- ... --- ,.-- ............. _ -...

Which twomeaning?

53

l I

i

_ -- __ - - - - - - __ - - - -- -. 1 -- . .. .. .. .. ..- - . ... . � . - .r .- -- ... .- � ..1�

II1

IIiII

I

I


We have turned this figure over and then spunit around. Which of the patterns below is theresult-a, b, c, or d?

a

toS©

b C

ECH TRW L.O

What is the missing letter?

/15

Sd

12

Supplythe missing number.

-1I~

0 la C r- (O - _j- LU

BCA

Change the arrangement of the spokes tocreate a complete word on each circle.

. 10 11 21 31 41 1401 16 512

What number should this series begin with?

' 0SURFACE - ACPLAYING - PYSINGING - GIFOLDING - DFLISTEN - El

Which line differs from the other four?

03

Supplythe missing number.

54

( �e

(i)


-- ...... . - ... , . - ..(.x.........

AS

a,


ac

lon;3. ....... . . -.. ................ ...

-. ..... .. ...... --.---.. .

(Ip)

4

Supplythe missing number. I

BRAGGINGFRAGILITYBOREDOMDELICACYPURITY

Which two of these five words are similar inmeaning?

b

5�Nm

55

"' 7

(aiI

I

I

IIJ

I

i

IIJ

I

I

I

IIik

Which of the numbers in this figure is the oddL man out?

......... ..... ..- ..........- -...-..--...... A..-.

SPISMISIPSILEINLAGOVRHACIZONAMA

The letters of these five words have beenjumbled up. Concealed in four of them arenames of rivers. Which is the odd one out?

Which letters does the series continue with?

58 ft 12 te 27 tn 81 .~~~~~~~~~. .... ......... .... ,.A.. .... . .... ......... S.-.--

E M isto Hand C I is to F

as D G is to ...


0A

* 4

as / '/I 1.:

(a

isto At h

A tA

is to

.. I

I<M]

I GUARANTEE THAT IT IS UNTENABLE TODENY THE OPPOSITE OF THE VERACITY OFMY AFFIRMATIONS

Does this mean "I am lying" or "I am tellingthe truth?"

0

1 \ V A\. V A~ \U

/" ,\ '* %.v 777/

7, 7/

bA A V

C )It

SEOROKHCELMAPROLPTURCARNSELITTHDENBIWED

The numbers in this figure go in pairs. Withwhat should 732 be paired?

Which word contains the name of a tree?30

02

___ 4

1

7

5 V

MEGRUNIAPLONPOTTNERGHEESUOROARDHCADAPSE

30

3632

----- 121

Which of these words conceals the name of af lower?

0DLIBIDOROMEGREATEXIT3

- DOUBLE- EMIR- EGRET- TAXI

Supply the missing number. Which line is the odd one out?

56

4 A v

A / A AIa/ /% /E

a

0a

226

732

aK 56'

es, games of logic

I . .- . . ................

We have turned this figure over and then spunit around. Which of the figures below is theresult?

a44&* **34**

a b c d .

j Which number goes in the bottom square?, .,.... ...- .. . - ... .... ..... - - L-~ ... _. ._ . . ... .. . . . . . .. . .. . ..

Supply the missing letter... .... . . . . . . ... . . . . r. I . - : . . . - - .. . . ........ .......... . - - -. .

.I. .. .. I . . . ., . . , .. .. . .- - =. -I-I---- -

is to 7!

as is to

+ A I'a b c

~~~~~~~~~~~~~~~. ...... . . A.... ..... ....... .. . ... ... -.... A. ...l

I C, .

ALL CATS ARE FURRYSOME CATS ARE NOT WITHOUT A SENSE OFHUMOURALL FURRY CREATURES ARE SARCASTIC

Which of the four sentences below contradictsthese statements?

I a) Certain sarcastic creatures are without anyhumour.

b) No furry animal is without a sense ofhumour.

c) Even if sarcastic, no furry animal has asense of humour.

d) Without humour, no sarcastic creature is. ' ....... . -... -

57

.. ... ..... - . .... - -- --- -I

I

Lnigmas, nucile

I

i

!I

(2

I

II

EunU

We have turned this figure over and then spunit around. Which of the figures below is theresult-a, b, c, or d?

F- a. .I -,

18jI

is

Supply the missing number.

GREEN DOGS ARE LIVE ANIMALSALL LIVE ANIMALS NEED FOOD

**UU .

E m.U.K

0 .l'-f lu

U .bE EU

* El

dUE U

Assuming the above statements are correct,which of the following is true?

a) My dog is green because it needs food.bi All green dogs need food.c) Certain green dogs do not need food.d) Some green dogs are not live animals.

0DAMASCUSFERRETTENDERNESSSMILE

- ET- GU- UT- UG

Which is the odd line out?

Exact answers

1) 10. The leaps between each number and the next go 2,3, 5,8, and 13: each represents the sum of the two previous ones.

2) T. The series is in reverse alphabetical order, with gaps of 1,2, 3,4 letters. Thus: VUtSRQPONMLKJIH

3) Figure d.

4) AUTOMATIC and INVOLUNTARY.

5) a, the only one having two internal points where three linesintersect.

6) H 0. Two letters before and two letters after.

7) d: d is formed of two circles and a rectangle, while the othersconsist of two rectangles and a circle.

8) Eliminate OIPTK and leave the circular words FLAG, MALT,COMB, GOAL, DUCK.

58

0B


53

- -il --

12"I--. ,

04


9) 1133: each number is double the previous one, less thenumber of the sides of the polygon in which it stands.

10) FERVENT and ZEALOUS.

11 E Figure a. Copythe original figure on a piece of paper, turnthe paper over and give it a 180 degree spin. Then place it oneach possible answer.

12) Change position of OAALG and SILIE to get: POST, TAIL,BALD, CLIP and AGED.

13) 0: the sum of the digits in each number is successively 1, 2,3, 4, and so on.

14) 9: the first letter of each group is followed by the letters thatcome two before and three after it in alphabetical order.

15) 20: the three outer circles could have been turned so that 1,2, and 3 coincide to give 6, and hence:

+ 57Y9482

68\6/201 9

16) The second; on the other lines the fetters on the rightrepresent the two first letters in the alphabet to appear in theword on the left. For instance, in SURFACE, A & C comebefore S, U, R, F and E in the alphabet.

17) 4: this gives the series 1, 2, 4, 7, 11, 16, 22, 29, wherein thenumbers increase by 1, 2, 3, 4, 5, 6, 7 respectively.

18) Figure d: figure a has six regions: figures b and c each havean isolated region that only touches one other region.

19) 34: we turned the three outer circles and added theirnumbers-25 is the total of the three lowest numbers +1;the inner circle thus adds I to each total. Therefore:

711571,5

+ 38 13 9 4 11

20) FRAGILITY and DELICACY

21) 362: the product of the figures in the other numbers isalways 12.

22J The fourth: CHAIR (the others were MISSISSIPPI, NILE,VOLGA, and AMAZON).

23) ee: each number is followed by the first and last letter ofhow it would be written out in full. Thus 58 = fifty-eight, ft.

24) C: the numerical position in the alphabet of the fetter on theright corresponds with the difference between the twofigures on the left e.g.E = 5,M = 13; 13- 5=8whichcorresponds with H.

25) c: by fitting the two left-hand figures one inside the other,the square on the right is obtained; to begin with, however,one is turned over and spun 180 degrees, and the other spun90 degrees in a clockwise direction.

26) The third: POPLAR.

27) 17: each upended T gives on the right the product of the twoleft-hand numbers + 2.

28) "I am lying.." "TO DENY THE OPPOSITE OF THE VERACITY"= to deny the falsehood = to affirm the truth; "IGUARANTEE THAT IT IS UNTENABLE" - it cannot bemaintained.

29) 43: in each pair, one of the numbers is the product of thefigures of which the other is composed + 1.

30) The first: GERANIUM.

31) The third: GREAT-EGRET (in the others, the right-handword shares the same consonants as the left-hand one, butin reverse order).

32) Figure c.

33) 19: in each square with three lines leading to it is the productof the northwest and northeast squares, less that of thenorth.

34) L: the two upper circles have been spun around and thenplaced on top of each other. Q placed on S becomes R (theletter between them in the alphabet), F on J becomes H(similarly midway between them alphabetically). The sameprocedure gives L from I and 0.

351 b, being symmetrical on a vertical axis.

36) Sentence c: there are some furry animals-certain cats, atany rate-with a sense of humour.

37) 8: the semicircle multiplies by 3, the circle divides by 2, andthe concave figure multiplies by 4.

38) b: all green dogs, since they are living creatures, needfood.

39) The last, because in the others, the right-hand letters arethose that in the alphabet follow respectively the first andthe last letters of the word on the left.

40) Figure b. Copy the figures onto a sheet of paper, turn it over,and spin it 90 degrees anticlockwise, then compare.

Exact answers Intelligence Quotient

13-16 116-125

17-20 126-130

21-24 131-135

25-28 136-140

29-32 141-145

33-40 145 plus

59 -


From the laboratory to everydayexperienceTests place people in artificial situations. They tendto analyze psychological characteristics and evalu-ate them one by one. This seems rather "unnat-ural," given that all human actions engage an indi-vidual's whole personality, however, this kind ofanatomizing of intelligence is necessary if we wishto gain some idea of how the mind "works."

The tests we have just undertaken to discover ourI.Q. have been geared towards a variety of personalcharacteristics, including skill in recognizing simi-larities, differences and relationships betweenfigures, mastery of words, ability with numbers, andpowers of logical thinking.

Such character traits are "tested" continually ineveryday life. Under "normal" circumstances (notin exam conditions) we are required, for instance, torecognize road signs, choose the best route to get toa certain spot, express ourselves competently andgrasp the possible significance behind certain ex-pressions, and follow a train of thought or the logicof development in both ideas and actions.

Thus it seems clear that the laboratory test situa-tion concurs well with the demands of daily living. Itis fair to say, then, that intelligence is exercised in agreat variety of ways and by a great many kinds ofstimuli both concrete and abstract.

And the results we shall achieve in the games inthe following paragraphs will be functions of ourintellectual level (though not of that alone, as willemerge later).

Historical digressions

The wisdom of the Jewish King Solomon (c. 973-933 B.c.) was famed throughout the known world.He was indeed a noble and peaceful king, dedicatedto justice and to establishing good relations with hisneighbour countries. His renown was due not toprowess in battle or any very remarkable achieve-ments, but to his cultivation of wisdom.

In the Bible there is a fine account of the visit, insplendid pomp, of the queen of Sheba, "to test himwith hard questions." According to this account, thequeen of Sheba had travelled a considerable dis-tance-from far-off Ethiopia-to meet Solomonand test his intelligence. Solomon "answered all herquestions; there was nothing hidden from the kingwhich he could not explain to her,"

This is perhaps the best illustration of the extent towhich, throughout the Near East, intelligence wasmanifested in riddles and enigmas.

History shows that the earliest human civiliza-tions grew up alongside rivers (the Nile, the Tigrisand Euphrates, the indus), and that they managed toemerge, evolve, and flourish for so long by dint ofsheer hard labour. River waterwas a true life sourceonly once it had been carefully regulated and aslong as it was continuously husbanded. Yet even inthese early communities, despite the toughness oflife, men found time to play games of intelligence.Among the more interesting archaeological findshave been rudimentary chessboards, which musthave been used for games similar to our own chessand checkers.

Throughout history riddles and puzzles, too, havedeveloped as a vehicle for intelligence.

In particular, the ancient Greeks enjoyed thisrather intellectual kind of pleasure. It is recordedthat the comic poet Crates of Athens (fifth centuryB.C.) livened up his plays with games, curious anec-dotes, and riddles that the audience had to answeron the spot. It was an original method of encourag-ing theatergoers to participate actively, and it musthave proved a considerable draw to the Atheniansof his day. Sadly his works have been lost, andfrom the few disconnected fragments that remainit is impossible to make any worthwhile recon-struction.

Riddles as the Greeks understood them wereusually short verse compositions, rich in deliber-ately obscure imagery, ambiguities, and puns, con-taining an idea that had to be puzzled out andexpressed in a straightforward manner. The aura ofobscurity and mystery with which they were im-bued placed them in the sphere of the sacred. Theywere part of religious experience. Throughout theMediterranean world, people would ponder theenigmatic utterances of the oracles and sibyls thatthey consulted to know the future or to help solvesome crisis. Such utterances would then be paro-died in the theaters, in a witty, popular mode, incatchy songs with references to ordinary everydayobjects and experiences-not infrequentlyobscene. Hence a riddle could either be of greatsolemnity, a true enigma, or a jokey kind of comicquestion-and-answer.

Oedipus and the Sphinx: a tragicprecedent!

Enigma in Greek meant "thought or problem ex-pressed in an obscure way," a riddle of the solemnvariety.

The existence of enigmas was a reminder of allthat was inexplicable about life and suggested a

60


The earliest civilizations evolvedalongside rivers (the Nile, theTigris and the Euphrates, theIndus). Society survived only atthe cost of immense labour.Much effort of the human mindwas concentrated on regulatingand exploiting the natural watersupply, on which life depended.We are still impressed today bythe devices invented and thehuge networks of irrigationchannels built, especially inMesopotamia, to compensatefor the irregularity of the watersupply (almost tropical in springand very scant in summer) fromthe rivers Tigris and Euphrates.The picture above, from a reliefin the palace at Nineveh (theancient capital of Assyria),dating from the seventh centuryB.C., shows an Assyrian parkirrigated bywaterfrom anaqueduct supported by arches.

Note the pathway leading uphillto a royal pavilion and altar.Below: a way of raising wateruphill by means of theshade'- a kind of bucketsuspended from a long polebalanced at the other end by acounterweight. This device (stillin use in certain places) wasknown both in Egypt and inMesopotamia. One might thinkthat the toughness of life inthese early nver civilizationswould leave little time for leisuregames. However, archaeologistshave found evidence that in fact,despite the hardships ofeveryday existence, people stilldevoted some of theirintelligence to entertainment.Thus once again we are made torealize that playing is a naturalneed, even if mostly expressedat one particular stage of ourlives.

degree of coercion and menace, boundaries beyondwhich only the light of intelligence could venture. Tosome degree, the play of intelligence can dispose ofand put to rout the ghosts and monsters with whichhuman imagination peoples the realm of the uncer-tain and the unknowable. One story from ancientGreek mythology in particular is worth recalling inthis context: the legend of Oedipus. Myths andlegends were, in the ancient world, a means ofexpressing human sensitivity, dilemmas, and waysof perceiving reality.

The legend relates that Laius, king of Thebes, hadseriously offended the god Apollo, who, to punishhim, condemned him to a "childless death." To anancient Greek, for whom children and the continua-tion of the family line were a form of earthly immor-tality, there could be few harsher punishments. In amoment of folly, Laius fathers a child, Oedipus. Buthaving been told by an oracle that he would die bythe hand of this child, he immediately tried to doaway with it. He exposed him on Mt. Cythaeron, tobe devoured by wild beasts. Discovered by ashepherd, however, Oedipus was taken to the kingof Corinth, who, having no children of his own,adopted him.

Years later, Oedipus travelled to Thebes and onthe way became involved in a petty argument thatthen became a fight with a regal-looking dignitaryand his escort. Oedipus defeated them all and killedthe old dignitary, who in fact was his father, Laius.Meanwhile the inhabitants of Thebes were beingterrorized by the Sphinx, a monster with the head ofa woman and the winged body of a lion, who hadbeen sent by Dionysus as punishment on the city fornot accepting his cult. To all who passed by it, theSphinx addressed a riddle and mercilessly slaught-ered those who were unable to answer it. Oedipus

61


Left: detail of the great Sphinx174 m long, 20 m high} at Gia inEgypt. As in the Greek myth ofOedipus, it symbolizes all that is

alone managed to give the correct answer, as heentered the city. The following account of theirmeeting comes from the prologue to OedipusRex, atragedy written in the fifth century B.C. by theAthenian playwright Sophocles:

There is a creature that walkson two legs, on four legs, and on three legs,who has only one tenor of voice:it alone among all creatures that moveon the earth, in the air, and in the water,changes its shape. But when it walkson four feet, thenit has least strength in its limbs.

To which Oedipus replies:

Listen, whether you wish to or not, 0 goddess ofdeath,

evil-winged goddess, my voice shall pronounceyour end: it is man you mean, who walkson the earth first of all as an infanton all fours, away from its mother's womb,then in old age leans on a stick, ason a third foot, his neck bowingunder the weight of agel

The Sphinx then kills itself by hurling itself from aprecipice. The grateful Thebans elect Oedipus king

62

dark and obscure in humanexperience, and which the lightof human intelligence alone cantransform.

and give him Queen Jocasta as his wife.Thus the prophecy of the oracle is fulfilled: with-

out knowing it, Oedipus has killed his father andmarried his mother; and the children of their unionare born to equally fated and tragic lives.

It is at this stage that Sophocles' play begins,consisting essentially of Oedipus' step-by-step dis-covery of the truth about his past and his trueidentity. The horror of the realization leads him totake terrible vengeance on himself, putting out hisown eyes-the eyes that had not recognized hisfather or mother should no longer be allowed tolook on the children of incest. For the rest of his lifehe would have to make do only with the eyes of hismind, of his intelligence. When Jocasta hears thetruth, she, too, kills herself unable to bear it. Intelli-gence takes one to the truth, but the truth can behorrifying and repugnant, and may be rejected, asboth Oedipus and Jocasta reject it in their differentways. This is the stuff of myth!

Later tradition evolved a shorter version of theSphinx's riddle: "What walks on four legs in themorning, two legs at noon, and three legs in theevening?"

The answer remains "a human being!"In more recent times, too, the old Greek legend of

Oedipus was given a new lease of life by SigmundFreud (1856-1939), who used it to describe thecomplex of unconscious feelings related to earlychildhood, of desire for one's mother and jealousyof one's father (or vice versa).

Enigma

In the myth of the Sphinx, life itself is at stake.Anyone unable to solve the riddle dies. But then theSphinx itself, after terrorizing the citizens of Thebes,must disappear. And it is the unconscious parricide,Oedipus, through whom the city is saved.

Through this legend we come to a better idea ofwhat enigma meant for the ancients.

Over the span of life of any individual there werecrucial moments, decisive decisions to be taken,steps experienced as trials, which, if overcome,would raise the protagonist's life to a higher level ofconsciousness and wisdom. Initiation rites in primi-tive societies, for example, serve rather the samefunction, marking the passage from childhood intoadult life. Enigmas are in some sense the heirs of,and a metaphor for, those ancient rites: forming a

(continues on page 68)


These curious-looking devicesare elaborate "toys" fromancient times, which still arousewonder and delight. Right: thesinging fountain of Hero ofAlexandria; Below, left: Hero'smachine for opening the doorsof a ternple. Below, right: theclock of Ctesiiius. Themechanical principlesincorporated in these devicesreveal that the Greeks hadmastered many of the laws ofphysical phenomena.It was these same principles thatled to the invention of the steamengine, the commercial use ofwhich was so fundamental to theIndustrial Revolution in theeighteenth century. In antiquity,however, machines such asthese never had any practicalfunction, They were sheerexpressions of intelligence atplay, designed and made forpure entertainment. Singingfountains, little theaters, and"magic" self-opening doors hadno real use. They tended to becommissioned by the wealthy tocreate a sensation at feasts. Tous it is strange and in a way"unnatural" that these bizarreand endlessly fascinating toysdid not lead on to machines thatcould have alleviated the burdenof material existence (work inthe fields, in mines, onconstruction sites, at the loom,and so on). Probably it wassimply that slave labour was socheap and plentiful thatmachines were neverconsidered a necessity. At thesame time, a general contemptfor these slaves went hand inhand with a contempt for allmanual work. This must in turnhave meant that people spentless time thinking about basictechnology than about more"worthy" subjects (astronomy,

and pure maths).Let us now look briefly at howthese three machines worked.The singing fountain (above):the water spouts from the lion'smouth into a covered bowl I1),which, as it fills, expels the airthrough two tubes Z)) leading towhistles concealed in thebirds-who thus start to "sing."When the water level reachesthe tubes (3) it overflows intoanother bowl 14). In this bowlthere is a fioat 151 with a cordattached, running around a shaft(fi, at the top of which an owl is

perched. At the other end of thecord is a weight I7): so, as thewater and the float rise, in 14),the shaft and the owl turnaround. Then as 1l and 141 areemptied (perhaps by slaves), theftoat I51 drops again, and the owlturns back to its originalposition.,Hero's machine Ibelow, left):when the fire is lit on the altari1l, the air inside expands,building up pressure in thesphere 12), which is part filledwith water. The water is thuspushed through a syphon 131into a container or bucket (41,which, under the weight, dropsand turns the door hinges (5) bymeans of a cord 161 and acounterweight Q). Then, whenthe fire on the altar goes out, thesame process occurs in reverse'the air pressure returns tonormal, the water in 14) is suckedback into the sphere 12), and thecounterweight 171 rotates thehinges the other way so that thedoors shut again.The water clock of Ctesibius(below, right): water drips into afunnel 1ll leading to a cylinder121 containing a float 13) thatsupports a figure (41. As thewater level in the cylinder rises,the float rises with it, causing thefigure to rise, too, and indicatethe time on the column (5). Atthe same time the water alsorises in the syphon 161 and falls

into a drum (7) divided intocompartments. As eachcompartment fills, the drumturns slightly. A cog on the drum(6) turns three other cogs, one ofwhich is mounted on the shaftsupporting the column.

63

I A)'I,

., < ! - L "t ' . J .\

K;•i 4

SOLITAIREAn attractive legend attributesthe invention of this game to aFrench nobleman imprisoned insolitary confinement in theBastille during the Revolution (atthe end of the eighteenthcentury). He is said to havedevised this pastime to divert histhoughts from the sadpredicament that he had onlytoo many hours to ponder.However, it is known for certainthat in fact solitaire alreadyexisted at the beginning of thecentury. It is mentioned, forinstance, by the Germanphilosopher Baron GottfriedWilhelm von Leibniz in a letterdated JJanuary 17,1716, alreadyunder the name "solitaire."Indeed, it would appear to be avery similar game that the Latinpoet Ovid (alive in the firstcentury B.C. and the first centuryAD.o) mentions; and again, it waswidely played in ancientChina-hence its still frequentalternative name, "Chinesecheckers." Whatever its origins,it came very much into vogue inFrance in the eighteenth century.Undergoing something of arevival today, it still goes mostlyby its French name of solitaire.(The "solitaire" played withcards is known in English as"patience.")Solitaire is played on a smallboard with 37 holes, and thesame number of appropriatelysized pegs Isee opposite page),or else with 37 roundeddepressions and a similarquantity of little balls.The game starts with theremoval of one or more pegs,thus leaving one or more holesfree. The remaining pegs arethen moved by jumps, overtheirneighbours, into the free holes.The pegs that are jumped areremoved from the board, ashappens in checkers. They canmove horizontally or vertically,but never diagonally.The interest of the game variesaccording to the pattern formedat the start, and the patternwhich is ultimately aimed for.The simplest version, thoughalso the most mechanical, is toremove a single peg at thebeginning and to end up withjust one on the board-perhapsin competition with a friend, tosee who can play it out quickest.Let us imagine, however, thatthe board is numbered as infigure a, and that we remove thelast peg (37). Peg 35 moves to37, disposing of 36, then 26 canmove to 36, disposing of 32. Ifwe indicate such a move as26-.36 (32 out), it then becomespossible for us to make any ofthe following moves:

fig. a (1 )2 03 i

(4 )( (t ) (i)

(9') (10)C 1) (12 (13 1 (4

1 , 17 1a) C 09) 200(S) 2(4 205 OM 2

030) (i)1 0 (in3kC 4)

(F () @)6 o

30-.32 131 out)36-26 (32 out)34-32 (33 out)20-33 (27 out)37-.27 (33 out)22-.20 (21 out)20- 33 (27 out)29-27 (28 out)33- 20 (27 out)8- 21 (14 out)

12- 14 113 out)14- 28 (21 out)2-12 (6 out)3-.13 (7 out)

12-14 (13 out)15- 13 (14 out)13-27 (20 out)

26-12 (19 out)28-26 (27 out)32 - 19 (26 out)4- 6 IS out)

18- 5)11 out)5-. 7 (6 out

19- 6 (12 out)7-. 5(6 out)1-.11 (5 out)

16-18 (17 out)18- 5 l 1 out)

9- 11 (b00ut)5-18 (11 out)

18-31 (25 out)23- 25 (24 out)31-18 (25 out)

The end result would be one pegin hole number 18: a somewhatdull gamerIt can be made more interestingand de-manding by deciding thatthe final peg should end up inthe center hole, number 19,rather than 18.The reader will easily be able towork out how to reach thisresult, starting with the removalof any peg-say, the central one,number 19.Here now is a possible sequenceof moves to achieve the sameresult on an "English board"(see figure b below), whichdiffers from the other in having 4fewer holes, thus permitting aslightly simplified game. Thistype of board is particularlygood for children. The end resultis a single peg left in the centerhole. Renumbering the board asin figure b, here are the moves:

fig. b 1 2

4/ 5 ( 6;

7': (8 .1 (9.10) 11) 12. .13)

14 15,. 160) 1 18) C 192 (20:'

21,'. 22,: 231 '24; (25) (261 ('27)

(023, (29' ( 30,

31 j32, 33j

5-17 (10 out)12-'10 (11 out)

3- 11 (6 out)18- 6 (11 out)1- 312 out)3- 11 (6 out)

30- 18 (25 out)27-.25 126 out)24- 26 (25 out)13- 27 120 out)27-25 26 out)22-24(23 out)31- 23 128 out)16-28 (23 out)33- 31 (32 out)31- 23 (28 out)

4- 16 (9 out)7-. 9(8 out)

10- 8(8 out)21- 7 (14 out)

7- 9 (8 out)24- 10 (17 out)10- 8 (9 out)8-.22 (15 out)

22- 24 (23 out)24- 26 (25 out)19-17 (18 out)16-18 (17 out)11-25 (18 out)26-24 (25 out)29-17 (24 out)

There are many problems with acertain outcome (with a preciseprefixed goal) that can be playedon both boards. One version,playable on the French board, isto remove all the pegs except forthe 9 central ones (6, 12,19, 26,32, 17,18, 20, 21),which arethenleft forming a cross on theboard, and to reach a single pegat the end in the minimumnumber of moves. One possiblesequence would go as follows:

12-. 2(6out)26-12 (19 out)17-19 (18 out19-. 6 (12 out)

21-19 (20 out)2- 12 (6 out)

12- 26 (19 out)32-. 19 (26 out)

Another problem with a certainoutcome is to start off with thecenter hole empty and end upwith one peg in the center, andall the outer holes filled-like a

teacher in the middle of a ring ofpupils.Possible moves for thiswould be as follows.

21-19 (20 out)34- 21 128 out)32-.34 131 out)30-32 (31 out)17- 30 124 out)4- 17 (10 out)6- 4(5 out)8-. 6 (7 out)

21- 8(14 out)18- 20 (19 out)

20-33 (27 out)33-A31 (32 out)31- 18 125 out)12-14 (13 out)17-19 I0Bout)26-.12 (19 out)1 1-13 (12 out)14-12 (13 out)

6- 19 (12 out)

It is also possible to devise gamesfor which the outcome is notcertain. Any number of pegs isremoved at random (2, 3 5..)and the aim would be to end upwith a single peg left in any onehole. If several boards of thesame type are available (eitherall French or all English),competitions can be arranged,with each player removing thesame pegs to begin with, and thewinner being the one withfewest pegs remaining after allthe possible moves have beenmade. The games that do have acertain outcome can also beplayed competitively, of course.It is always worth appointing areferee to check that noshortcuts are being taken!Both boards mentioned here areavailable in shops, but you caneasily make your own bycopying the squares in figure conto cardboard and playing withcounters.

fig. c

1 2 3

4 51 6 71 8

9 10 11 12 13 14 15

16 17 18 19 201 21 22

23 24 25 26 27 28 29

30 31 32 33 34

. 35 36 37 l _ . .

64

FOX-AND-GEESEMany different games stillcontinue to be played on thesolitaire board (see pp. 64-65).Certainly one of the best isfox-and-geese. This game hastwo features of interest: first, ithas been a traditional royalfavourite in England (EdwardIV-1461 1483 had twocomplete sets with silver giltpieces, and the young QueenVictoria used to enjoy playing itwith Prince Albert); second, it isa "hunting game" that varieswherever it is played, accordingto the natural and socialenvironment of the locality. Forinstance the indigenous Indiansof Arizona call itpon chochoti, or"coyote and chicken"; to theJapanese it is yasasukranmusashi. which means "eightways of hunting out soldiers;"the Spanish know it as cercarlaliebre "hunting hares"), and soon. Although some of thesegames are played on differenttypes of board, they are broadlysimilar in concept, and all sharethe basic outlook expressed byArnold Arnold: "You cannothunt with the hounds and runwith the hare." Unlike solitaire,fox-and-geese is for two players,one of whom has only 1 peg (thefox), while the other has 17 (thegeese). To start, they arearranged on the board as shownin the colour plate opposite(p. 671 (which depicts balls inhollows, rather than pegs inholes). The fox is the first tomove. Throughout the game itcan move in any direction, killingas many geese as it can byjumping over them (as insolitaire). The geese can onlymove one at a time, eitherforwards (down) or sideways,but not backwards (into theprevious position) and notdiagonally, to try to hem the fox

I

in. They win if they succeed indoing this, and the fox wins if itmanages to eat at least 12 geese.

AsaltoHere is another game played onthe same sort of board, thoughwith lines marked out, linkingthe holes, and a particular groupof 9 holes having specialsignificance (see figure a). Asaltois Spanish for "assault," and it isthe same game as the Britishofficers andSepoys iso called inthe last century, after the IndianMutiny-1857-58-when Indiantroops in the Anglo-Indian army,the Sepoys, rebelled againsttheir British officers).This, too, is a game for two

players, one with 2 pieces (theofficers) and the other with 24(the Sepoys). To start off, theofficers are in the fortresscompletely surrounded by the"mutineers" (see figure a). Theycan move in any direction, whilethe others must go eitherforwards (down) or diagonally.Officers capture Sepoys byjumping them. Sepoys can onlyeliminate officers if they fail totake a possible prisoner.The Sepoys win if they manageto hem the officers in or tooccupy the fort: the officers winif they succeed in capturingenough Sepoys to make itimpossible for them to force theofficers to surrender.

Other versionsNumerous variants are possibleto play on the English solitaireboard, Perhaps some of thecolour plate illustrations of othergames in this book will evenhelp provide ideas for new ones.With a little imagination, forinstance, it should be possible toadapt alquerque (see p. 34) andbagh-bandi (see p. 37)for thisboard. Easiest of all is to adaptthe genuine version of Chinesecheckers (see p. 70): all one hasto do is arrange the variouslycoloured pieces as is suggestedin figures b, c, and d, dependingon the number of players (two,three, or four), then follow therules of Chinese checkers.

67


sort of narrowing and intensifying of experience,which many are not able to cope with. All that isinvolved is a question, an obscure, ambiguousproblem; but the person faced with such a questionhas to answer it. Hence the sense of urgency andpanic that enigmas and riddles can create-feelingsthat are themselves a kind of test from which theindividual emerges qualitatively changed. Out of allthis esoteric jargon frequently appears a style oflanguage, enigmatic to outsiders, possession ofwhich includes one in a secret circle, holding thekey to interpreting and understanding the world,inanimate matter, and animate beings. For the un-initiates, such language has by the nature of thingsto seem mysterious.

To gain a better grasp of enigma, let us analyzethe following riddle: "What has feet but cannotwalk?" Put thus, it sounds like a schoolboy trickquestion-hardly very "enigmatic" in the grandsense. However, all the ingredients of trueenigmam" are there: "feet" is not used in the normalsense but refers to something else, in a differentcontext, which the imagination needs to work hardto identify. For instance, the way we speak of the"foot" of a mountain offers, through metaphor, apossible alternative context. Thus "a mountain"could be the answerto the riddle.

This is one form of interpretation. But, as it stands,a riddle is meaningless to anyone who does notunderstand this way of thinking and referring tothings. Another vital aspect of solemn "enigma" inthe riddle "What has feet but cannot walk?" is theseriousness of the context-how dramatic or eventragic-in which it is asked.

The philosopher Aristotle (384-322 B.C.) relatesthat the famous poet Homer (c. ninth century B.C.), towhom the Iliadand the Odyssey are attributed, onceasked the god what his native country was and whohis parents were. The god's reply was incompleteand enigmatic: he said that his mother's nativeisland was lo, that it was a mortal place, and that heshould beware of a riddle put to him by certainyouths. Homer realized that if he wished to shedlight on his origins, he would have to go to lo,always bearing in mind the god's warning. Onceon the island, the poet saw some fishermenapproaching the shore and asked them if they hadanything to eat. As Aristotle tells it, they had notmanaged to catch anything/ andthiswasthe answerthey gave: "What we caught, we left, and what wedid not catch, we have with us!"

Aristotle explains that, having caught no fish, theyoung men had busied themselves picking off theirbody lice. Thus the lice they had caught, they hadleft behind, while those they had not managed toremove were still on their bodies. It would appear

that Homer, unable to make head ortail of the riddle,died of dejection.

Here, too, then, failure to solve a riddle is associ-ated with the dire problem of penetrating the darkmystery of individual origins-a problem that canactually lead to death. Essentially, a riddle is a kindof contest, a battle with question and answer asweapons, in which life itself is the prize. Not to beable to solve a riddle that "consumes" us withcuriosity is tantamount to death-as Homer and thecitizens of Thebes found out to their cost. It is note-worthy also, though, that in the tradition of riddles,the opposite may sometimes occur: the ability toconceive an enigma that no one can resolve by onewhose mind and awareness of the true nature ofthings confers life-saving power. A not infrequentexample of this is the prisoner condemned to deathbut offered one last chance by the court: he pro-pounds a riddle, and if the judges are unable toanswer it, he is released.

The intelligencebehind riddles

For centuries popular culture found in riddles avehicle of expression appropriate to its own tastes,its own imagination, and its own down-to-earth kindof realism. Many examples survive from the MiddleAges, suggesting a very widespread oral and writ-ten tradition. The courts of kings and nobility wereparticularly favourable to the genre of the riddle. Butat the same time, monks and clergy, too, in thesolitude of their monasteries, often abandonedserious study in favour of these nugae (frivolities),as they called them in their learned Latin. Still,however, itwas amongthecommon folk, in populartradition, that riddles really come into their own.Much dialect literature consists of collections ofproverbs and riddles. And it is not hard to under-stand why. Think back to the days Inot so very longago) before television and radio; in the evenings,especially the long winter evenings, people wouldgatheraround thewarmth of the hearth to exchangethoughts and stories, seated under the oil lamphanging from the ceiling. On such occasions taleswould be narrated, the news of the day would bediscussed, and frequently riddles and guessinggames would be bandied about as a good way ofsharpening one'swit, aswell as providing entertain-ing, competitive games. For a long time popularintelligence and sensitivity were expressed by thesemeans.

Now let us have a look at the structural andlinguistic features of a riddle.

68


First and foremost, it has a metaphorical mean-ing, nearly always dealing with external, materialreality, and relating to the moral values of the worldof the uneducated peasant; second, its purpose wasto highlight and give greater effect to the contrastbetween appearance and reality, between thatwhich has to be guessed and that which at first sightappears; finally, the better the riddle, the harder itwould be to solve. Lively images and colourful,imaginative parables tend to sink in much betterthan abstract, nonfigurative language.

What is that .. .? (Some sample riddles)

Having said all the above, we shall now attempt tosatisfy the reader's legitimate curiosity with someexamples of riddles that are either very popular or ofparticular historical interest.1) What walks all day on its head?2) What grows biggerthe more you take away?3) What kind of ear cannot hear?4) He who makes it, makes it to sell,

He who buys it, does not use it,He who uses it, does not know it!Still in the same vein:

5) What is it that, having it, cannot be given away?6) He who does them, forgets.

He who receives them, remembers.

7) He who does them, remembers them.He who receives them, forgets them.

8) 1 have a comb, but am not a barber,I mark the hours, but am not a bell ringer,I have spurs, but I do not ride.

"What is it that is twisted and cuts off that which isstraight?" King Alboino asked Bertoldo.

And Bertoldo replied, "A sickle!"

The oldest collection of riddles dates from 1538and was compiled by Angiolo Cenni of Siena, ahumble blacksmith. One of his riddles will conveysomething of the skill and poetic content of thepractice of riddle making:

Answers to the riddles

1) a nail (in a horseshoe) 4) a coffin 7) favours

2) a hole 5) death 8) a rooster

3) an ear of corn 6) acts of spite 9) breath.

9) I give at once both heat and coldand death to all to whom I give life;with my strength I drive away heat and cold.See whether my power is infinite;if I be not one with you,you would have neither death nor life.

From the sixteenth century on, riddles becamesomething of a cult in poetry. New riddles wereenjoyed by all social classes, and old ones wererefined and polished, made more literary, andgenerally adapted to suit the salons and courts ofthe rich and powerful.

Alongside the development of neoclassicalpoetry in the sixteenth century and later, there grewa whole genre of literature that one might withoutany pejorative overtones call "enigmatographic:"riddles came to be composed in highly accom-plished verse forms (sonnets, ottava rima, terzarima, and so on) worthy of the best literary tradi-tions.

They continued to enjoy ever wider vogue in theseventeenth and eighteenth centuries. With thespread of the first newspapers in the nineteenthcentury, there came also periodicals devoted toriddles.

It is hardly worth commenting on the presentage's love of the enigmatic and of puzzles of allkinds: newspapers and magazines clearly demon-strate that it is now very much a mass pursuit.

How many hares have the huntersbagged?

One well-known riddle, belonging to the dialectliterature of Italy, is of special interest in that it couldbe considered typical of the genre. Originally writ-ten in the local dialect of Mantua, it runs roughly asfollows:

Two fathers and two sons went hunting,and each killed one hare ...How many hares does that make?

The answer is three, not four. The men who wenthunting were grandfather, father, and son: the "twofathers" being the father and grandfather, the "twosons," the father (son of the grandfather) and theson (the grandson of his father's father).

Thus they were three in all, so there were onlythree hares killed.

In its original form, the riddle is in rhymed verse,while the problem appears to be a simple mathema-tical question. The solution is not merely to identifyan individual character or object, but to give thenumber of hares caught by the hunters.

69


How is it possible?

The trick in the last riddle rests on degrees of familyrelationships that we do not normally ponder. Anyone person may be a father in relation to a son and atthe same time a son in relation to his own father.

Following an earthquake, father and son wereburied under rubble, severely wounded. When therescuers reached them, they were immediately seento be in grave danger and were rushed to the hos-pital. The father died on the way, and the son wastaken at once into the operating theater. Thesurgeon on seeing him, however, to the great sur-prise of all said:

"I cannot operate on him, he is my own son!"How so?The solution is rather banal: the surgeon was the

wounded boy's mother and could not risk the re-sponsibility, if therewas the slightest error, of beingthe cause of her son's death.

What is "entertaining" about this example is notthe oddity or the difficulty of the answer, but its veryobviousness, once given.

Deceived by the normal male connotations of"surgeon," it is easy to forget that women can alsobe surgeons, and thus that "the surgeon" in thiscase might just as well have been the boy's mother,who was not mentioned at the beginning as havingbeen trapped in thefalling masonry.

How many are we in our family?

Gone now are the days when small companies oftravelling players would advertise their perform-ances with short anticipatory sketches and scenesand question-and-answer routines in the local townor village square. But here is a riddle posed by onesuch group of actors to the spectators they hadmanaged to attract.

"I have as many brothers as sisters." (The actor ishooded and cloaked so that it is impossible to deter-mine which sex he or she is.) Then onto the stagecomes a woman, who in forthright tones declares:"I am the sister that has just been mentioned, and Ihave twice as many brothers as sisters. So howmany are we in our family?"

Anyone who guessed the correct answer wouldreceive a free ticket for the evening show.

A simple process of reasoning will reveal theanswer. Let us suppose that the cloaked figure whospoke first is a sister. Then the sister who spokesecond should have said the same as the first-forexample, should also have said she had "as manybrothers as sisters." However, that is not what shesaid. Hence the first speaker must be a brother.

What conclusions would that lead to?There must without doubt be one brother more

than the number of sisters.Now let us analyze what the sister who spoke

second said: according to her, she has twice asmany brothers as sisters-thus half as many sistersas brothers. Half the number of brothers is then two,the number of brothers is four, and there must be(including the last speaker[ three sisters, making inall seven brothers and sisters!

How old is Peter?

One day Peter was invited to a meal with an uncle towhom he was close.

"Tell me, Peter," his uncle began, "I've forgottenthe date of your birthday."

After a moment's thought, Peter jokingly repliedwith the following riddle:

"The day before yesterday I was fifteen, and nextyear I shall be of age (eighteen)."

His uncle was stunned by this answer. He smiledand straight away went and bought a superb pre-sent for such a witty reply.

What was the date of Peter's birthday, and howcould his reply make sense?

If one thinks about it, Peter (who was sixteen yearsand one day old) must have been visiting his uncleon the first day of the year: if "the day beforeyesterday" he was fifteen, that must mean that hisbirthday fell on the previous day-December 31.This is the only workable date. On December 30Peter was still fifteen, on the thirty-first he becamesixteen, at the end of the year just starting he wouldbecome seventeen, and thus "next year" he wouldbe eighteen.

A logical riddle

Solving a riddle often means pursuing a correct lineof reasoning through to its conclusion. Sometimes agame or a problem can seem insoluble simply be-cause we lack the patience to exercise our reason.The game that comes next is not a traditional riddle,but it is a problem to which the answer is notimmediately apparent. It is necessary to weight thedata and then pursue a line of argument.

From a pack of cards three are removed, one black(clubs or spades), and two red (diamonds or hearts).Two players are involved. The three cards are prop-erly shuffled, then dealt as follows: one, facedown,to each player; and the third, also facedown, ontothe table. At a sign from the dealer (who is also thereferee) each player then looks at his or her card.

70


4!A A

V

tV

The winner is the first one to guess the colour of theother player's card. Thinking ourselves into theposition of one of the players, our reasoning shouldgo thus:

1) If I have been dealt the black card, then there isno problem: my adversary must have a red one, andso I cannot lose.

21 Ift however, I have been dealt a red card, Ishould wait (though not too long) to see what myopponent says: if he does not speak immediately,he cannot for certain have the black card, so he musthave the other red.

The winner here is the player who first works outthe logic of the game and outwits the other. (Thegame, of course, cannot be played more than oncefor obvious reasons.)

The eyes of the mind

Sooner or later even convicts have their hour ofrelease! A prison governor, noting the good con-duct of three inmates, thought to offer them achance of parole. Of these three, one had normal

Weights and scales

What counts in this game is skill at combinations.We have three pairs of balls, similar to those usedfor putting the shot: two red, two white, and twopink. The two balls of the same colour are indis-tinguishable except for the fact that one of the pairis three times the weight of the other. Hence there isone set of light balls of identical weight (red, white,and pink) and one set of heavy (red, white, and pink).The game is this: to establish, with the use of scaleshaving two pans, each capable of containing notmore than two balls at a time, which is the heavierand which the lighter ball in each pair of the samecolour-taking only two weight recordings! Howshould the balls be combined on the scales?

71

I

eyesight, one had only one eye, and the third wascompletely blind. Calling them into his office, thegovernor posed the following conundrum:

"I have here five caps, three white and two red. Ishall put one cap, either red or white, on each ofyourheads. You cannotseethecolourof thecaponyour own head, but you can see the caps on theothers' heads!"

At this juncture one might think that the trick wasunfair to the blind prisoner. However, it was not toprove so.

First of all the governor called the prisoner withnormal eyesight and asked him if he was able to saywhat colour his cap was. He answered no. Theone-eyed prisoner was then also asked, and he, too,confessed that he could not give an answer.

Not even thinking to ask the blind prisoner, thegovernor was then about to send them back to theircells when the blind man stepped forward andasked if he, too, could take part in the game like theothers. With a smile, he said:

"I do not need eyes: given what my companionshave said, I can see, with my mind's eyes, that mycap is white!"

This was indeed the case, and true to his word thegovernor granted him his liberty.

How had the blind man's thinking gone?It must have been more or less along these lines:

If I have a red cap, the one-eyed prisoner-seeingthat the normal-sighted one was unable to tell thecolour of his cap-would have thought that had he,too, had a red cap, then the first prisonerwould havebeen able to know for sure that his was white. Andsince this had not been the case, he would haveconcluded that his is white. Now since he had notsaid white, either, it must mean that I have a whitecap!


Wr WR

wr 0

wR 0

wp 1

wP: 0

0

0

1

1

Wp

I

0

0

0

There are a number of solutions. We suggest thefollowing: Put two balls of different colours-say,one red and one white-onto one of the measuringpans and a white and a pink on the other. The scalescould react in one of two ways:

1) If the scales remain in equilibrium, remove thetwo balls of different colours (this will then showwhich of the white balls is the heavier); having onceidentified the heavier of the two white balls, one canthen work out which was the heavier of the twocoloured balls with which the white ones were in-itially weighed; and hence it can be deducedwhether the unused balls were the heavy or lightones.

2) If, however, the scales drop to one side or theother, it means that the white ball on that side mustbe heavier than that on the other (otherwise thescales would stay in equilibrium); yet it is not poss-ibleto establish immediatelythe relative weights ofthe red and pink balls; the next step, therefore, is toplace on one of the weighing pans the red ballalready used in the previous step and balance itagainst the twin of the pink (or vice versa); depend-ing on what happensthen, the relative weights of allthe balls will be deducible. Let us analyze thevarious possibilities that may occur, to clarify theprocedure.

W (capital) = the heavier white;w (lowercase) = the lighter white;R (capital) = the heavier red;r (lowercase) = the lighter red;P (capital) = the heavier pink;p (lowercase) = the lighter pink.

Remembering that one of the pans has provedheavierthan the other, and thatthe white ball on thisside must necessarily be the heavier rather than the

72

lighter, the possible combinations are as follows inthe chart above.

How, though, are we to read this chart? First of all,o = impossible combinations, given the conditionsof the test. For instance, the combination Wp-wR isout of the question because it would have held thescales in equilibrium, and in the present case we areassuming that the scales are tipped down on oneside. Equally impossible is the combination WR-wRsince there are not two heavy red balls. Let us nowthen consider a slightly more complex example: thecombination WP-wr i.e. the bottom pan containingthe heavy white and pink balls, and the light whiteand red in the other; as soon as WP sinks down(during the first step of the procedure), I can identifyW and w; I then weigh the unused red ball (in thiscase R) against P, and sincetheyare of equalweight,the scaleswill stay in equilibrium. Two assumptionscan then be made: if the two balls are p and r,then inthe first step, since in the second the unused red ballcame into play, it has to be R. Yet that would havegiven us, in the first step, Wp-wR, which by the verynature of the puzzle could not have been the case:the 0 in the chart opposite Wp-wR indicates this;such a combination would have kept the scales inequilibrium rather than weighted on one side.

Ping-Pong balls

In this gamewe have to extract balls from a box. Thetrick behind it, however, is entirely a matter ofreasoning, not of calculation.

A team of Ping-Pong players has ordered a supplyof six phosphorescent balls (to stimulate andstrengthen their visual reflexes): three are redand three white.

The supplying firm sent three small boxes: one

.w.O.

IX

WP

1

1

0

0A& 11.�MkWWI'I


containing two red balls, the second containing twowhite balls, and the third containing one white andone red. To recognize which was which, the sup-pliers had labelled each box: ww meant two whiteballs; rr two red; wr a white and a red. So far sogood. Unfortunately, however, the labels had some-how got muddled, and every box was mislabelled.

Here is the game: Removing one ball at a timefrom any box, what is the minimum number ofremovals necessary to establish the contents of allthree boxes?

It is in fact possible to do it in a single operation.Let us suppose we have removed a white ball fromthe box labelled wr. Since the label is wrong, theother ball cannot be red, so it must be white: thuswehave identified the box containing the two whiteballs.

Let us now examine the box labelled rr. We knowit cannot contain both red balls, as the labels arewrong. It must therefore contain a white and a red.Hence it is also possible to assume that the last boxis that containing the two red balls.

This solution depends on the information that thelabels are all wrong.

One last question: If, after the removal of the firstball, we had turned to the box named ww, would asingle operation still have sufficed? And if we hadtaken our first ball from the box marked rr? (Youshould each be able to work out the answers.)

In the world of opposites

A sociologist came to hear of a strange societycomposed of two rigidly opposed groups: one al-

II - I M ,

ways told lies, the other always told the truth. Feel-ing he should investigate this he set off for thiscurious enigmatic land.

The two roads

Oursociologist was making forthe main populationcenter of this land when he came upon a fork in theroad. Which way should he go? As he stood in aquandary, two people appeared. His first feelingwas one of relief. But such feelings had soon to bechecked, for he remembered somebody warninghim before he left that he would indeed encountertwo people-and one would be a liar, the other atruth teller!

There was no telling from mere appearances.There must be some way out of this predicament,

thought the sociologist.After a moment's hesitation, he asked one of the

two strangers a simple question, waited patientlyfor the answer, then set out confidently in the direc-tion that had been indicated to him.

Here is the riddle: What was the question?Ponder the situation. Could he not have asked

either stranger-no matter which-the followingquestion: "Excuse me, would you please tell mewhich road your companion would show me, if hewere to show me the wrong one?"

Let us analyze the logic of this question. There aretwo possibilities: the person to whom the questionis addressed may be a liar and the other one a truthteller, or it may be the other way around. Let usbegin by supposing that the person addressed is atruth teller. Since he always tells the truth, will henot also have transmitted the request faithfully inthis case? His lying companion meanwhile will havegiven the wrong answer-or the right direction tothe town.

Supposing now that it was the liar who wasaddressed in the first place: since he cannot but lieand reverse everything, he will change the verycontent of the question and ask the truth teller:"Which is the right way to the town?" thus resultingagain in the correct answer.

In other words, the truth teller, who always tellsthe truth, will indicate the correct road to the town.Either way, then, the sociologist can be confident inthe road he takes.

A variant

We have deliberately introduced the reader to theproblems of logic by means of a very popular oldstory, known sometimes as "The Fable of the Liar

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and the Traveller," over which generations of logi-cians, philosophers, and amateur fans of conun-drums and riddles have exercised their minds.

Here we encounter a curiosity of logic that forcenturies has proved intellectually stimulating-aparadigm of logical difficulties that are unapparentor apprehended only at an intuitional level.

The different variants of the fable testify to itsusefulness.

According to one such variant, the traveller onlymeets one rather than two strangers. He does notknow whether it is a liar or a truth teller he has met;nevertheless, after posing a simple question, he isagain able to continue on his way, confident that heis taking the right road.

Once more the problem is: What was thetraveller's question?

One solution could go as follows:"If you belonged to the other group, which road

would you direct me onto to reach the town?"Let us suppose that the stranger is a truth teller:

putting himself in the place of a liar, he would thengive the wrong road.

Remember, the truth teller always tells the truth.Thus, having been asked to answerfora liar, he sayshonestly what the liar would have said, namely thewrong road. Supposing nowthatthe stranger wasaliar: he would put himself in the place of the truthteller, and then say the exact opposite of what thetruth teller would have said! Thus in both cases theroad indicated would be the wrong one.

Confident of this, the traveller could then happilytake the other road.

Shrook!

Sure of being on the right road, the traveller-sociologist now prepared mentally for the quan-daries that lay ahead. The problem at the roadjunction taught him one thing about this society:people at least acted according to rigorous logic. Hethus resolved to think along the same lines. Thiswould enable him to understand the innerworkingsof a rather singular human community!

Suddenly in front of him he spied two people. Atonce he thought this would be a good chance to testhis resolve. He determined to find out which groupeach belonged to. Courteously approaching them,he addressed the most affable looking of the two:"Are you one of the group that always tells thetruth?"

"Shrookl" came the rather confusing reply.What does "Shrook" mean? he thought to him-

self. Then, just as he was on the point of addressingthe other stranger, he was forestalled.

"He said, 'Yes,'" interpreted the second stranger."But he's a terrible liar!"

What are the two strangers? Are they both liars?Or both truth tellers? Or is one a truth teller and theother a liar? In which case, which is which? Thesolution is much simpler than might appear: allrests on patient reasoning.

Let us suppose the first stranger was a liar, whowould therefore have answered, "Yes"-just as atruth teller would also have answered, "Yes." Ineither case, in other words, the initial answer had tobe "yes."

Thus in saying that his companion had said,"Yes," the second stranger was telling the truth. Histranslation of the bizarre answer "Shrook" was reli-able. He must be a truth teller.

Satisfied with this conclusion, our intrepidsociologist continues on his way.

Two buffoons

One of the most revealing indicators of the nature ofany society is its educational system: the way itprojects its own notion of itself toward the future, itsaims, its programs of development.

Well aware of this, the sociologist-traveller had,even before setting out, decided to make a specialstudy of local schooling. He had indeed a kind ofplan, and top of the agenda was the establishing oflinks with teachers. It was not hard to find out whichwere the major educational establishments, and offto one of these he betook himself. Having found theway to the staff room, he was greeted by an unusualscene: elderly people, breathing hoarsely and dres-sed without taste, some positively in rags, mingledwith younger individuals, who were equally odd-absurdly dressed, as if for carnival. It must be tocatch the pupils' attention decided the sociologist,doing his best to hide his feelings of disgust.

He was anxious not to waste time by becomingtangled in a web of lies: thiswould have jeopardizedhis whole venture. Now a further division amongthe inhabitants of the world of opposites (besidesthat of liars and truth tellers) was made, into politi-cians and ordinary citizens respectively. (For exam-ple, "All liars are politicians, and all truth tellers areordinary citizens.") Wishing for an immediateanswer, he approached a young man and a some-what older man who were reclining on armchairswith a perennially tired air and inquired: "Is one ofyou an ordinary citizen?"

The younger looking of the two replied, such thatthe sociologist was able to get a true answer.

What was the young teacher who had replied-truth teller or a liar? And what about his older

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colleague? Within the terms of the game alreadygiven, we have enough information to know forsure. However, it is necessary to argue from certainhypothetical assumptions. First, we know that theteacher's reply was sufficient to tell the sociologistwhether he and his colleague were politicians,ordinary citizens, or one of each. Let us thereforeimagine the younger teacher's reply as being, "Yes,one of us is a truth teller."

Would this have sufficed for the sociologist todiscover what the young man really was? Let usreflect: If he was an ordinary citizen, he would ofcourse have answered, "Yes"-the truth, in that atleast one of them was a truth teller; if he was apolitician, however, his "yes" would have meantthat they were both politicians. Hence "yes" alonecould not have allowed the sociologist to discoverwhat the two men were. Yet the fact is (as we knowfrom the story) that he diddiscoverfrom the answerwho was what. And if "yes" was not enough for himto have made that discovery, then the answeractually given must have been "no."

Now let us consider: Who could have said "no"-an ordinary citizen or a politician? Were the youngteacher an ordinary citizen, he would not have beenable to say "no" and yet still be telling the truth.Thus he must necessarily be a politician,telling a lie.His "no" would indicate that at least one of themwas a truth teller, and if it was not the speakerhimself, then it had to be the elder teacher. Havingestablished this, then, the sociologist could confi-dently address his further questions to the truth-telling ordinary citizen-the elder teacher.

The meeting with the prime minister

An interview with the prime minister would con-siderably aid our sociologist's work, and he eventu-ally contrived to arrange one, after much pushing.

During his preparations for this interview, itoccurred to him that he should first find outwhetherthe prime minister was a truth teller or a liar. Onlyonce he knew this would he be able to formulate hisfollowing questions. However, there was a furthercomplication; for he had discovered that in this oddcountry there was yet another division of indi-viduals-those who answered every other questionwith a lie and every other question with the truth.There were not many of these, but such as therewere tended to be found in the upper echelons.Hence the prime minister might very well be one ofthese. . . "alternators."

First of all, therefore, the wily sociologist deviseda method of establishing beyond any shadow ofdoubt what the prime minister was. After a while he

wrote two questions on a sheet of paper: what werethey? In fact, it was merely the same question askedtwice: "Excuse me, are you an alternator?" (Thereare other possibilities here, too.)

If the prime minister replied "No" on each occa-sion, thatwould make him a truth teller; if each timehe replied "Yes," then he would be a liar; and if hereplied "Yes-no," or "No-yes," he would revealhimself to be an alternator indeed.

That information would probably suffice for thesociologist. But say he wished to find out whether(should the PM after all be an alternator) he says thetruth first and then lies, or vice versa?

Are there a couple of questions that would elicitthat information?

The solution would appear obvious: one needonly ask him two questions about indisputablefacts. One could, for example, ask: "Do cats havefour paws?"

"Yes" times two would signify a truth teller; "no"times two a liar. A "yes-no" or "no-yes" wouldindicate both an alternator and the type of alterna-tor. And this could significantly affect the sociolo-gist's inquiries.

At the Assembly of the Wise: all truthtellers or all liars?

How was the country of liars and truth tellersgoverned?

In the course of his inquiries, our redoubtablesociologist managed to gain access to the supremehouse of government, the Assembly of the Wise.This was a chamber of representatives, formed ofhighly able people, deputed by others to shoulderthe burdensome responsibilities of governing.From the very name of this governing body, thesociologist realized that he might be up againstproblems. Nor were his fears unfounded.

Having gained permission to enter the house ofthe assembly, he eventually turned up during animportant session. All the members were present,arranged around a circulartable. At first sight it wasimpossible to tell the liars from the others.

It has to be remembered, in the interest of truth,that in this strange country, the liars are also politi-cians, and truth tellers ordinary citizens. Thesedesignations were used, for some reason, in politi-cal relationships, too.

Now our ingenious sociologist began, as oppor-tunity presented itself, to ask everybody therewhom they represented and whether they belongedto either of the "parties." To his surprise, all themembers of the assembly claimed to be ordinarycitizens and, as such, truth tellers.

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Is it possible, thought our hero, that only the truthtellers are represented in this house?

Clearly something was wrong: a bit of cunningwould be needed to get to the bottom of this! So hethought up a plan: he asked each member seated atthe circular table if his left-hand neighbour was atruth teller or a liar. What was his astonishmentwhen he heard once again an identical reply fromeverybody, though this time the opposite of whathad been asserted the first time. Each member'sleft-hand neighbour appeared to be a liar!

Not one to be defeated so easily, he returnedto his room, sat down, and pored over the data. Itwas not easy to establish the composition of theAssembly of the Wise and the percentage represen-tation of politicians and ordinary citizens. By nowaccustomed to thinking logically, however, he hadalmost reached conclusive results. Yet one detailstill remained for him to find out: the number ofmembers present that day at the assembly.

Without that information he would not be able tocome to any conclusive decision as to the politicalmakeup of this society.

How many members?

Our indomitable sociologist picked up the phoneand got himself put through to the president of theassembly. He asked his question and was told thatthere had been in all fifty-seven members present.Experience had taught him, though, not to trust thefirst reply he might receive, all the more since he hadnot yet been able to gather whether the presidentwas a truth teller or a liar. Picking up the phoneagain, he called the secretary, who sat beside thepresident, and asked: "Is it true that there werefifty-seven members in the assembly?"

"No, no," cried the secretary vehemently. "I'vealready told you, the president is a typical politician,he's the biggest liar of the lot; there were sixtypeople around the table. I counted them myself."

Whom should he trust? Can a president of anassembly lie so shamelessly?

From the secretary's emphatic, angry tone, thesociologist was inclined to give greater credit tohim. But previous experience had made him wary,and he would only trust to the rigorous certainty oflogic. Turning to a citizen who he knew for certainwas a truth teller, he appealed for help. Withouttelling him the identity of the liar, the man he askedreplied, "Only one of the two is an ordinary citizen."

So who was telling the truth. This informationalone was enough for the sociologist to be able toestablish the precise number of the members of theassembly: then what was the number of persons

seated around the large round table?Let us try analyzing the situation together.Asked singly, the members had all declared them-

selves to be truth tellers; while, according to theirright-hand neighbours, every one of them was a liar.These replies already can lead us to two hypo-theses. Let us suppose, on the basis of the first reply,that all were truth tellers-in that case they all liedwhen they said their neighbour was a liar, andtherein is a contradiction, since truth tellers herealways tell the truth; alternatively, let us supposethat all were indeed liars-then they would havereplied correctly in saying that they were them-selvestruth tellers, buttherewould again have beena contradiction when they declared their left-handneighbours to be liars.

We thus have to drop any notion that all themembers of the assembly were the same, be it liarsor truth tellers. We can only conclude that they werea mixed bag of liars and truth tellers. Now there isone more bit of information we can perhaps derivefrom what we have been told: the order in whichliars and truth tellers are seated around the table.And perhaps this will then enable us to work outhow many members the assembly had and thus todecide which of the two-the president or thesecretary-were barefaced liars.

Let us then suppose that (going clockwise, fromleft to right) we encountered two truth tellers andtwo liars: the first truth teller, in claiming his left-hand neighbour to be a liar, would not have beentelling the truth, so that will not do! The sameobjection holds if we suppose there were two liarssitting next to each other. Thus, in the Assembly ofthe Wise, liars and truth tellers must have beenseated alternately: no other arrangement wouldsquare with what was said by each about them-selves and about their neighbours. If, therefore, liarsand truth tellers alternated, there must have been aneven number of members-for otherwise therewould at some point around the circle have had tobe either two liars or two truth tellers together. Andas we have seen, the answers given by the membersthemselves excludes this possibility.

The diagrams on the opposite page depict ahypothetical large round table with truth tellers (thelittle circles marked T) and liars (the circles markedL) seated around it: in I there is an even number ofmembers (for convenience sake, only a few, not thefull contingent), while 11 (again less than the fullquorum) depicts an odd number.

As can be seen, where there is an odd number ofmembers, two liars end up next to each other, whichwould mean that one or other would be obliged tosay that his left-hand neighbour was a truth teller.

It emerges then that the president was telling an

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outright lie in claiming there were fifty-seven (anodd number members. As the secretary was sittingnext to him, he must be a truth teller: thus thenumber of members must truly have been sixty!

The game never ends

To draw out all the possibilities implicit within agame, a riddle, or any kind of problem is not only amatter of methodological habit, it can also provevery useful. Logic in this respect shares the samemethod of procedure as mathematics. If our aimwas simply to find a solution in the simplest andmost direct way, our game might be consideredover. We can, however, make the solution morecomplete by following a further line of investiga-tion: by asking what conditions could have madethe game insoluble.

Let us suppose that the president of the assemblyhad claimed there were fifty-eight rather than fifty-seven members. He would still have been lying, butwe should not have been able to establish his dis-honesty forcertain; indeed, we could have assumedhim to be honest without thereby creating anylogical inconsistencies.

We can be sure he was lying only because fifty-seven is an odd number, while it was logicallynecessary that the total be even.

We can say, then, that the problem would havebeen insoluble had both the president and thesecretary claimed an even number of members,rather than conflicting even and odd numbers.

A serious game

Within human society it is not possible to make anyclear distinction between good and bad people-between liars and truth tellers. Individuals withinthemselves are never all one thing or the other.

Since our game rested on the assumption thatpeople were thus either entirely honest or entirelydishonest, there is no question but that it was mereimaginative entertainment. It might thus seem thatto reduce the complexity and variety of real humansociety to two clearly demarcated groups is purelyfanciful. Nevertheless, there may be more to it.

Strange as it may seem, there are times whentaking an unrealistic viewpoint (putting onself in asense outside reality) actually helps towardsreaching a clearer picture of reality-to grasp other-wise elusive aspects of it. Reality (human, social,physical . . ) is too varied and complex for us tograsp wholly in all its aspects. This does not mean,however, that it cannot be grasped. Generally onebegins by isolating certain individual aspects, andfrom this basis one can gradually build up a morecomplete picture of whatever area one is investi-gating. To begin with, therefore, only isolatedmoments and separate elements of reality areconsidered. Yet this reduction and simplification ofthings should not be mourned as a loss, but ratherseen as a necessa ryfirst step. To some extent, this isthe way of the rational method in the sciences.Physics, for instance, started as a rational study ofexternal reality, concentrating on only some of theinnumerable material phenomena of the natural

(continuedon page 82)77

CHINESE CHECKERS-ALMAChinese checkers is widelyplayed today by both young andold alike. It is a game thatrequires a certain degree ofmental application. Though anAmerican invention, dating fromthe second half of the lastcentury, it owes its name to thefact thatthe country in which it ismost popular is China, where itis called xiaoxing, tisoqi. Someconfusion can arise between thisgame and solitaire (see p. 64),which is also occasionallyknown as Chinese checkers.Real Chinese checkers is playedon a hexagonal star-shapedboard, with 121 holes or hollows(depending on whether pegs orballs or marbles are used-seepage opposite).Two, three, four, five, or sixplayers can participate. If thereare two or three players, eachhas 15 pieces; if more, then eachhas 10, all the same colour, adifferent colour per player. Tostart off, the pieces are arrangedin the 'points' of the star, asshown in figure a (in which threeplayers are assumed to be

involved). The object is to moveall your pieces into the star pointdiametrically opposite that inwhich you started. The firstplayer to manage this is thewinner. Pieces can be moved inany direction, one at a time.They can eithertake one step orjump another piece, but not bothin one go. Jumping can continueover any number of pieces, bothrivals and pieces of the samecolourfinishing atthefirstempty place bordered byanother empty place (see figureb). (Note: A popular alternativerule is to jump over only onepiece at a time. However, severalof these single jumps may bemade in one move.) Pieces thathave been jumped are notremoved from the board, sincethat is not the point of the game.There are differing versions ofrules (such as starting off with 10rather than 15 pieces, or 6 ratherthan 10), but there are no basicdisagreements.Chinese checkers is oftenconsidered to have derived froma much less well-known game

fig. b

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called Alma, which was also used). Each player moves ininvented in the second half of turn, one piece at a time, not inthe nineteenth century and has all directions, but on/ysimilar rules. It takes its name diagonally, such thet a piecefrom the riverAlmna in the on a light square will alwaysCrimea (where the great battle of remain on light squares. As in1854 was fought), on the start of Chinese checkers, each piecewhich the rules were devised, can move either a single squareAlma is sometimes written with or jump (once or as often as thean h at the beginning, after the layout of the other pieces allows)ancient Greek word hi In, both aneney and friendly pieces;meaning "jump"-obviously a pieces that are jumped are notmore direct reference to the removed from the board. Thegame itself. winner is the player who firstThe board is square, with 16 manages to rearrange his piecesplaces around each side, in the corner opposite that inalternately dark and light (see which he started.figure cI. It can be played with A rather complicated variant oftwo or four players: if two, each Alma is the numbered version,has 19 pieces; if four, 13-all the in which every piece issame colour, though each player numbered in order and arrangedhaving a different colour (for on the board as shown in figureinstance, white, black, red, and d. The rules remain the same,yellow). To start, each player but the object is to reconstitute aarranges his pieces in a corner of symmetrical arrangement ofthe board, as shown in figure C one's pieces in the opposite(if there are 19 pieces, the 19 corner of the board, with all theadjacent corner squares are numbers in position.

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NINE MEN'SMORRISThis is a simple game, playedthroughout the world in differentforms and under differentnames. The American andEnglish name nine men's Morrisbecomes three, six, or twelvemen's Morris according to thenumber of pieces used. Theearliest known traces of thisgame have been found in thetemple of Kurna, on the westbank of the Nile at Thebes(fourteenth century B.C.).Remains of boards were alsodiscovered during excavationsat the lowest level of Troy, andsimilar remains have been foundin Bronze Age tombs in CountyWicklow, Ireland, in the vestigesof a funerary boat at Gokstad,Norway (tenth century B.C.), andin many other places. It is alsomentioned in Alfonso X ofSwabia's Libro dejuegos (see p.34), in the Jewish Talmud, andeven in Shakespeare's AMidsummer Night's Dream.

fig. a

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during the maneuvering oncethey are all in play. In the lattercase, pieces are maneuveredsimply from one square to thenext. The winner is the firstplayer to reduce his opponent's"army" to 2 pieces orto make itimpossible for him to form a line.

Three men's MorrisHere the board is slightly moredeveloped than that fortic-tac-toe, but again it is playedwith 3 pieces (see figure bh. Therules are the same, but thepieces are positioned on thepoints of intersection of thevarious horizontal, vertical, anddiagonal lines see figure b).

Six men's MorrisOn this board the diagonal lineshave disappeared, and anothercentral square has beenintroduced (see figure cl. Eachplayer has 6 pieces (hence thename of the game). Again, therules are similarto three men'sMorris. Each playertries toforma line along one of the sides ofeither square.

fig. b

Nine men's MorrisThis is the most popular versionthe world over. Each player has 9pieces, and the board consists of3 concentric squares joined bystraight lines (see oppositepage). The rules are as describedearlier, with the followingconsiderations:1. Once all the pieces are in play,they move one at a time fromone point of intersection to anyfree neighbouring intersection,the object still being to form arow or mill of 3 pieces.2. Pieces can be aligned alongthe same axis as many times asdesired (though of course tomake the game moredemanding, it can be ruled thattwo lines cannot be formed onthe same spot, as in a cross or"T" shape.3. As before, enemy pieces thatare part of a line of three cannotbe taken.4. Once a player has only 3pieces left, he can move them(and jump) one at a time withouthaving to follow the lines asdrawn.

fig. c

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fig. d

Twelve men's MorrisDifferent from the above only inhaving a more complicatedboard and in each player'shaving 12 rather than 9 pieces(sea figure d).

Morris with numbersAn interesting variant because ofits similarity to magic squares(see pp. 169-171). A board isdrawn on a sheet of paper andsubdivided into 9 circles (seefigure a) or squares (see figure f6,and one plays with numbers.One player uses 0 and evennumbers 10, 2, 4, 6, 8,10), theother the odd numbers (1, 3, 5, 7,9). As the former has one extrapiecece" he starts by writing oneof his numbers in any of the 9circles or squares. Then thesecond player does the same,writing an odd number, and soon, each number being allowedonly one appearance. Thewinner is the playerwho firstsucceeds in making a sequenceof three numbers, in anydirection, totalling 15.By complicating the board

fig. e

Some of the most interestingvariations of this game are asfollows:

Tic-tac-toeThis is perhaps the simplestversion of nine men's Morris,played by two players on a9-square board, with 3 pieceseach (see figure a). The boardstarts off bare. Each player inturn then places 1 piece on anysquare, the aim being to create aline of 3 consecutive piecesl"tic-tac-toe"l, either horizontal,vertical, or diagonal. The winneris the first player to achieve this.The game is rendered moreinteresting if 5 pieces are givento each player ratherthan 3: inthis case, having created a line of3 pieces, the relevant player cantake one of his opponent'spieces (though not one that isalso part of a complete line).Pieces can be taken either duringthe process of introducing the"armies" onto the board or

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somewhat, and thus increasingthe quantity of numbers at eachplayer's disposal, numerousvariants of this game can bedevised; and new ideas canalways be drawn from thedescription of magic squareslater in this book (see pp.169-171).

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world. Other disciplines have evolved along similarlines.

By necessity a starting point is reductive. Thefinalaim, however, is always to achieve as complete apicture as possible of the reality under examination.

In a way we have done the same, postulating asociety formed exclusively of liars and truth tellers-simplifying reality, abstracting ourselves from it.But by doing so, we have been able to familiarizeourselves with a method, a viewpoint, that is onlyapparently remote from everyday reality: namely,that of logic.

The story of the sociologist-traveller in the worldof opposites is a serious game because it is fun-damentally an exercise of logic-that is, it is basedon a discipline that forms the groundwork of scien-tific reasoning.

A liar from antiquity

One illustration of how, even in antiquity, in theearly days of philosophy, thinkers sought a betterunderstanding of natural and human reality bydreaming up hypothetical, unreal stories isthe para-dox of Epimenides.

Epimenides of Crete was a curious figure. Weknow little about him, and most of what we do knowis legendary in character: for instance, he is said tohave lived to the age of 299, with odd bouts of sleep,one lasting fifty-seven years when he was meant tobe tending his father's sheep!

A native of Knossos (the site of the celebratedlabyrinth of Minos), he is thought to have beencalled to Athens in the early decades of the sixthcentury B.C. to purge the city of plague. His philo-sophy is usually classed as pre-Socratic; that is,simply, that it predates Socrates. It should berememberedthatthe philosopherswho lived beforeSocrates had little notion of themselves as "philo-sophers" in the sense in which we now understandthe word. They were more like priests, wizards, orsages, or devotees of purificatory rites (asEpimenides of Crete would seem to have been).

The paradox that has immortalized the name ofEpimenides in the history of philosophy suggestsalready a certain mastery of the discipline of ab-straction. In its original form, it runs thus: "AllCretans are liars."

Why does this simple statement have such signi-ficance? There are many Cretans, but let us acceptthat they are all liars: now, since Epimenides him-self is a Cretan, he, too, must be a liar! Hence hisstatement is true-which it cannot be, since a liardoes not tell the truth. Thus Epimenides is a liar, andhis statement is not true. To deny a universal state-

ment such as "All Cretans are liars" istantamount tosaying that there is at least one Cretan who tells thetruth.

To conclude, all that can be deduced fromEpimenides' paradox is that 1) Epimenides is a liar,and 2)there is at least one Cretan whotellsthetruth!

The antinomy of the liar

What is known historically as the paradox ofEpimenides is not a paradox in the strict sense, eventhough it has given rise to a whole series of para-doxes, all on thetheme ofthe liar. Usuallya paradoxis a statement thatflies in the face of common senseor the rules of logic. In philosophical jargon, this ismore properly known as an antinomy, and althoughtechnically there is a distinction between the twoterms, for our present purposes they are nearenough identical in meaning. So as we trace theprocess of explication and clarification ofEpimenides' paradox in the history of philosophyand logic, we shall speak of the "antinomy of theliar." An antinomy is generally a statement that,either when asserted or when denied, leads to con-tradiction. The antinomy of the liar is usually pre-sented in the following form: we are to imagine aliar saying, "The statement I am making at thismoment is false."

Is the proposition true or false?The answer is a clear self-contradiction: if it is

false, then it is true; and if it is true, then it is false!It is possible to resolve this antinomy only

through a complex analysis of language, whichwould take all night! Suffice it to say that suchstatements lack any real base. In other words, themistake is in the very notion of a statement such as"this proposition is false," since "this proposition"conveys no information and thus has no content.

An invitation to logic

What point isthere in imagining a societyasabsurd,as remote from real life, as "the world of opposites,"inhabited only by liars or truth tellers?

Only a cursory, superficial view of the matter willfind it confusing. In fact, as we have already seen, itrevealsverywell a certain method of thought, awayof approaching problems, that is much more downto earththan might appear. Thewaythesociologist-traveller has to reorientate himself to find out thetruth is to some extent similar to the logic of thescientific method. Notions of truth and falsehood lieat the heart of the type of knowledge we call "sci-ence." More precisely, what partly distinguishes

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those propositions that we call scientific is their verynature of being able to be true or false.

Examples of propositions might be:

3 is a number. (True)Snow is white. (True)Birmingham is in Spain. (False)

A moral injunction, a question, or an order are notpropositions in the sense understood here, Thelogic concerned with propositions that are capableof only two values, true or false, is known as two-valued logic. There are now also other types of logic,known as multivalued, to do with propositions thatare capable of other values. However, such areas ofthought are only relatively recent developments.

The logic with which we have been concerned inthe "sociologist" game is basically very traditional(Aristotelian and medieval), in the algebraic andformal mould in which it has been shaped for thelast two centuries (and known therefore as "math-ematical logic").

Mathematical logic has so far proved to be themost powerful instrumentforformulating a rationalsystematization of the various scientific disciplines.Within the realm of ideas it has furnished conceptsand methods appropriate to the basics of mathema-tics and has shed much light on the rational struc-ture of that discipline. At the same time it has alsosupplied the other sciences with models for theirrespective ordering and analysis of the raw materialof experience. This has been particularly so in phy-sics, chemistry, biology, medicine, psychology, andlaw. Finally, it is worth reminding ourselves of therole of logic in computer electronics, cybernetics,and automation in general. One need only lookaround one to realize how greatly both "pure" andapplied logic affect our daily lives, our work, ourvery culture!

True/false: an old dichotomy

We can in some ways say that civilization beganwhen mankind started to transform his environ-ment according to his own requirements, throughagriculture and the domestication of livestock. Solong as life was just a matter of hunting and gather-ing whatever was at hand, nature was a self-enclosed, passive reality. Now, however, mankind'sattitude towards it changed; it became a dynamicreality, in which he could intervene and makechanges according to his own desires.

This change implies a certain qualitative shift ofthinking: it was now understood that some of theworkings of nature could be comprehended. More

generally speaking, the transition from a nomadicexistence to an agricultural way of life also saw atransition of thought, a first elementary grasp ofreality: out of the variety and infinite multiplicity ofnature, man had begun to select and distinguishcertain constants. It is interesting to see how theearliest "thought categories" went in pairs of oppo-sites-hot/cold, male/female, death/life, bad/good,sacred/profane, and so on. Anthropologists notethat a similar way of thinking in opposites is alsotypical of primitive cultures even today.

It is therefore perhaps no surprisetofind the samemanner of thinking in terms of opposites in earlyWestern philosophy, among the first Greek think-ers: Heraclitus of Ephesus (c. 550-480 B.c.) sawobjective reality as a complex of opposing elementssuch as-day/night, winter/summer, war/peace.The opposition true/false likewise belongs to thisancient way of thinking, although over the course ofhistory it has acquired rich depths of meaning,finding ever more precise definition, down to thelogical systems of thought of our own day. Such"opposition" forms the basis of the logic we appliedin our games and stories in this chapter, which forthis reason is known as two-valued.

Games of logic

Logic has undergone a long process of evolution,acquiring its own symbols and conventions, so thatit has now become a rigorous and precise discipline.It has its own complex jargon and its own rulesaccording to which it functions. Hence it has beenable to alter certain aspects of normal language thatit sees as obstacles. Normal spoken language isnecessarily vague, nuanced, and has more than oneway of referring to the same object or concept,because of the infinite variety of situations, indi-vidual states of mind and spirit, and objects ofperception with which it has to deal. Yet this greatquality in everyday language proves a hindrance inlogic, which has to define everything precisely andunambiguously. Just as in mathematical calcula-tions symbols and formulae are used, similarly inlogic, "calculations" can be done in a more abstractway, without reference to the meaning behind thesymbols. Hence the name "mathematical" or "for-mal" logic. The "formal" character of logical cal-culation lies in the fact that the same lines of reason-ing can be followed to describe certain properties ofnumbers, a problem in geometry, in physics (dyna-mics, optics, and so on), or any other science. Abook of recreational games is hardly the place togive a detailed picture of this symbol languageitself. The questions posed in the next few para-

83


graphs are problems of logic, which require nospecial style of calculation. However, it will be dif-ficult to understand their logical structure, and sosolve them, without a minimum of systematicanalysis, without ordering the component state-ments under categories of F (false) and T (true).

Let us begin with a simple problem, in which thereasoning out of the solution can be diagrammati-cally illustrated (see figure below).

What colour are their clothes?

Rose, Violet, and their friend, Bianca, which means"white" in Italian, were very close and often used togo out together. On the last Saturday before theholidays, they arranged to have an evening out.

"What a coincidence!" said Rose as they met up,all in high spirits. "Our names are Rose, Violet andBianca, and look at us! We're wearing just the right-coloured clothes-violet, white, and rose pink."

"So we are!" Bianca replied at once. "But look,none of us is wearing the colour that goes with ourname."

"That's true," added Violet, who up till then hadremained silent, listening to the others.

Now, Violet was not wearing white. Given thisfact, can you say who was wearing which colour?

Simple reasoning will give us the answer. Know-ing that Violet is wearing neither violet nor white,she must be wearing pink; Bianca is not in pink, nor

8

w

p

V

R

T

F

F

V

F

T

F

can she be in white, so she must be wearing violet;and Rose therefore can only be wearing white, sinceBianca is in violet and Violet in pink.

Let us now try to express this in a chart, using thesymbol B for Bianca and wforwhite, VforViolet andv for the colour violet, and R for Rose and p for thecolour pink.

First, draw a chart with three squares on eachside, both vertical and horizontal (see figure below).

Knowing that Violet is wearing neither white norviolet, we can mark F (false) in the square corres-ponding to that option (the intersection of the hori-zontal columns w and v with the vertical column V.

Since Violet must be dressed in something, wecan safely mark T (true) in the pV box. On the basis ofthis information, we can fill in boxes pB and pR, andthere is then little problem about finishing: Biancabeing in neither white nor pink, she must be inviolet; thus Rose can only be in white.

A mixed bunch

Six people find themselves travelling together on acrowded train. Three are distinguished-lookinggentlemen (Mr. Valence, Mr. George, and Mr.Brown), while the other three are ill-dressed, lout-ish-looking characters with the occupations of thief,mugger, and layabout. By coincidence the threelouts share the same names as the three gents. Tomake our discussion clear, we shall not refer to anyof the louts as "Mr.," but rather by their occupationsor surnames only.

What is the name of each individual lout?We can work it out, given the following informa-

tion:i) Mr. Brown lives in Bristol;ii) the mugger lives in a largetown exactly half way

between London and Bristol;iii) Mr. George has five children;iv) of the gents, the one who lives in a town closest

to the mugger has three times as many childrenas he;

v) the gent with the same name as the muggerlives in London;

vi) Valence can beat the thief at billiards.To facilitate our investigation, let us construct a

chart with the names of the three gents (V = Va-lence, G = George, and B = Brown) along the topand the professions of the louts (t = thief, m =mugger, and I = layabout) down the side. (Seefigure at the top of the page opposite.)

As in the previous game, an F (false) in theappropriate name-profession box will mean thatour reasoning concludes that option to be impossi-ble, and T (true) the reverse.

84

-

= -


Of the gents, one lives in London (v) and anotherin Bristol (i), at the same distance from the mugger.Thus it is the third gent who lives closest to themugger. This obviously cannot be Mr. Brown, wholives in Bristol, nor Mr. George, who has five chil-dren-a number indivisible by three (iv). Thuswe are left with Mr. Valence: Mr. Valence livesclosest to the mugger. Mr. George, then, lives inLondon and therefore shares his surname with themugger.

Now let us start filling in the chart: we can put a Tin the Gm box, and this will then allow us to fill infour other boxes: two Fs-in the m (mugger) rowcorresponding to the names Valence and Brown;and two Fs in the thief and layabout options in theGeorge column.

Let us now continue our process of reasoning. Weknow that Valence can beat the thief at billiards (vi);thus the thief (who, we know already, cannot becalled George) is not called Valence or he would beplaying himself at billiards. He must therefore becalled Brown, which would leave us with the nameValence for the layabout.

We now only have to fill out our chart with two Ts(one in the Bt box, the other in the VI box) and thedue number of Fs.

Who is the guard?

Richard, Bob, and Nelson are the guard, driver, andsteward on a train, though not necessarily in thatorder. On the train are three passengers with thesame names (and again, forthe purposes of disting-uishing them, from the employees let us call them"Mr."). Exactly one passenger and one employeeeach live in the three towns of Bristol, Tonbridgeand London. Who is the guard?i) Mr. Nelson lives in Tonbridge;ii) the driver lives in London;iii) Mr. Bob long ago forgot all the algebra he ever

learned at school;iv) the passenger with the same name as the driver

lives in Bristol;v) the driver and one of the passengers (a well-

known physicist) worship at the same church;vi) Richard can beat the steward at Ping-Pong.

Here, too, it is vital to make out some sort of visualaid, to see at each stage how much information wehave been able to deduce.

Having thoroughly digested the data, let us seewhat we can make of the problem. Here in fact weshall need two charts: one (see left-hand figure p.86) with the towns of residence of the three railwaypassengers along the top and their names down theside; and the other with the job names along the top

t

m

I

G B V

F T F

T F F

F F T

and the employee names down the side (see right-hand figure p. 86).

Let us start with the simplest deductions: clue vitells us that Richard (not Mr. Richard) is not thesteward, so we can write F in the box in the right-hand chart corresponding to Richard/steward. Clue itells us that Mr. Nelson lives in Tonbridge, so weshould put a Tin the relevant square in the left-handchart. Having done this, we can then write F in theother squares on the same row and column toindicate that Mr. Nelson cannot live in London orBristol, since he lives in Tonbridge, and that sincethat is where he lives, Mr. Bob and Mr. Richard mustlive elsewhere.

Proceeding further, let us now take a look at cluesii and v together: from these, we gather that one ofthe passengers, a well-known physicist, lives inLondon-since he worships at the same church asthe driver, who we are told lives in London.

But what is his name?From clue iii we learn that it cannot be Mr. Bob,

who has forgotten all the algebra he ever learned atschool; nor can it be Mr. Nelson, who we alreadyknow lives in Tonbridge. It must therefore be Mr.Richard, so we can mark T in the Mr. Richard/Lon-don box in the left-hand chart, and F in the othersquares belonging to that row and column.

Examination of this chart will now tell us a newfact: since it is the only square still empty, we can besure that Mr. Bob lives in Bristol. Thus the chart canbe completed with a T in this box.

The final deductions are not difficult. Clue iv tells

85

,--


Mr. Richard

Mr. Bob

Mr. Nelson

Tonbridge London

F T

F F

4 -

T F

Bristol

F

T

F

us that the driver's name is Bob (Mr. Bob, we know,lives in Bristol). Thus a T can be placed in the squarecorresponding to the combination Bob/driver, and Fin the other squares of the same row and column, asbefore. The steward must be Nelson; hence theguard must be called Richard.

Only one sort of logic?

The word "logic" is used in different ways. Here ithas always been used in its most common meaning-as a science, a coherent and rigorous discipline.Even someone with the vaguest notion of what logicis probably thinks of it in terms of reasoning andassociates it with coherent thinking; in this sense,"logic" is more or less interchangeable with"reasonable" or "rational."

Logic is concerned with the rules of correctreasoning: when these rules are those of inference,onetalks of "deductive" logic. The mental teasers inthe last few paragraphs were based on an intuitiveuse of such logic-a method of reasoning towards aconclusion by way of premises: the premises in ourcase being the stories themselves and the clues, andthe conclusion simply the solution finally arrived atas a result of a process of inevitable, necessarystep-by-step thought. Thus it may be said that theconclusion was a logical consequence of those pre-mises.

Let us follow the examples of reasoning below:

Premises /) The only type of beans contained in thisbag are black Mexican beans;/I) The beans in my hand are from thatbag.

Conclusion l//) Thus these beans are black Mexicanbeans.

guard

Richard T

driver

F

Bob F T

Nelson F F

i- i

steward

F

F

T

It is easy to see the necessary link between pre-mises i and ii and the conclusion, iii.

There are cases, however, when the conclusion isnot wholly correct, even though the premises arequite correct. There can be degrees of correctness:of probability.

Take the following example:

Premises /) Stephen is a young eighteen-year-oldathlete;fl) Stephen has pulled a muscle.

Conclusion /Il) Thus Stephen will not run the 200 intwenty seconds tomorrow.

Between premises and conclusion here there isnot the necessity that was apparent in the previousexample: it is probable, but not 100 percent certain,that Stephen will not run the 200 in twenty secondstomorrow.

A tough case for Inspector Bill

A spectacular theft from the vaults of a major bankhad occurred: almost every security box had beencleaned out, and the villains had skedaddled with atidy fortune. When Inspector Bill began his inves-tigations, the situation was not easy. The thieveshad obviously been skilled professionals, and theonlytracesthey had leftwerethosethatcould not beremoved. These tracks led to a street running along-side the bank: truck marks were still visible on thetarmac. Obviously, then, the thieves had used agetaway truck. From this slender clue, a long andpatient investigation eventually led to the arrest ofthree young men: "Fingers" Freddy, the lock picker;"Hairy" Barry (so-called because of his distinctivemustache); and "Handy" Tony, a quiet lad, thesmoothest handbag snatcher in town.

86

. . )


The inspector knew that one or all of them werethe guilty ones. He had only to sift through hisfindings and order them correctly to be able toidentify at least onevillain. Eventuallyhewas abletoset out the following facts:i Nobody was involved in the theft other than

Freddy, Barry, and Tony.ii) Tony never operates without Freddy (some-

times also with Barry, but never alone).iii) Barry cannot drive a truck.

Which of the three could be definitely accused?Inspector Bill proceeded along these lines: Let ussuppose that Barry is innocent; in that case, Freddyor/and Tony must be guilty; on the other hand, ifBarry is guilty, then he must have had either Freddyor Freddy and Tony together as accomplices; henceagain, we are pointed toward Freddy or Freddy andTony together. Thus, whether Barry is innocent orguilty, Freddy at least must be guilty.

Let us now concentrate on Tony and Freddy: IfTony is innocent, then Freddy must be guilty (given

a

If0

goeswith

0

that clue i tells us that only Freddy, Barry, and Tonyare involved); and if Tony is guilty, Freddy must stillnecessarily be so, too, given clue ii.

The inspector thus charges Freddy, since who-ever else might have been involved, he must cer-tainly have been guilty.

Here is an example of deductive reasoning inwhich the conclusion is a logical consequence ofthe premises).

A problem of logical deduction

We are so accustomed to reasoning in words thatwe forget there may be other ways in which thoughtcan proceed. We can be rational without using ver-bal language. As an illustration of this, here is a littlegame that can be considered a visual test in logicaldeduction. We first look carefully at a series offigures (a, b, c) that develop according to a "logic"that is purely visual:

b

0

00

then

c

** S* 0

The same sequence is continued logically in one of the figures in the series below, which togetherconstitute all the possible developments.

d e 9

0 0 00 00 0 0

0 0 00 0 000 0

J4

87

goeswith

0 0

lf


The problem is to identify which of figures d, e, f,and g follows best in the same sequence. It should Q anot be hard to see that it must be f. But let us explorethe deductions that lead us to that conclusion. First, ait is clear that the circles should not be all red, sincein the series a, b, c they alternate red-white-red. f [ If A

Thus dand e are out. Aswell asthe alternation of redand white, we then see that figure b has two circlesmore than a, and c two more than b: thus g, which bhas three circles more than c, must also be out. Onlyfremains.Anotherelementisthenalsotobenoted: goes goes bthe circles within the figures are increased symmet- with withrically, two by two; thus the two circles added to b,which go to make c, find a symmetrical increaseonly in figure fi

What sort of reasoning is this? then cBy selecting analytically certain common features then

in the first sequence of figures (a, b, c), we were thenable to seek the same features among the proposed the Ccontinuations of the sequence. We could represent A thenour reasoning process thus:

Premises i) If a, b, c have alternate red and whitecircles, and

ii) if a, b, c are constructed so that at eachstage two circles are added in sym- goesm etry; wt

Conclusion iii) then falone can follow on from a, b, c! d

Note how in the mental process that takes us from goes da, b, c (premises i and ii) to figure f (the conclusion) goeswe have made an arithmetical calculation, however with ,basic. And mathematics is only one of the lan-guages in which we express ourselves. However, -the fact that any calculation, or information, formu-lated in any kind of "language," can be representedverbally is taken by psychologists to show the grea-ter wholeness, versatility, and adaptability of verbal e elanguage. It is not possible to explore the verydepths of the human mind; but we may be certainthat it works in an integrated and integrating way, as i)though distinctions of language-which the pro-cess of analysis constrains us to use-did not exist.

Reasoning with figures

Here is a series of tests for anyone who wishes toexercise their faculty for deductive logic. Similar tothe previous game, they are taken from intelligence 7tests and arranged in increasing order of difficulty. AHowever, the reader should approach them as agame, for the sheer enjoyment of it. Thus there areno final scores-the results are solely for the read-er's own benefit, to interpret as he or she wills. (Theanswers are given on p. 92.)

88


a

If

b

goeswith Ij

C

d! goes

with C

e

[.

Li

ff

0S

a0

If

(0 a a

I If If

b b

goeswith

then

goeswith

C

withL-

goewiti

C C

then

d

then

d

* goeswith I 0

d

e e

89

nnne9�

f


Correct but not true, true but incorrect!Let us see whether the following lines of argumentare true or false:

a) Premises ) A// cats can fly.II) Fufi is a cat.

Conclusion ///) So Fufi can fly.

b) Premises 1) Summer is the hottest season.//) ButAt/antis never existed.

Conclusion /I1) So the Nile is the longest river in theworld.

We can start by distinguishing the two modes ofreasoning: in the first, a), the conclusion is not true,because the premises are not true, even though the

reasoning is impeccable. Meanwhile the second, b),has a true conclusion, but nobody could claim thatthe reasoning is correct. Why not?

Common sense should tell us that the validity orcorrectness of an argument and the truth (or false-hood) of a conclusion are distinct elements, since itis possible to reach false conclusions from correctreasoning and true conclusions from bad reason-ing. Thus it is necessary to distinguish truth from"correctness": one does not necessarily imply theother. Correctness is a requisite of the "form" of anyline of thought, regardless of the content (true orfalse) of the premises and the conclusion. Truth iswhat we are concerned with in the conclusion.Nevertheless truth and correctness are not totallydisassociated: forthe "formal structure" of an argu-ment is correct or valid so long as true conclusionsemerge from true premises.

Answers to the tests on pp. 88-90

1) e: if the line with the perpendicular section attached on theleft (a) goes with the line with a perpendicular sectionattached on the right (b), then the larger circle with thesmaller circle on its left (c) must similarly become (e)-inwhich the smaller circle has moved to the right of the largerone.

2) d: if the square containing a smaller square in its topright-hand corner (a) corresponds to a circle also containinga smaller version of itself in the top right (b), then a squarewith the bottom line intersected perpendicularly by the sidesof a smaller square ic) must correspond to d-a circleintersected at the bottom on the left by a smaller circle,(Note how, in translating into verbal language the mentalsteps we have already taken regarding these geometricalfigures, it all seems more complicated: what the eyeimmediately absorbs and transfers to other figures, nowbecome ponderous descriptions of exact characteristics.)

3) f: there is symmetry between a and b, which is continuedonly in f.

4) e: as b is the element common to all the geometrical figuresmaking up a (which is formed of two circles and a square),so e is to c. This test is slightly harder because it is notimmediately obvious which features of a recur in b; thussome reflection is needed to discern the same correlationbetween c and e.

5) f: the red ball is in the same position in both a and b-as it is,too, in c and f.

6) d: while the red section moves one-quarter anticlockwise,the red ball moves 180 degrees.

7) e: the pink arm moves 90 degrees anticlockwise, thered-and-white arm moves clockwise 90 degrees.

8) f: the circles in a and b rotate clockwise, and the little outer

squares anticlockwise; thus the correspondent to c must bef, in which the inner triangles have moved clockwise and thesmall outer circles anticlockwise. Another correct answerwould be e. For b can also be seen as symmetrical with a, soit would be fair to choose e as being the mirror image of c.

9) e: if we trace a vertical line through the middle of the twohexagons, we find that a and b are symmetrical; similarly, eis symmetrical with c.

101 d: the red sections in b are where the pink were in a; the pinkare in the places that were white, and white has become red!

Discussion of answers to the tests on p. 91

1) The only true conclusion is IV. The others contradict thepremises. To reach the logically correct conclusion, we haveapplied a form of reasoning similar to that employed inmathematics: if A is greater than B, and B greater than C, thenA is greater than C.

2) The suggested conclusion (Ill) is not necessarily correct, sincepremise Ii is irrelevantto it.

3) IV is the correct conclusion: since both Ill and V introduce anelement of closeness/distancethat is not to be found in thepremises.

4) Even intuitively we can sense that the correct answer must beIll: this kind of argument, which follows a typical form of "AllA is B; all B is C; therefore all A is C" is known as a syllogism.

5) The logically correct conclusion is IV, since if all children havebeen vaccinated against polio and only some against Germanmeasles, it follows that some must have been vaccinatedagainst both diseases.

6) The conclusion is not true: an opossum cannot change into aCapuchin monkey. It is a faulty syllogism.

92

-

Areas without bounds"I should have more faith, " he said; '7 ought to know by this time that,when a fact appears opposed to a long train of deductions i invariably

proves to he capable of bearng soame other rlrerpretation. "Sir Arthur Conan Doyle

Against the mechanical

It is easy to split the mind's capabi cities into differentareas, reflecting the activities by which area of abil-ity is measured. This can be of some practicalbenefit: for example, someone who is good atmotor-visual tests (for example, good at recogniz-ing shapes and colours, and manipulating them) islikely to be good at jobs in which such skills arerequired. Similarly, a high score in word tests maysuggest aptitude for work in which mastery oflanguage and ability of verbal expression are impor-tant. However, these distinctions are somewhat arti-ficial and limited in value. For any prognosis of thiskind to be fully reliable, it is necessary to take thewhole personality into consideration. The humanmind is not, after all, divided into watertight com-partments. Rather,the categoriesthrough which the"functioning" of the mind can be analyzed are cre-ated for the convenience of the observer (thepsychologist, for example). Our daily lives are aconstant interaction between present stimuli, pastexperience, states of anxiety or relaxation, hope,expectation, disappointment, desire for affirmation,submissiveness, and so on and so forth. And it isthis continuous, complex interaction (of which weare largely unaware, albeit to greater or lesser de-gree sensing it through feelings-likes, dislikes, andso on-that we cannot fully explain) that deter-

mines human behaviour. Hence it is that predictingsuccess or failure in studies, for instance, or in a job,is so often extremely difficult. And this applies toboth individuals and the groups to which they be-long, even when narrowly specific activities areinvolved. We do not of course wish to claim thatpredictions based on tests are unreliable; butwe dowish to warn against a mechanical kind of inter-pretation of test results. As we have just said, it isnecessary carefully to consider personality-interests, motivations, and the like-in order tounderstand (and predict) behaviour.

Such knowledge is only to be gained throughstudy of areas of experience that are notoriously"fluid," such as success, love, friendship, politics,art, and ethical and religious feelings.

Mind journeys

The games, exercises, and tests that we have pro-vided, relating to those aspects of behaviour thatseem most closely related to intelligence (verbal,numerical, and visual skills), all share one commonelement. All concentrate the mind to find a singlesolution, out of a whole range of possibilities. Iden-tifying the number with which a certain series con-tinues, or choosing one drawing out of several thatshares features in common with certain others,

93

Areas without bounds

illustratesthis point. When one thinks about it, it canbe seen that the mind works upon several data, then"converges" on a single given fact, thereby "con-cluding" the exercise. Our use of the word "con-verge" was deliberate, because psychologists talkof "convergent thought" to denote one of the waysin which the mind works. An opposite way of think-ing is called "divergent thought," whereby (as theterm suggests) the mind works toward a certain"openness": from a small number of data, it seeksto find the greatest possible number of solutions,from the most varied and original to the oddest.

This broad distinction was first made by theAmerican psychologist J. P. Guilford, who statesthat divergent thought is characteristic of creativityand the search for new and effective answers to theproblems thrown up by life.

tion, continuing the genetic line of his parents andphysically conditioned by work, diet, and so forth,and on the other hand the heir of a cultural heritage(the ideas, traditions, corpus of knowledge, andpreconceptions of the society of which he is part). Itis thus unrealistic to think of creativity as a facultyacting freely and untrammelled in the mind of theindividual. Being creative involves ordering theworld round about, and one's own personal experi-ence, in a new and original way.

It is not easy to be creative. We often have tostretch our mental boundaries, direct our imagina-tions, and understand how to change already exis-tent realities into new forms. And this implies theability to step outside ourselves, away from ournormal modes of thinking, to see and judgeourselves from a different perspective.

e== a- -. Ad =---a

= m * =.

Imagination and creativity

Creativity is commonly associated with imagina-tion. To "use one's imagination" means steppingout of the normal scheme of things and conceivingsomething new: work out the plot of a story, paint apicture, or even prepare a meal with friends, devisean interesting holiday, and, on a broader scale, buildone's future and happiness for both ourselves andthose with whom we live-all these activities de-mand the use of the imagination.

Usually, to "be creative" means to be inventive.Inventiveness is a faculty that takes many forms andsuggests intuition, the involvement of an indi-vidual's inner life, motivations, and interests. Assuch it can be discerned in the most varied spheresof human activity, not only in art or at work, but evenin disciplines that might appear to leave little roomfor imagination, such as geometry and mathematicsin general. There is a common tendency to reducecreativity to mere "fancy," but to be truly creative,entails more than living in an imaginary world: itmeans freeing the imagination from past experi-ences in which it has all too often become enclosedand seeing new solutions to old problems.

An individual is a complex biologicoculturalentity, on the one hand the product of natural evolu-

Two different ways of thinking: twodifferent ways of playing

Matches are useful objects that, over and beyondthe purpose for which they were invented, alsomake good materials for many games. This dualfunction can be taken to illustrate the differencebetween reasoning and imagination-broadly, thatis, the difference between "convergent thought"and "divergent thought."

With twenty-five matches one can make eightadjacent squares, as in the figure above. A typicalproblem of, broadly speaking, convergent thoughtmight be: How would it be possible to make eightsquares using three fewer matches? One possibleanswer is shown on the opposite page.

Working above all with the imagination, divergentthought transforms cold geometrical shapes intoreal, colourful objects, giving them movement andlife. The eight squares in the figure above can thusbecome the basis for a story such as that of therabbit breeder.

The rabbit breeder. Rabbits having proved a worth-while investment, an enterprising breeder makeseight cages (see figure above), each big enough tohold two rabbits-a male and a female.

94

III - - I - III- I III- 11 - I-

IJ I J I


1

11 1 1'1* =

Io ifU ... @

The eight cages were ready the day before marketday, but during the night three sides of one werestolen. Still wishing to have eight pairs of rabbits,the breeder did not know what to do.

Was there any way he could re-create eight cagesout of the remaining seven?

The solution is illustrated in the figure above.

Making up a storyThe way we were educated, the values taught us athome and in school, and the experience of everydayliving form a sort of "package" that we never lose.Certain ways of seeing things, of classifying andevaluating them, and our reactions to given facts,objects, and types of behaviour all owe much to pastexperience. As we have seen, an individual'screativity lends new significance to these presetattitudes. All too often, imagination and sensitivitylazily conform to such attitudes, at home in a worldfrom which habit has rubbed off the "edges" ofnewness and peculiarity. In relationships withothers, creativity means seeing the whole scope of aperson's character, recognizing the variety andpotential of those we meet, and always seeing themafresh.

Imagination and inventiveness can also make ussee new uses for familiar things-even when verydifferent from their intended function.

Geometrical figures such as squares and tri-angles, points and lines, generally evoke memoriesof school, mathematical formulae, and logical argu-ment. Yet imagination can invest them with a newvitality, bringing them more immediately to life.

The story of the exclamation mark

One day Peter picked up his geometry book andbegan to read the first page, which the teacher hadset for homework: "The point and the line are thebasic concepts of geometry . . .!" Often when hewas doing his homework, he would be seized with

L . -

boredom and would go outdoors to escape theeffort of schoolwork.

On this occasion, too, Peter at once started dream-ing over his book. Then, as if awakened from a longsleep, the point and the line began to speak:

"I represent unity and perfection, and am thebasis of geometry," the dot began.

"I reflect continuity, I convey a sense of the flow oftime; without me geometrical figures would haveno shape," the line replied defensively.

"But I give a sense of the single, fleeting moment;without me, time could not be measured," the dotwent on.

"I convey the idea of infinity, of that which con-tinues without ever stopping," retorted the line.

"But I give the idea of what is complete, the finite, Iput a limit to things," the dot replied again.

"But without me no lengths could be calculated,"countered the line.

"Without me, it would be impossible to count, anddo calculations," insisted the dot.

This altercation was becoming rather long drawnout when Peter, admonished by his mother, beganto reapply his mind to geometry. But it was soboring! Taking a pen, he started doodling on thebook-and all of a sudden out came a dot and aline-an exclamation mark!

95

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A square, a circle, and . .. a child!

Once upon a time, a square and a circle were seizedwith a terrible fit of boredom on their page in ageometry textbook.

"I am a perfect figure," the square began vainly."Acircle is really no more than a square blown up sothe angles disappear."

"On the contrary!" replied the circle defiantly.It was not the first quarrel between the two. And

once they began, they went at each other hammerand tongs.

"I symbolize stability, solidity, and perfect sym-metry," continued the square.

"I'm not impressed by your boasting," the circleretorted again. "If you want perfection, I am themost perfect curve, others are just poor imitations."

"My shape has lent stability and strength to themost enduring monuments," the square went on.

"But basically you're also the symbol of the static,the immobile, the ponderous, whereas I inspired thewheel, and we know what that invention has donefor history. I symbolize movement and dynamism!"

The square had to admit that the circle was rightand began to feel depressed, when curious noiseswere heard. It was a boy forced to study geometry,leafing through the book. As young students areoften wont to do, the boy started to liven upthe dullfigures in his book. The square became an earnest,strong-looking man with a flowing mustache andtop hat; the circle was transformed into the clear,smiling face of a pretty girl with beautiful, flowinghair and long eyelashes.

It is not hard to guess what happened once theboy abandoned his book: the square took a fancy tohis attractive neighbour, and the circle in turn wasnot averse to the attentions of the m ustach ioed facewith its eminently masculine conformation.

They remained happily in love foryears, and even(it is rumoured) have had a child!

The long history of matches

What history can an everyday utility object like amatch possibly have? We do not always realize thatour acts-even those that seem most mechanical-are the outcome of a lengthy process of evolution.Daily habit has stripped them of all sense of historicor cultural significance. This is true of the simple actof lighting a match. Yet the origins of that little splintof wood, tipped with a compound that will flare intoflame with the right sort of friction, goes way back.Its history is part of mankind's long endeavour toharness fire to his own uses. When documentariesshow us primitive societies in which, still in thetwentieth century, fires are litfrom sparks caused byrubbing sticks together, we are reminded that ourown ancestors once lived similarly, and that it is ourexperience of the effort and anxious strain (the fearof not succeeding in what one is attempting to do) ofsuch a life-style that has stimulated our develop-ment. That is not to say that the invention of thematch was the decisive answer to the problem ofharnessing fire: nevertheless it did represent a con-siderable advance towards easier control of one ofthe most vital elements.

The discovery of fire opened up new possibilitiesof life and mastery of nature to primitive man, notjust by making more food available (for instance,cooked vegetables became tastier and more digest-ible), but also by encouraging characteristicallyhuman behaviour-such as the sense of belongingto a group (tribe or clan), enhanced by being able togather around the warmth of a fire.

For thousands of years, the problem was how tomake fire readily available. It would take too long totrace the whole history of mankind's conquest offire. Butwe should recall that, as well as rubbing twosticks together, there was another way of creatingfire: from the very earliest times, flames weresparked also by knocking two flints together. TheSouth American Incas had another highly ingeniousdevice: with large burning mirrors they caught therays of the sun and concentrated them onto someflammable material. For the ancient Greeks andRomans fire was considered to be sacred, a verysymbol of the city. For all this, however, it was notuntil centuries later that a simple, quick, and safeway of "harnessing" fire was invented.

Advances in chemistry during the IndustrialRevolution (late eighteenth, early nineteenth cen-turies) eventually led to the production of a simplesliver of wood, one end covered with a com-pound (of nontoxic phosphorus) that when rubbedagainst a rough surface burst into flame. Theflame would then ignite the wood and so continueto burn.

96


Above: component elements ofthe bow-type device used by theEskimos for igniting fire. BeJow:the invention of matches (lateeighteenth, early nineteenth

centuries) not only provided aneasy means of lighting a flame,but also supplied a source ofmaterials for model making(bridges, ships, and the like).

97

Matches: to spark the imagination!

Let us begin with a completely banal statement:some of the commonest objects lying around atman's disposal were stones. That is hardly the sortof observation that will set the Thames on fire!However, it contains an important cultural hint.Stones could be used for all kinds of things-heapsof things, literally-but they were also an aid to-wards abstract understanding, since they were ahelp in counting. The Latin calculus (and hence ourown words "calculation" and the like) in fact meant"little stone." And of course little stones are theancestors of the abacus, and of many games andpastimes developed therefrom.

The history of matches has much in common withthat of stones: to begin with, they too were intendedfor practical use and have since come to be usedalso as aids to the imagination, in games of intelli-gence. This demonstrates how mankind seeks toresolve practical, material problems, yet at the sametime feels an irrepressible urge to give functionalobjects some unfunctional significance, by playingwith them.

It was above all in the twenties that games withi matchsticks became widespread. Newspapers and

magazines printed brainteasers, puzzles, and littlefigures all constructed out of matchsticks.

How did the little match girl die?

Hans Christian Andersen (the Danish writer, born ofhumbleoriginsin Odensein 1805, died Copenhagen1875) told the tale of "The Little Match Girl," who, onNew Year's Eve, hungry and shivering with cold,goes to sell matches in an archway. A few peoplepass by, glancing at her unfeelingly. Nobody buysany matches. Time passes and it becomes colder.Unable to resist the temptation, the girl tries towarm herself by lighting her matches one after theother. Gradually she uses them all up, until even-tually. as the last one burns out, she too dies.

Of cold? Well ...The actual history of matches perhaps provides a

less moving, more prosaic explanation. To beginwith, the flammable compound of the matchheadconsisted of phosphorus, which gave off a lethal gaswhen alight. Only later were matches invented thatused red phosphorus, which while it might havecreated a somewhat unhealthy gas, was at leastnot deadly. (The inventor of this was the SwedeE. Pasch; further improvements also came fromSweden.) Thus we may conclude that Hans Christ-ian Andersen's little girl probably died from a lethaldose of phosphoric gas.


Four balls, five coins, six matches ...

Four golfballs can be arranged in a pyramid so thatthey are all touching each other (see figure aboveleft). In the same way, five coins can be so arrangedthat each touches all the others (figure abovecenter).

Is it possible to arrange six matches so that eachone touches the other five? (They must not bebroken or bent.) The traditional solution is as shownin the right-hand figure above.

Animals and matches

From earliest times, the colourful and endlesslyvaried animal kingdom has stimulated man's im-agination. This is hardly surprising: primitive man'svery survival depended on animals, which providedraw materials (sharp bone for knives and the like,skins for clothing, bristles, and so forth) as well asfood. Thus in the prehistoric era, wal Is of caves werecovered with animal paintings. In the same way, itwas perfectly natural that when people began toinvent games with matches, designing formalizedanimal figures with them was one of thefirstof suchgames.

The top figure on the right is an attempt at por-traying a stocky wild boar rooting for food. Imaginenow the baying of a pack of boar hounds in thedistance.... The boar stops and raises its snout.. ..By moving just four matches, the top figure can bealtered to represent the boar sniffing the air, sensingdanger.

Which matches must be moved?The answer can be seen from the bottom figure on

the right.

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98


Art and creativity A first exerciseA work of art is a highly complex experience, com-bining aesthetic taste, sensitivity, culture, vision ofthe world, and individual sense of being. It is oftenthe product of hard discipline, accompanied bycontinuous objective, self-critical appraisal.

Artists are often thought to be "a bit mad," ex-troverted, sometimes rather wild, or at any rate notsimply run of the mill. In fact, being an artist requiresone particular quality: the development of inherentoriginality. The artist sees in a certain way and hasthe ability to order words, shapes, colours, andother perfectly normal elements of experience so asto create a unique and significant synthesis.

Scientists also exercise the creative spirit in theprocess of invention, of recognition of new func-tions and relationships, of acute observation ofnatural phenomena that are the common experi-ence of all and sundry; it is present in the way newtheories are expounded, and even more so in thedevelopment of advanced research programs, theevaluation of data, and the recognition of resultsthat go beyond-or indeed sometimes actuallycontradict-normal experience and commonsense.

Here it must De stressed that modern thought onscience and the scientific method sees imaginationas a fundamental component of progress.

However, it is not always possible to detect adirect relation between the spirit and scholarlyachievement: it is not the case that the "intelligent"schoolboy will necessarily be the most creativeadult. There are famous examples of scientists whowere not "star pupils," yet turned out to be majorinnovators (the physicist Einstein, who revolutio-nized classical physics, is a supreme case). Schooltends to teach a ready-made culture, in which thereis little call for personal imaginative contribution.Mastery of academic disciplines presupposesabove all a good memory and the ability to repro-duce classic notions. Individual verve, originality,and personal insight are rarely encouraged. Someresearch suggests that the most creative pupils arealso the most independent, the ones who kickhardest against rigid restrictions of thought andconduct.

We should therefore ask ourselves: Is there somebetter way of fostering creativity?

Family life and school can both in different waysstimulate the creative spirit, above ail by being opento the free self-expression of those growing up inthem: growing up without fear of self-expression,their interest and curiosity in life will be enhanced,and they will be helped to develop faith in them-selves and their own abilities.

The creative spirit manifests itself in the ability tosynthesize old forms and structures anew.

Figures a, b, c. d, e, f, g, h, i, and j are eachcomposed of one or more of the numbered figuresabove. The exercise is to identify the componentnumbered figures in each case (write the relevantnumbers beside the lettered shapes). The shapesare not necessarily kept the same way around orthesame way up. (Solutions below.)

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pulse, character, reaction to stimuli, and individualsensitivity are all fundamental. Exercises claimingto develop the creative spirit have been devised. Onthe other hand, it is hard to evaluate a faculty asmany-sided as creativity. Such exercises consist offilling in half drawings (see figure on left) as fancydictates.

Evaluation of the results is purely personal.Perhaps the reader would like to make some com-parison between his or her own "results" with thoseoffered below.

We suggest that the left-hand "results" below areless original than those on the right.

Success

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Re-creative games!

Can the creative spirit be enhanced?It is not easyto answer this, as so manyfactors are

involved in this aspect of "intelligence." As hasalready been said, imagination, motivation, im-

We live in a society-or rather, a type of civilization-in which individual success is one of the keyvirtues. Encouragement to seek success is fed to usin various ways from our earliest childhood. Gamesvery often are contests, with winners and losers,that make "winning" a primary aim. The massmedia presentation of certain sports (football beingan obvious example) promotes certain values, ofwhich success is the greatest. Yet above all it is theeducational establishment that promotes this idealamong the younger generation, Classification (forinstance, in exams), scholarships, and selection pro-cedures all in a sense give a "foretaste" of the life ofadult society.

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Is there an "intelligent" form of the successfullife?

We have seen how individuals absorb the successethos through various channels. Now our society isbased, and survives, on the mechanics of profitmaking, which is an economic value, It is thus easyto identify success with economic success. The de-sire to be well off and aspirations to wealth arelegitimate and positive goals. Yet there are waysand ways. And intelligence expresses itself in abalanced affirmation of oneself and one's abilities-"balanced" in the sense that there should be noclash with other values that contribute to humanhappiness, values that consist fundamentally inone's relations with one's fellow beings. An indi-vidual's instinct of self-affirmation, of self-expression at work, at leisure, and in relationshipswith others, can only be of positive value, a worthymeans of self-fulfillment, so long as it is guided byreason and imbued with realization of true auton-omy. This ideal can thus lead to individual happi-ness, and happiness also for others.

The game of success

To be successful in life, certain qualities are re-quired. One of these in particular is very valuable:the following game will reveal whether you have itor not!

The figure above contains randomly positionedcircles such as one sees, say, on a shooting range.Each player takes a sharp pencil and looks carefullyatthe circles for as long as desired, paying particularattention to their position. Then, eyes closed, andstarting with the pencil raised above your head, tryto pinpoint the center of each circle with the pencil,keeping your arm out straight-and your eyes ofcourse still shut! You have five goes only, and onlybull's-eyes count.

How many bull's-eyes did you manage to scoreout of five?

If only one, that does not prove that you lack thismagical, unknown quality. If you scored more thantwo bull's-eyes out of five goes, this would suggestthat you probably had your eyes open. To hit thevery center of the target with a pencil at arm's lengthis so difficult that it is more a matter of chance thanskill if you succeed.

It is highly likely that anyone seriously claiming tohave scored more than two bull's-eyes was in factcheating.

The most honest answer is much more likely to beno score at all. And it is this last answer that willreveal the right attitude for success. In striving for agoal, for success in one's job or an individual task, orfor the highest score in any competition, there canbe no justification for dishonesty, either towardsoneself or towards others. Honesty will generatetrust. And if we are untrustworthy, our life will bebound to prove a failure.

101

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Work

"Less work for young people and women.. . ." "Somany made redundant... .""Unemployment: whatis the government doing about it?" Every day wesee the same headlines in the news. The economyseems to survive on the brink of crisis day by day,with unemployment, takeover bids, and redundan-cies affecting the lives of more and more people.

Such emphases in the news highlight, with trueobjectivity, at leastthe central importance of work inthe life of the individual. There is no shortage ofinformation on the disastrous material and moraleffects of lack of work or job insecurity. All themore-since it is usually those in the weakest posi-tions socially who are affected, thus increasing thepsychological pressure-on young people seekingtheir first job, and women. Many experts agree thatunemployment and drug abuse are closely linkedamong the young.

Work is clearly a vital necessity for any individual:it provides contact with others and a sense of useful-ness-and hence of true participation in a givensociety or nation. Work provides a clear way ofsupporting oneself and one's family and hence is amajor prerequisite for personal and familial happi-ness. It can both affirm an individual and be a meansof self-fulfillment, and it can at the same time behumiliating, degrading, and frustrating, to such adegree that it sours existence.

Our day-to-day activities largely condition ourdaily "routine": a job takes up most of the daytime,determines our leisure time, and in effect sets thetone of our life. Our attitudes, our creative ability,our aspirations, indeed our whole intelligence areall directly influenced by the work we do.

Contemporary psychology sees work not merelyas a means of survival, but also as a reason forliving-an important way for the individual's per-sonality to grow, through to some extent "findingitself" therein. It can afford a newsatisfaction in life.

Of course, when financial necessity is the deter-mining factor, then these other individual factors,relating to quality of life, come second. There is nolonger necessarily the satisfaction that is to be de-rived from pursuing a career for which one hastrained and in which one is interested. Those whofind themselves obliged to take any job that comesalong tend to end up with tedious repetitive work.Such jobs are hardly very self-fulfilling, and it is noteasy to pursue them with interest, to find indepen-dence, or to become involved at personal level.Work of this kind is described as being "alienating."And if alienation of this kind is more than just atemporary experience, it becomes a sort of prison,from which any form of escape is welcome.

Capacity for synthesis

Different activities can sometimes be related to eachother by means of common features. The abilityto design new combinations of familiar objects,instruments, elements of a whole, figures, and evenabstract concepts unites the engineer and thearchitect, as well as the technician, the mechanic,and the carpenter, who work at a more concretelevel.The abilityto recognize newforms and models,to relate parts of a whole to a different whole, toapply old solutions to new problems, and to collateinformation and reach coherent and substantial re-sults, all suppose a skill in discernment, categoriza-tion, assembly, and manipulation of data.

This process can be described as analysis andsynthesis-and is common to many human activi-ties for example to the poet in the act of writing, andto the painter in the act of creating a picture.

Analysis-synthesis and breakdown-recomposi-tion are thus two sides of the same process. Thetests that follow below can be seen as games: thecomplete shapes (blank) are the product of a certainordering (synthesis) of the (red) shapes below,which represent a breakdown Janalysis) of the com-pleted composition. With a pencil, sketch the sepa-rate (red) shapes within the blank one that go tomake it up. (Answers on p. 104.)

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Technical aptitude

In our present society we are constantly coming intocontact with the results of technological progress.Few of us live without a modern stove or refrigera-tor; cars, typewriters, televisions, and the like arenormal features of modern life. Such inventions insome ways actually dictate our very environment.Even those who unconsciously reject them, andthose who dislike "materialism," still find them-selves having to face everyday tasks like tighteninga screw or loosening a bolt. A certain technicalability-manual and practical-along with a mental"kit" of elementary knowledge are features of "nor-mality" at virtually every social level. Some people,however, are vastly more competent at handlingtools and mechanical instruments than others andcan understand complex technology (again, a differ-ent sort of imagination is at play here) and designengines and electronic circuits that baffle most ofus. Ingenuity and an ability to appreciate spatialpossibilities and the play and flow of different forcesare required. Not only does one need to be able tovisualize but also, in one's imagination, reconstructthe component parts and the movements of en-gines. For example: a kind of spatial logic, of shapesand forms, is also involved.

The games below are an exercise to test yourtechnical aptitude (that is, not just your skills withtools, but the ability to grasp the physical principlesunderlying all mechanics).

To start with, let us suppose that X, Y, and Z in thediagram above are cog wheels: X having twentyteeth and moving Y, which has 40 teeth and which,in turn, moves Z, which has one hundred teeth.(Assume all teeth are the same size.)

1. If X turns in the direction indicated by the arrow,Y will move:a) in the same direction as the arrowb) in the opposite directionc) partly in the same direction as the arrow,

partly anticlockwise

2. If X turns in the direction indicated by the arrow,Z will turn:a) in the same directionb) in the opposite direction to the arrowc) partly in the same direction, partly anticlock-

wise

3. If Z does one complete turn, X will do:a? 1T5 of a turnb) turnsc) 11/4 turns

120

4. If X does one complete turn, Z will do:a) 1/5 of a turnb) 5 turnsc) 1'/4turns

5. If X does one complete turn, how manyturns willY do?a) 2b) 1/2

c) 20

6. If a fourth cog is inserted between X and Y, thiswill make Zturn:a) fasterb) neither faster nor more slowlyc} according to the size of the fourth cog

How is your mechanical skill?

To hold sometypes of jobs, especially in largefirms,applicants are often required to take tests devised toascertain whether they are suited to the relevanttype of work. Once natural aptitude has been dis-covered, then of course there are other tests to findout one's level of acquired knowledge. Our exer-cises on the next page are simple tests of mechanic-al skill that nevertheless demand a certain degree ofconcentration and careful thinking.

Answers to the tests on this page

lb 2)a 3)b 41a 5Bb 6)b

Answers to the tests on p. 106

1)a 2lthesame 3)a 4)a 5)a 6)b 7)a 8 c

105

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a b c d e f g h i j k I m n o p q r s

(see figure d. Thus an impasse(dead point) is created. This isknown in Japanese as seki. Insuch cases the pieces stay on theboard, and the game continueson another section.

* Leaving 2 intersections freecan also be a trick-an "escaperoute." In figure e we see two

* Another common impasse isknown as Ako (illustrated Insequence, from top to bottom, infigure 0. If it is white's turn tomove, it would be possible toput a piece on h13 and take theblack piece on g 13. But then ifblack were to place a piece ong13 in turn, the situation wouldmerely oscillate to and fro ad

fig. e

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examples of "eyes," where bothgroups of black pieces are still inplay and will count in the finalscoring. Of course, were white tomove into those emptyintersections (ill, ki 1, orel, gi),it would be nothing short ofsuicide, since it would be totallysurrounded by black and wouldthus automatically be taken.

fjq.f

infinitum. There is thus a rulethat black is not permitted toplace a piece on g13 (thus takingwhite h13) but has to wait forwhite to make at least one moremove-prior to which he has tohave moved elsewhere on theboard.Other examples of play areshown in figures g and h,

108

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a b c d e f g h i k I m n o p q r sillustrating how a single move -6 pieces on d4, dlG, di16, p4,might be made. p10, and p16;The number of possible - 7 pieces on d4, dlD, dI6, p4,combinations in go is extremely plO, p16, and j10;high. It has been worked out as - 8 pieces on d4, di0, d16, p4,10 to the power of 750-1 plO, p16, j4, and jt6;followed by 750 naughts. Here - 9 pieces on d4, d10, d16, p4,we are only trying to outline the plo, p16, j4, j10, and jB6;main rules. Other rules will be - 13 pieces on d4, diO, d16, p4,readily apparent to the reader pl, p16, j4, j10, j16, g7, g13, m7,during the course of a game, as a and m1 3;logical consequence or corollary - 17 pieces on d4, d10, di6, p4,of what is explained here. plO, p16, j4, ji0, jl6, 97, g13, m7,

m13, c3, ci7, p3, and q17.I The outcome of agame of gowill depend largely on the skilland experience of the twoplayers. If equally matched, theycan fight on for a very long timeand even decide to end with atruce. However, if one or theother is even slightly moreskilled, the less skilled player canbe completely outmatched.Thus, throughout the longhistory of the game, handicapshave been invented;predetermined weak positionsfor black to start off in, indicatedon the board by the boshi pointspreviously mentioned.Black of necessity always has adisadvantage, since it makes thefirst move. But white'sadvantage can be increased byblack's established scale ofstarting handicaps, which are asfollows:

- 2 pieces on d4. and p16;-3 pieces on d4, pIG, and p4

,- 4 pieces on d4, p16, p4, anddl6;- 5 pieceson d4, p16, p4, dl6,and jiG;

These positions are fixed. Onlythe inbuilt handicap of placingthe first piece on the boardalways black) enjoys freedom ofchoice of position. Whateverthestarting position, the next moveis always white's.

* The game can end wheneverthe players wish it to, or when allthe pieces have been played,orland when there is no moreterritory to be taken. The winneris the player who has encircledthe largest area (the most pointsof intersection on the board). Incounting up the score, eachplayer takes the intersectionsenclosed by his or her pieces. Letus take a concrete example. Inthecolour plate on p. 110we seethe state of play at the end of agame of 241 moves, asdescribed by Roger J. Giraultlsee bibliography atthe end ofthis book).The sequence of moves waslisted as follows 1+1, +2, and soon, indicating the number ofpieces taken):

191817161S1 41 31211

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4

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a b c d e f g h i j k I m n o p qwhite black white

2 D174= R56= R108=D510- C512 = 0414= 0416= 1R418 = PS20 = C6 + 122 = R324 - 0726- C928 = C1330 C1832 = E1734= B1836 = G1738 = F1340 - B1242 = 81144 = L346 -K448=B550 - F452 =F354 = ES56 = G458 - G560 = G662 =H664 J766= K668- El 570 = B1472 = A13+ 174 = G1476 = El678=D1380 = D1482 = H784 = F786= R14

87 N989= PlO91 = S1793 = R1795 - 01797 = E1099 = K9

107 = D15103 = H17105 = J17707= C10109 = JiB177 = L15

113= L10115= F12117= M10119= K15121 = N17123 = N8125- D3127- = C2129 : B21731 - M14133 = 09135 011137= M16139= Mi1147 = Rt1143- R8f45= S8147= K8149 = 16151 = 02153- 57165 = 013157- M7159 = L14161 = 01263 = P1

165 = LB + 1167 = P2169 = 011771 - J16

7 - R163 0 Q35= P177- E39 = C4

71 = B413 = Q575 = P477= P3

79 = 0521 = P623= Q5 + 125= N427= Cl529 = C1731 = D1633- B1735 - E1437- E1239 =C12417 =D1243 = M1745= L447- L549 = G357 - E453 F255- F557 = H459 HS61 = H263= J665= K767 = J569- B1371 = C1473 = F1475 =F1577= G1579 - E1381 - J883 = H885 = F9

111

r s

88 - P1490 = S1692 -01694= P1696= Fl198- H10

100 = H14702=B13104 = H18106- J15108= B10110- N16112 -K10114= K11t16= Lii718= M15120 = L13122 = 08124 = D4126 = C3728 = 83130 = A4 + 2132 = Ni1134 = R9136 = N14138- NIS140 - M1 2142 = R2744 = P7146 = L7148 = N7150 = 01752 = R7154 = S6156= R15158 = G18160 = N13162 = J13764 = M6166 = Q1168= P12170 = K14172 = H15

-

black white

173-D9 174 = Cl 1175- G1i 776- G10777= H11 178 G12179 = -H12 180 = G13787 - F10 1a2 = J11183 = F8 184 = D8185 = Q2 186= Rl787= N12 188- M1389 = S12 190= Q897 = S10 792 = J19193- K19 194- H19195 = K18 196 = N67,7 = E8 798 = D7199 = S15 200 = S14201=TI6+l 202=PS-+ 1203= 06 204= N5205 = 03 + 1 206 =07207= 04 208= P13209 = E7 W = E6211 - A2 212 - 09213=010 214=D10+1216 = Q14 216= 015277 = 016 218- 015219 - 017 220 = T14227 = R13 222 = T15223 - S16 224 - T12225=Tl1 226=T13227- 012 228- T7229 = S9 230 - J12231=E11 + 1 232=M5233 = M4 234= F16235S T 236 -T6237 = PS 238 = 5239 = E9 240 = D11241= 013 END

At the 241 st move the playersdecided to stop, and as at theend of every game of go, theythen had to "recognize" whatterritory each had taken, toestablish who was the winner.This was agreed as follows:

- with one move at a turn, eachplayer completed his "fences"lhl3, iG0, g8, g7, and so on), sothat the situations of seki werealso automatically resolved;- then in the same way theyfilled up the neutral positionsthat neither could claim Ig16,h16i n14, o1l, a3, and so on);- then they removed all thepieces that had then been"taken" (j6, j4, k3, and so on).

After this process of"recognition," the board lookedas shown in figure i, and eachplayer under the careful scrutinyof the other counted the numberof points encircled by his pieces(that is, not the occupiedpositions, but-as we havealready said-those inside"enclosures" of the respectivecolour). The result was 68 pointsto black and 60 to white.However, as white had taken 6black pieces during the game,and a further 7 during the finalsorting out, black had to deduct13, to bring his score down to 55.White then had to subtract fromhis score of 60 the 5 pieces taken

fig.,

by black during the game andthe 7 taken during the"mopping-up" operations,bringing his total down to 48.Black was therefore the winner.Written out like this, the gameseems much more complicatedthan in fact it is. With familiarity,though, go becomes much

simpler and more exciting. It isactua lly easy to learn, but veryhard to play well. Luck has littlepart in the game. It is all acontest of logic and mathematicalimagination, and this is whatmakes it such an educationalexercise-helping thedevelopment and vigour of

one's ability to grasp problemsand see how they can be solved.According to Girault, "Only oneother activity can be comparedwith go: that is, writing." Only inwriting is there a similar thoughindeed even more endless)complexity of combination ofsymbols.

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Love

Intelligence as expressed in our relationships withothers inevitably also includes our feelings."Others" are those with whom we come into con-tact in our day-to-day dealings, but they are aboveall our friends, the "partner" with whom we shareour life, and our family.

We have consistently described intelligence as a"unified and unifying" or "integrated and integrat-ing" human faculty: hence it is not strange to thinkof it as also being at work in the sphere of feelings.

It is a widespread belief that love is one of thesupreme experiences of individual existence, theclimax of a whole process of physical and psycholo-gical maturation. It consists in self-giving, for self-enrichment; and as such, it might be thought to bethe final stage of a long process of acquiring per-sonal and social self-reliance.

Yet love truly means risking one's own freedom toreach full "selfhood."

The experience of love is not therefore a slightthing. Far from simply affecting one in any normal

sense, or being the "final stage" of any "process," itbecomes the basis for a totally new departure.When two people meet, fall in love, and decide tolive together, it is as if they wished to start life fromthe beginning again. And this of necessity creates adynamic of change, towards new levels of maturity,deep in the character of each person involved.

Here, intelligence takes the form of, above all,sensitivity, intuition, good sense, and balance.

A sad story

Geoff and Sally were a young couple, seeminglyhappy. They married a few years ago and now havea two-year-old son, John. Everything seemed well-he with a job that enabled them to live comfort-ably, she now having given up her job, looking afterhouse, husband, and baby. To all outward appear-ancesthey madea happyfamily,yetthings were notquite as they seemed. For some time Geoff had beenreturning from work in a bad mood, looking unwell,as if suffering from something. His relationship with

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Sally had changed, there was more tension; quar-rels began when Geoff started questioning Sally,almost interrogating her. Gradually the atmospherein the home became unbearable. One sleeplessnight Geoff, feeling he could no longer go on likethis, decided to take a drastic step: beside himself,he took a gun, shot his wife, and then killed himself.

We need only open a newspaper any day to read asimilar horrific story.

Jealousy

In the story just recounted, Geoff's worries com-pletely dominated him, to the point of deranginghim and leading to a murderous and suicidal state.Jealousy in its worst forms can indeed lead to mad-ness; reality is distorted, the person closest to onebecomes the object of wild, accusatory fantasies,and the outcome is tragedy. The person sufferingfrom jealousy has no doubts as to his sanity, there istotal certainty that the beloved is unfaithful, or is atleast prepared to be so. Suspicion, endless de-mands for proof and for oaths of faithfulness, harry-ing insinuations and inquiries, even obsessive su-pervision and, in some cases, trauma and tragedy,all can be unleashed.

One is reminded perhaps of the chastity belts thatmedieval crusaders used to lock their wives up inbefore they leftforthe Holy Land! Essentially it is thesame defensive mentality, growing from fear oflosing something or someone very dear.

A jealous person is emotionally insecure and un-stable, with a need for certainty; yet unconsciouslyby his behaviour he renders himself impossible tolove and provokesthatwhich he mostfears: separa-tion from (abandonment, betrayal by!) the beloved.A clear contradiction underlies such behaviour. It islike somebody terrified of an approaching or evenmerely potential fire, who drenches his clotheswith some other highly flammable substance.

In the same way as it is normal for most people toexperience love, no one is whol ly free of the dangerof jealousy, which arises quite naturally therefrom.Just as envy is an indirect admission of the goodbelonging to an-ther, jealousy too is a sign of pas-sionate and sincere love. It grows from awarenessthat all human good is fragile, and must be care-fully protected. Kept within reason, and admittedtactfully and reservedly, jealousy can even bepleasantly flattering to the one to whom it is con-fessed. It can be a sign of a real love, rather than amere abstract passion. Like other natural feelings,it can be quite healthy when thus handled: ameans of more complete self-expression and self-understanding.

Othello and Desdemona in ascene from Shakespeare'stragedy. The very feelings mostvulnerable to instincts of

jealousy can, if intelligentlyexpressed, be creative andfulfilling.

The following test consists of thirty questions,related to different hypothetical situations. At theend is a suggestion how one might evaluate one'sown tendency to jealousy.

Who do you think is more useful?- a road sweeper A- a private detective B

Of these two characters, which do you like best?- Othello B- Hamlet A

Your cat purrs at somebody you dislike. Do you:- like it the more for being sociable? A- vexed, chase it out of the room? B

You are eating alone, and a piece of fruit falls fromthe plate. Do you:- pick it up from the floor? B- sweep it away? A

A friend has a terrible headache. Do you:- try to comfort them? A- recommend an effective remedy? B

Which hurts most?- a slap on the face from the person you love? A- a look of admiration by the person you love to

someone of the opposite sex? B

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Which would you prefer to invent?- an invisible video camera B- a motor able to run on seawater A

Do you feel happier:- in a twosome? B- with nine or ten others? A

If someone had seriously wronged you, wouldyou prefer:- to seek revenge? B- to dismiss that person from your life? A

When you see two people of opposite sex alonetogether, do you assume:- they are having a chat? A- they are lovers? B

You have realized that the peach tree in your gardenhas a fruit-laden branch overhanging your neigh-bour's garden. Next year will you:- prune the overhanging branch? B- suggest to your neighbour that you both share the

jam? A

A new national anthem is adopted. Do you:- find it better suited to the modern age? A- rather regret it? B

A young man or woman whom you do not knowbecomes your boss. Do you feel:- that strings have been pulled? B- that a breath of fresh air was what was needed? A

To your mind, faithfulness is:- a weakness? B- a virtue? A

When you are introduced to someone of the samesex, does he or she appear:- as a friend? A-as a rival? B

Which would you prefer?- a photo of your partner alone, but blurred- a good photo of your partner in a group

BA

If you were to go to a desert island with your partner,would you rather:- find it peopled with helpful inhabitants? A- manage alone together? B

What does the word "doubt" immediately suggestto you:- a search? A- a trick? B

If you catch someone winking at another person ofthe opposite sex, do you instinctively think:- there is something between them? B- it is a sign of friendship? A

As a guest in an unfamiliar house, you lose yourbearings during the night. Do you:- still try to find your own way? B- call for help? A

You are invited to a card game. Do you:- jokingly assure everyone that you will win?- say you are no good?

AB

Just before an important rendezvous your face ac-quires an unsightly pimple. Do you:- Try to conceal it with hand gestures? B- confess openly that you ate too many strawberries

the previous evening? A

You have received an invitation to dine with thequeen. Do you:- phone the embassy to find out all about it? A- assume it is a mistake? B

When your partner dances with somebody else, doyou prefer them to:- dance a modern dance? B- dance a set-figured dance? A

A mosquito flies into the room where you are sitting.Do you:- get up and fetch an insect spray? A- try to kill it with your hands while sitting? B

You visit Napoleon's bedroom. Are your thoughts:- of the nights he spent there with women? B- of his nocturnal musings upon war strategy? A

Someone you cannot see calls you. Do you:- respond without any sense of worry?- feel a momentary fear?

AB

While sunbathing on a boat, your suntan lotion fallsinto the sea. Do you:- at once dive in for it? B- try to fish it out with a net? A

You are on your own, looking at the moon. Do you:- hope your distant beloved is also looking at it? B- wish you had a telescope to see the craters? A

Would you prefer your partner to make a businesstrip alone to:- the West Indies? B- the North Pole? A

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Now we come to the "answers" to this game,which will varyfrom individual to individual. Havingfinished all the questions, count the number of Banswers you chose, then read the paragraph belowthat is appropriate to your "score."

0-5: You are not at all jealous. On the contrary youare tolerant, disinterested and unselfish. You do notbelieve in eternal love or grand passion. Ideasshould, you feel, be flexible, never rigid, and you arenot even concerned for your own comfort. It is notclear on what you base your life, and you areperhaps too disillusioned and cynical.6-12: You are not overconcerned to preserve whatis yours, either materially or emotionally, and thisresults in a certain coldness and a touch of cynicism.You are notabove having a high opinion of yourself,and your own ideas are in your eyes the soundest.Everything else you consider too uncertain andvague even to be deemed a problem.73-18: You have many doubts and uncertainties.You think one thing one moment and another thingthe next. Sometimes you are a bit ingenuous, atothers just indifferent and a little cynical. Jealousycomes and goes, and can flare up without realjustification. You are not good at going beyond yourown range, having little confidence in yourself andothers. Do you try to protect your own sensitivitiesand do all you can not to expose yourself? By doingso, you risk concealing yourself from others and somaking it impossible for others to get to know you.Even a little jealousy can be a good thing at times, asit can make us more interesting.19-25: It would seem that your attachments areprimarily to the realm of feelings and ideas ratherthan to material possessions. You can hardly becalled altruistic, and you suffer greatly fromjealousies. At times this is well concealed, while atothers the slightest thing will suffice to set off theprocess of self-defense that renders you suspicious,greedy, or stubborn, according to the circum-stances. Try to be more reasonable: it could keepyou from many bouts of excess.26-30: You appear to have an extreme sense ofproperty, attaching yourself immoderately to thepeople and things in your life. This almost alwaysdistorts the way you see things and hence under-mines your ability to make wise choices. You areperhaps more taken up with keeping than with re-newing, and this can create a suffocating sense ofanxiety-of desire for possession-that merelycontinues to mar your relationships with others.Remember that the first person to be hurt byjealousy is the one who has such feelings.

Games and friendship

In specialist jargon, "intelligence" is often associ-ated with certain particular forms of behaviour relat-ing to the ability to solve problems. We know,however, that it is fundamentally a gift of "integrity"-in that human beings are "integrated" wholes. Itcould even be said that there is intelligence in any-thing that is human: thus also in the way we get onwith our fellows.

The search for happiness is common amonghuman beings; and part of leading a happy life isenjoyment of friendships. This, however, entails areadiness to be open to others-a disposition andsensitivity that can indeed be described as intelli-gent. The attentiveness and love shown in the main-taining of an old friendship, the constant opennessto new friendships, the personal effort of ensuringthat every moment (the slightest gesture, or word,or request . . ) could potentially increase the senseof familiarity that is the basis of friendship, allthese are expressions of intelligence. Of course,friendship is not wearisome in the sense that aserious problem may be, even though one's whole

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being may be involved. It is more normally associ-ated with an increased sense of freedom (we feelfriendly towards those who encourage this feelingwithin us) and relaxedness. It cannot be other thanintelligent to wish and discover how to foster thesense of harmonythat makes the company of othersso agreeable, to keep alive the pleasure of beingtogether and expressing mutual happiness, and ingeneral to create opportunities for human rela-tionshipsto grow and strengthen. A meal or a drinktogether, or even a game played together withothers, can become a special thing, involving allkinds of emotions, through which new and deeplybased relationships may be formed.

Games remove obstacles, psychological barriers,uneasiness, creating among people who do notknow each other very well the sense of pleasure incompany that is the basis of friendship. They arealso a good means of coming to know oneself andother people better, often providing a chance tounderstand sides of somebody's character that hadpreviously been unexpressed. In such ways, gamesbring people together rather than separating them,making us wish to keep up contact and enjoy anew aform of pleasure enhanced by the presence ofothers.

Who is it?

Some games have the simplest imaginable rules,are easyto play, and are ideal forms of light pastime.For instance, games of question and answer: oneversion might be to guess the name of a famousperson by means of ever more specific questionsabout them. To prevent the game from dragging ontoo long, however, there should not be too manyparticipants. Here is one way of playing. One mem-ber of the group goes aside while the others conferabout which famous person to select. When thequestioner returns, he or she addresses the ques-tions to each member of the group in turn. (Thegroup should therefore not be too large! Otherwise,of course, it can be subdivided into smaller groups.)

The questioner usually starts by establishing cer-tain basic facts: is the famous person male orfemale, alive or dead, imaginary or real-life?

It might also be worth their while writing down allthe information they manage to gather.

In one game the following questions and answers

Opposite page: good food anddrink, in good company, and indue proportion, are expressionsof an intelligent approach to life.

came forth. (Try to guess the final answer beforereading it at the end. Q = question; A = answer.)

0: Male or female?A: Female.0: Realorimaginary?A: Half and half!0: Deadorstillliving?A: Dead.Q: Did she die recently or centuries ago?A: Some centuries ago.0: In the Middle Ages or the Renaissance?A: In the seventeenth century.0: Was she English or some other nationality?A: Spanish.0: Was she aristocratic?A: Yes.0: Was she beautiful or ugly?A: Beautiful.0: Didshe haveahappylife?A: No, unhappy.0: Was that her own fault?A: No, more other people's fault.

Right: a nineteenth-century printof the nun of Monza, one of themost complex characters inManzoni's novel The Betrothed.

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0: Was she in the church, or was she a lady of theworld?A: In the church.0: Is it a character in a tragedy or a no vel?A: A novel.0: Was the novel written recently?A: In the last century.0: The authorisAlessandro Manzoni?A: Correct.

Anyone familiar with Alessandro Manzoni will bynow have gathered that the book concerned is /Promessi Sposi, and the half-real, half-fictionalcharacter must be the nun of Monza.

The game need not go on until the answer hasbeen reached. It could be decided beforehand, forinstance, that no more than a certain number ofquestions should be asked, or that no more than acertain amount of time should be allowed. Also, ifthe questioner suggests two wrong names, then hemight be deemed to have lost. The winner would bethe person to guess correctly in the shortest time orwith the fewest questions.

Guess the person

There is another rather similar "famous person"game, a bit more complicated to play, in which thename is guessed through clues for each letter. Onemember of the group-either in turn or picked atrandom-goes aside for a moment, while the othersagree on thername of some famous person, eitherfrom the past or in the present. The number ofletters in the name should ideally be the same asthenumber of people in the group. The guesswork isnot a matter here of moral qualities or physicaldetails. Each player has to take an active role, takingone letter of the chosen name in order, then playingthe part of some other famous character whosename began with that letter (each in turn speaking inthe first person, to give the guesser a clue, as illus-trated below).

The player who went aside, and who must nowguess the answer, does not ask questions. He or shejust writes down the initials of any names he man-ages to guess. If after somebody has spoken theirclue he thinks he can guess who that person ispretending to speak as, he writes the initial of thename he has guessed in the appropriate place on asheet of paper. (He has been given beforehand asheet of paper containing a number of emptysquares, corresponding to the number of letters inthe name he has to guess-these letters in turn, ofcourse, corresponding to the number of people inthe group and their order of sitting.) The name

slowly becomes evident and can often be guessedbefore all the squares have been filled in.

In one of these games ten students had chosenthe name Machiavelli, and took as their clue namesMagritte, Ariosto, Caesar, Hitler .. . and so on.

The first one (Magritte) began:- "1 paint dreamlike pictures."

Then the second followed:- "My poem was the rage at every court in Europe!"

And the third:- "I brought my legions to the Rubicon. ..

The fourth:- "I wanted to conquerthe world!"

And so it went on. It is not uncommon to need togo around a second time, to give the person gues-sing another chance. So then "Magritte" expanded:

- ... Iike a man with an apple face...."Then "Ariosto" said:.... the main character was one of Charlemagne's

paladins...."And "Caesar" continued:

and I crossed it, saying, 'The die is cast!'. ."Hitler":

- "I invaded Poland."Part of the fun of the game is for the "actors" to

portray their adopted character without saying toomuch. By the nature of the game, there is no limit tothe amount of time it can take or to the number ofclues the group might have to provide.

What is it?

Question-and-answer games are harder when thefinal answer is an object rather than a person: itcould be an animal or anything concrete or abstract.

It is important to ask the right sort of question atthe beginning,to obtain precise answers.

.Here is a game played by a group of children (0 =question; A = answer).

Q:A:Q:A:Q:A:Q:A:0:A:Q:A:

Is it concrete or abstract?It's concrete.Is it natural or man-made?It's man-made.Is it made of iron, or wood, orsomething else?Of wood.Is itpolishedorrough?It's polished.Is it ornamental or something you wear?In a way, something to wear.Do you wear it above the waist or below?Above the waist.

At this point the questioner stopped to think.Made of wood, polished, worn-in a way-above

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the waist. . . . It must be either a piece of woodenjewelry or a small cross around the neck.. . or (whynot?) a pipe!

So then he asked the decisive question:

0: Do you wear it around your neck or put it in yourmouth?A: Usually you put it in your mouth.

-Then it's a pipe!

The guess was right. But the group was still notfully satisfied; more details were wanted.

The questioner then continued:

0: Is it any sort of pipe or a particular kind?A: A particular kind.0: Does it belong to a famous person, eitheralive ordead?A: A famous local person, alive.0: Is the person a public figure?A: Yes.0: Is he a seniorpublic figure?A: He was a senior public figure.

Finally the questioner got the answer: it was thepipe the former mayor of the town was always seensmoking.

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Yes, no

This game can be made more demanding by impos-ing another restriction: the answers can only be'yes" or ''no." Since this obviously means morequestions have to be asked, a time limit is set,making the game livelier and more fun.

Here is an example of this game, played by agroup of boys and girls one evening:

Q0 Is it a concrete object?A: No.0: /s it abstract?A: Yes.Q: Is it an ideal?A: No.0: Is it a belonging?A: No.0: Is it a quality?A: No.0: Is it a virtue?A: No.0: Is it a defect?A: Yes.Q: Is it a feeling?A: Yes.0: Isitnormallyfeltformore than oneperson?A: No.0: For one person?A: Yes.

At this juncture the person guessing realized thefeeling must either be hate or jealousy. It was impor-tant that the final questions should leave no roomfor ambiguity in the answers. Thus they becameincreasingly specific:

0: Do you feel this feeling towards someone whohas wronged you, materially or otherwise?A: No.0: Towards someone you still love?A: Yes.

So "jealousy" was the correct answer.

The hidden trick

Every now and again a little variety and noveltylivens games up and stops them from becomingboring. In this game, what has to be guessed is not aperson or a thing, but (at least so it is said at the start)a whole story. The fun and novelty of this game liesin an element of surprise: the whole structure of thegame is crazy! How and why, we shall discover later,in the answer. One player, who obviously must not

know the trick of the game, has to reconstruct theplot of a whole story. Any questions are permissible,but the only answers that can be given are again"yes" and "no.' The game might start thus:

Q: Does the story have more than one character?A: Yes.0: Are the characters young?A: No.0: Does love come into the story?A: Yes.Q: is it a contemporary story?A: No.0: Is it set in the last century?A: No.0: The Middle Ages?A: Yes.0: Does a lot happen?A: Yes.0: Is it boring?A: No.0: The characters are well-known figures?A: Yes.0: Are they universally famous?A: No.0: Are they well known only to highly educatedpeople?A: No.

At this point the person trying to reconstruct thestory may feel a bit confused. After a little reflection,however, he may realize that it is a "set-up job," andthere is in fact no story. In fact, the group hasdecided together to answer "yes" to any questionstarting with a consonant and "no" to any thatstarted with a vowel.

The game continues until the person asking thequestions gives up or manages to grasp the prin-ciple behind the seemingly self-contradictoryanswers. The more imaginative and persevering thequestioner, the more entertaining it is for the group-hearing a random half story emerge through thequestions, from the questioner's own mind.

One playerfound himself faced with the followingblind alley:

0: Is there more than one character?A: No.Q: Is that character male?A: No.0: In that case she must be female, then?A: No.

On this occasion the questioner then went on toconstruct a glorious tale about a hermaphrodite,which is of course neither male nor female!

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The analogy gameThere are different variants of the guess-the-famous-person game. What changes is the style ofthe questions and answers. This version is some-times known as the analogy game because the cluescome through character analogies with plants,animals, and objects of all sorts. All the questionsstart, "If he/she were . . ." or, "If he/she had been

." and the answers, "He/she would be .. It issaid to be particularly popular in the forces, espe-ciallywherethere is national service,whentherearelong hours of inactivity, combined with a lack ofhome comforts, and fantasy tends to be given freerein! Well-known film stars of the more physicallyalluring kind are thus ideal for this game. In theexample given below, the name is that of a famousand beautiful woman:

If she were ... She would be..... i an animal? a cat.

a plant? an orchid.. . a season? a hot summer.... an article of clothing? a skimpy bikini.

a colour? a blonde.. .a country? Ia belle France.

a drinkn? a soothing pint.. a piece of furniture? a luxuriant bed.. . . a car? a red Ferrari.

(The answer on this occasion was Brigitte Bardot;see photo.)

Who said...?

Friends who see a lot of each other or who livetogether get to know each other's good and badpoints. An observation about ourselves made by afriend whom we respect and by whom we feelaccepted helps us understand ourselves and im-prove our weaknesses, rather than irking us. Evenwhen such observations are critical, the same ap-plies. In fact, between good friends a game can bemade out of each other's little foibles. The game weare now describing consists in a free expression ofhow friends see each other, or at least certain sidesof each other. Everybody who plays has to answercertain questions. In turn or at random, one by one,each goes aside, so as not to hear what the othersaretelling each other about him or her. One memberof this group acts as secretary, noting down thedifferent comments that are made. Each member ofthe group has to contribute one (not more) observa-tion on the victim's character. The secretary recordsall that is said and also who said what. Then the

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"#victim" is invited to come over and listen to thesecretary read out the various judgments expressedon him or her. After leaving a little time for reflec-tion, the secretary then rereads the list slowly, itemby item, and the "victim" must decide who madethat particular comment in each case. For instance,the questions might go thus:

- Who said you are easily annoyed?Who said punctuality is not your strong point?

- Who said you have a mania about tidiness?- Who said you are lazy at work?- Who said you are bigoted?- Who said you are too aggressive with women?-Who said you never have a comfortable rela-

tionship with men?- Who said you are a greedy pig?- Who said you are always losing things?- Who said you are not physically very wonderful to

look at?

There are not really any winners here. It is more agame of mutual familiarity and confidence among agroup of friends. However, it can be made moreentertaining by imposing a penalty for every wrong-ly guessed name-every time the "victim" attri-butes an observation to the wrong person.

Also, the game can be differently organized sothat everyone is actively involved. Thus, for inst-ance, each player writes his or her name on a pieceof paper, then folds it in four and hands it to thesecretary (who still plays the main role). The secret-ary carefully mixes up all the bits of paper, thenhands them around to the players again, goingclockwise. Usually each member of the group gets apiece of paper with somebody else's name on it,then duly writes a comment on the character of thatfriend. If by chance you getyour own signed piece ofpaper back after the shuffling, you can write somejudgment on yourself. After this the secretary readseach name aloud, as it comes, with the attachedcomment. And again, the "victim" has to identifythe author of the comment and pay a penalty if he orshe is wrong.

Other variants can be dreamed up to improve thegame still further and to help everyone becomemore involved.

Blind associationsThere are many games that groups of friends canplay. Here is one of the most amusing. Everyone sitsin a circle, each person with a sheet of paper andsomething to write with, and simply writes the firstthing that comes into their head-even if it makes

no sense. All that bits of paper are then folded, justto cover what has been written, and passed to theright-hand neighbour. Everybody thus receivesfrom their left-hand neighbour a similarly foldedpiece of paper. Without reading the covered-upwords written by the neighbour on their left, eachplayer then once again writes whatever comes intohis head. This continues until the sheets of paper arefolded up completely. Obviously, at no stage shouldanyone read what has already been written. It isusual to stop after the sheets of paper have beenaround the whole circle. They are then throwntogether, jumbled up, and redistributed, after whicheach player reads out all that is written on his or hersheet. Some of the associations of words andphrases can be extremely funny. But since everyonecontributed to each sheet, there cannot be said to beany winners.

The sort of randomness produced can be hila-rious:

- Will you come and have a meal with me?- The moon is blue!- Like a cabbage ...- Two's company, three's a crowd!- She sells seashells ...- Why?

A love story

Different things can be written on the sheets ofpaper-descriptions of an event, a meeting, an ex-change of wit, and so on. The game then has to becarefully arranged. First of all, each player is given asheet of paper and told to write on the top, "The. . ."plus some suitable adjective to refer to a man (hand-some, charming, stumpy, audacious, timid, and soforth). Then the paper is folded, so that the wordsare invisible, and passed on to the next player, who(without looking atthe adjectivewritten by hisor herneighbour) adds a man's name. It can be all themerrier if the name is that of one of the peopleplaying the game! Other possibilities might besome common acquaintance or a name of somefamous person. The third writer then adds the verb"meets," followed by an article and an adjectivesuitable for a woman (beautiful, attractive, hideous,alluring, and so on). The fourth step is to write awoman's name-again, of course, without seeingwhat has been written beforehand. And here, too, ifthe name is that of one of the people taking part inthe game, it can be all the more entertaining. Ingeneral, the more players there are, the livelier thegame is. The story need by no means stop here:

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other actions can be introduced (such as they greeteach other, they shake hands, they turn their backson each other, they insult each other, and so forth);then "he said to her" and "she replied," and so on. Ifthere are more than seven players, it is not hard tothink up further ways of continuing the story. Onesmall group of girls and boys came up with thefollowing "love story," which prompted muchlaughter:

"The handsome Anthony met the clubfootedBertha. They kissed. 'Did you bring the money?'asked handsome Anthony. 'I smashed the platesthis morning!' replied clubfooted Bertha."

In itself it is notthatfunny, but itwasfarfunnier incontext: for one of the group happened to be thelocal priest-whose name was Anthony!

A macabre game

You only need two people with pencil and paper forthis type of game. One person thinks of the mostobscure word possible and writes the first and lastletters, leaving blank spaces for the others. Theother player tries to fill in the blanks, letter by letter.If he guesses correctly, the first player duly writesthe letter down in the correct space. If his guess waswrong, however, his opponent starts to draw acrude diagrammatic sketch of a man on a gallows.(This game is often, in fact, called "hangman.")Thus the game continues until either the word has

3

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been guessed or the drawing of the gallows (seefigure below) is finished. Hours can be wasted play-ing this game! And to establish a winner, it is worthkeeping track of how many times each player issuccessful. On each occasion it is the winner of theprevious bout who thinks up the next word. As thedrawing can easily have fewer or more stages, youcan agree on a maximum or minimum length fortheword to be guessed. If the same letter occurs twicein the same word, it need not be guessed twice.

The same-letter game

The important thing in this game is to work out howto win, and then follow one's strategy through to theend. It is possible to play with many participants.Each player has a sheet of paper and a pen. Onethinks of a word-either selected beforehand orpicked at random-and tells the others how manyletters it has. Each of the others in turn then sug-gests a word with the same number of letters. If inthese suggested words one or more letters coincide(for example, if the same letter occurs in the sameposition as in the word to be guessed), this has to bedeclared-though only the number of "same let-ters" is declared.

To illustrate this, let us start an imaginary game.Imagine that the first player has thought of the

word "window" and thus told the others that thereare six letters. Each of the other players then writessix blank spaces on their sheets of paper (see figureabove). Then they begin to suggest other words ofsix letters: the first, "carpet," to which the responseis nil, since there are no same letters. Each playerwrites "carpet" and crosses out the letters with aline. The second player then suggests, say, "cor-don": this time the answer is two (the d and o-notn, since the n in "window" comes third rather thanlast). Now, since "carpet" resulted in a nil response,we can immediately eliminate the c and r of "cor-don" as well, since they correspond to the c and r in"carpet," which we know are wrong. The task re-mains to discover which of the other letters of "cor-don" are the two correct ones. The following player

723

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should pick a word as close as possible to "cordon.""Burden" brings a response of one-which is prob-ably either the d or the n (since "cordon" and "bur-den" share r, d and n in common, and r has alreadybeen eliminated). We can now try another word. Ifthe next suggestion is "shadow," the response willbe three: and since the only same letter between"burden" and "shadow" is the d, the fourth blankspace at the top is probably d.

Usually, once one letter has been discovered, thegame becomes easier. It is important, however, totry to identify a cluster of letters, rather than indi-vidual ones separated from each other. In ourexam-pIe, the o ("cordon"-"shadow") is a likely candidatefor the blank after d. The word will in fact often beguessed before all the letters have been discovered.

Various different approaches can be adopted, butthe basic principle remains that each player benefitsfrom the words suggested by those before him. Thewinner is the first one to guess the word, and it is heor she who then thinks up the next.

A meal in companyYou are having dinner with friends. Some of theother guests you know well, others little or not at al 1.

As the meal progresses and the mood relaxes, con-versation turns to familiar themes, and the usualopinions and feelings are aired. Then a member ofthe opposite sex, who you hardly know says some-thing-nothing very remarkable, perhaps-that forsome reason attracts your attention. Their meremanner has somehow become more "interesting"in the context of a friendly evening with good foodand good wine. You feel a wish to cultivate thisacquaintance, to dedicate yourtimeto following upthese feelings. It is the start of a relationship....

Probably many will recognize in this descriptionthe beginning of a friendship or love. Human rela-tionships, whether close friendship or compan-ionship, thrive on casual, relaxed occasions, whenfood and conversation are shared and time spenttogether. Games can certainly strengthen friend-ship, but a meal shared can introduce a new warmthand new interest into a friendship.

An invitation to a meal can be a good way ofending a job of work together, a friendly way ofrepaying some debt or obligation, a means ofstrengthening new ties, and an auspicious start to anew undertaking. Careful preparation and taste, anelement of surprise, good food and wine all worktogether to create an occasion when all our tact,sensitivity, and intelligence come into full play.

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An ancient remedyGames, good company, and shared meals all helptomake our life on this earth more agreeable. How-ever, it is worth devoting some attention to one ofthe traditional vital ingredients of enjoyable relaxa-tion-wine.

Two thousand years ago, the poet Horaceobserved that wine makes everybody more elo-quent. The ancients well knew the liberating effectof wine and its importance in developing humanrelationships. During feasts and banquets (at whichRomans reclined on couches-the triclinium),friendship burgeoned in an atmosphere at oncewarm and lively, with fine food and wine, singingand dancing. Reclining on a couch was also a goodsymbolic expression of relaxation from the press-ures of daily affairs.

Horace once again extols the pleasures of"freeing the spirit . - with sweet Lyaus" (Lyausbeing another name for Bacchus, the god of wine,son of Jupiter and Semele, the inspirer of poets andthe dispeller of care).

Horace's work continued in a tradition dear to theG reeks-evoking the delights of feasting, with winehaving special human significance.

Wine drinking goes back to the dawn of history.Christianity, continuing the Judaic tradition, con-firmed the ritual importance of wine and thus en-couraged the spread of viticulture. Indeed, we readin the Bible how Noah was the first to plant vines,drink wine, and get drunk.

Most people speak more freely and are more atease physically after a glass or so of wine.

Obviously we are talking here about moderateconsumption. It is commonly accepted that whenwine becomes a means of escaping the worries andtensions arising from serious problems, when onestarts desiring to drink more and more, and to drinkon one's own, then it becomes a destructive drugand starts to cause serious problems. Once an indi-vidual ceases to be in control of his or her owncravings, wine drinking ceases to be an "intelligent"occupation.

Wine and good food, as the ancients realized, helpto make life happier and more pleasant, halting-momentarily, at least-the race of "time's wingedchariot" that bears us inexorably towards our end.

At the "Full Moon" innIn the great Italian novel IPromessiSposi, one of thecharacters, Renzo, stays at the Hostelry of the FullMoon in Milan. It had been an action-packed, ex-hausting day, and the young man was hot and

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thirsty. Summoning the innkeeper, he orderedsomething to drink. One glass followed another,and not surprisingly Renzo became drunk. Growingsomewhat boisterous, he was taken to bed, wherehe slept soundly. The following morning he wasawakened by loud knocking at the door.

Tolhisastonishmenthefound himself in policecus-tody, handcuffed and about to be led to prison. Hemanages to escape, but nevertheless those glassesof wine proved more than he had bargained for!

Illustrating this story, we might imagine that thetop shape in thefigure above represents the inn: canyou now change the arrangement, by moving sixmatches, to represent two of the wineglasses Renzoemptied? (The solution is shown in the lowerfigure.) (Similar arrangements are possible by mov-ing just four matches.)

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fig. b fig. c

fig. e

SEEGA-DERRAHAfter mancala (wai; see p. 129),the best known of African boardgames, there are many othersthat reveal in their simplicityhow different tribes exercisedtheir wits in devising pastimes.The class of games with whichwe are concerned here does notuse a board of alternating darkand light squares, such as isused forchess orcheckers. Theboard in this case consists of anumber of plain sections alone.Anthropologists haveestablished that up until the lastcentury many Africans played agame called seega, using aboard of 25 plain sections. Thisis a game for two players, eachhaving 12 pieces, either white orblack, Turn by turn they place 2pieces at a time in any 2sections, though not the centralone. After all the pieces are onthe board (see drawing onopposite page), the last player tohave placed a pair in positionmakes the first move. (it isthusimportant to agree at the outset,by lot, who starts.) Pieces can bemoved one at a time, from onesection to a neighbouring one,though not diagonally, in anattempt to "trap" the opponent.An enemy piece is taken (andremoved from the board) bybringing 2 of your own piecesalongside it, 1 eitherside-rather as if it were asuspect being arrested by two

fig. a

policemen (see figure al. If apiece moves to a space between2 opponent pieces, however, it isnot taken, since it has (so tospeak) declared that it is notguilty by placing itself betweenthe two "police."If the player who has the firstmove cannot make a move, hehas the right to removewhichever of his opponent'spieces he likes. (In a variant rule,his opponent must take a secondturn.) It is then his opponent'sturn. During the game,whenever a piece takes anenemy, it can move again if it isable thereby to take anotherenemy, any number of times.The winner is the player whomanages to capture all of hisopponent's pieces or, if there isan impasse (excluding the startof the game), the one with mostpieces on the board.

More demanding variants of thisgame can be played on boardsof either 49 or 81 sections (seefigures b and c), with 24 or 40pieces per player respectively.

"Block"One version of seega still playedin Africa is called "block" or"jump," which starts with thepieces set out as in figure d.They are moved singly, onespace at a time, and neverdiagonally. Opposing pieces aretaken in single or multiplejumps-rather as was explainedfor alquerque (see p. 34)-but itis not obligatory to take a piecewhenever possible.

DerrahDerrah is still played today inNigeria and North Africa, on asimilar board, though with 42sections Isee figure e). It cannot

really be considered a version ofseega: it might more fittingly beconsidered an ancestor oftic-tac-toe (see p. 37). Played bytwo people, there are 12 piecesper player, each player having adifferent colour. Having decidedwho shall begin, they then inturn place on piece at a time untilall are in play. Then the firstplayer continues, moving asingle piece one space (notdiagonally). The idea is to formrows of three, either horizontallyor vertically (again, notdiagonally), removing an enemypiece each time such a row isachieved. However, rowsformed during the initial placingof the pieces do not count; nordo rows of other than 3 pieces.The winner is the player whofirst makes it impossible for hisor her opponent to make anotherrow of three.

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126

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WARIWart is a highly entertaining andvery subtle game played by twopeople, like chess or checkers. Itis also known by various othernames and is widespread inequatorial Africa. In certain areasit still retains its old sacred statusand is reserved for the priestlyand noble castes. We havechosen to use the name warnbecause this is the most familiarin Europe (especially Britain), aswell as in Florida and someislands of the West Indies Isuchas Antigua}. It is a game for bothadults and children. Othernames for the same game areawele (W. Africal) gabata(Ethiopia), suba or wuri Itheblack communities in the UnitedStates), and most commonlymancala Imhncala in Syria,mancila in Egypt, mancala inFrance)Like chess and checkers, wari isa game of pure skill, with noroom for chance. Though it mayat first appear simple, it will soonbe apparent how absorbing itcan become, demanding skill,quick reasoning, and tacticalability. And it retains thesequalities even when reduced tothe most basic form (wheninstead of the board, simplewooden or plastic bowls, brokenshards of pottery, or even-asAfrican children make dowith-holes in the earth areused, with berries, small shells,or pebbles as pieces). As wasnoted above, wari is a game fortwo players, the aim being laswe shall see) to take as many ofthe opponent's pieces aspossible. The board lies betweenthe two players such that eachhas a row of 6 bowls or troughsin front of him. Troughs a-f(figure a) belong to player S(Southi, and g-l to N (North). Tostart, each player places 4 ballsor pieces in each of the troughson his (or her) side. It must thenbe decided one way or anotherwho shall make the first move.At each move, all the halls in oneof your own troughs aredistributed around the board,one per trough, starting at thenext-door one, in acounterclockwise direction Iseethe arrows in figure a and figurec), without missing any trough,thus frequently landing in theopponent's territory. (in theillustrations, the balls beingdistributed are red, the onesalready in each trough black.)When there is only one ball tomove (when there is only oneball in the trough), it is simplydropped into the neighbouringtrough-which may well be anenergy one, as for example infigure, f-g. Examples: if S

starts from trough d, he will go e,then f, then over to theopponent's side, around to h. Atthis point, N could take the 5balls in g and distribute themaround to I, without going intoenemy territory (figures c and d).By now the basic principle of thegame should be clear: the ballsin troughs a-f can only bemoved by S (South), and thosein g-l by N (North).To take balls, the last one of therelevant distribution must havelanded in the enemy camp. Ateach move there are fourpossibilities: al the distributioncan finish on the distributor'sown territory, in which case noballs are taken; b) it can finish inenemy territory, in a trough witheither 1 or2 balls already in it(thus bringing the total in thattrough to 2 or 3 respectively)-inthis case, allthe balls in thattrough are then taken,the rulebeing that when the last troughto be filled in any go ends upwith 2 or 3 balls in it (no moreand no less), then those balls canbe taken; c) the final trough ofany distribution may be emptyor may already have 3 ormoreballs in it, in which case the ballsare nottaken (according to therute just stated): dl when the lastball of a move brings the total inthat trough to 2 or 3, those ballsare all taken-but this is notnecessarily the end of the move:if there are also 2 or 3 balls in theprevious troughs, the playerwho has just moved continues totake balls until there is a troughcontaining either only 1 ball ormore than 3 (see figures e and ).Figure a gives an example inwhich S sets off from e and takes9 balls (2+3+2+2), namelythose in g, h, i, and j. Figure fshows two less fortunateexamples: if S plays the 6 ballsin f, he or she will be stopped byk; or if the 4 balls in e, by h. It willonly be possible to take the ballsin I and i (3) respectively.There are three forms of defensewhen threatened:a) flight-when your opponentis threatening the ball or balls ina trough on your side, the troughin question can be emptied(figure g: in this example, S isunder threat from trough i, soempties d, transferring those 2balls to e and fl;b) reinforcement-when atrough containing 2 balls isthreatened (figure h: here Sreinforces d, which is threatenedby N's trough k. by moving the 3bails in b, thus augmenting dito3 balls, thereby making itsecure) c) overkill-athreatening trough on theenemy side can be renderedharmless by a kind of 'overkill'tactic, by increasing the number

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129

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Of balls in it (ffigure it. In this trough to do the round of theexample. N's offensive, board, continue beyond the firstrepresented by the 7 balls in k, is trough, and end up back indefused by S distributing the 8 eneny territory. Such aballs in c (k thus ends up with 8 maneuver is not necessarilyballs, which would mean it advantageous: a) completewould end up in its own success-S distributes the 19territory, in g). balls ind (figure Ar),finds theOne important rule should be opponent side almost finishedobserved: When there are (all thetroughs either empty orenough balls in any one trough containing only one ball) andto do a round of the whole board ends the game by taking 15and beyond the trough again, balls, thus emptying the enemythat trough must be missed out territory,; b) middlingThis being an exception to the success IS moves the 20 ballsgeneral rule given at the inc (figure /), reaching, but isbeginning). Moving the 15 balls halted by i, so can take only 7in d figureer, S will then jump balls (3+2+2); c} completefrom c to e, and end the move in failure-unable to make anyh, taking 6 balls (1+1 +1 in h, and other move, S distributes (figure1 + 1+1 in 91. m} the 21 balls in b, and finishes"Building a house/' means in 1, where, since there are now 4having enough balls in a single balls there12 +1 + 1),the move

fig. k

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ends; d} complete plan of action (this requiAresfailure-unable to do otherwise, much practice) and to see eachS moves the 18 balls in f I(figure move (each distribution of ballsn), so ending up in a, which is his from any one trough) in the lightown territory, hence no balls can of this strategy, anticipating asbe taken. far as possible your opponent'sThe game ends when one strategy.player's territory becomes In the same way as go (see pp.completely empty. However, 1D7- 11 2p, wari is intellectuallythat player is not necessarily the stimulating and tests one'sloser. The winner is the one who powers of strategic thinking.has taken the most balls by thetime one or other side ends upempty. (Balls still remaining introughs do not count either wayin the final scoring.To understand the real attractionof this game it is necessary toplay it often. You should not bediscouraged if at first it seemsinconclusive, or if it is difficult todecide how to move.Remember, it is vital to have a

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The point U

U numbers andimagination

I have often admired the mystical way of Pythagoras. and the secret magicof numbers.

Sir Thomas Browne

Mathematics and reality

Numbers evoke a feeling of exactness and preci-sion. "Two plus two equals four" is a universal wayof suggesting that there are some incontestablecertainties, However, this is in a sense a "biased"view, when considering arithmetic, algebra, analy-sis, and mathematics in general.

In fact, numbers have always been rooted in thetraditions, the technology, and the trade of differenthuman societies down the ages.

Sacred symbolism (such as the symbolic perfec-tion of the numbers 3 or 7, or the "golden section"-a certain precise ratio of the different lengths ofthe sides of a rectangle) often underlies thearchitecture of temples and other holy places. Tradeand cultural contact with the East led eventually tothe abandonment of Roman numerals in favour ofthe Arabic system. There are indeed many exam-ples of how, far from being a remote abstract realm,mathematics is essentially bound up with the worldof concrete reality.

Here it is necessary to draw a parallel between thecommon misconception of mathematics and theway that intelligence and creativity, and motivation,feelings, and desires are commonly related. Wetend often to separate reality into neat compart-ments. But to do so is as misguided as to separate

mathematics from the human contexts in whichthey have evolved.

The world of the psyche (in its broadest sense,including mind and soul, as well as "psyche" in theusual meaning) does not have precise lines of de-marcation. Just as we are all "intelligent" to somedegree or other, we are also all "creative," each withour own rich world of impressions and feelings. Thevarious elements of our being interpenetrate andinteract, and it is this process of inner working thatcreates an individual personality, with all its greatstore of potential.

A synthesis

Ingrained habits, the stimuli of the environment, theteaching and example of important individuals inone's life (parents, relatives, school and universityteachers, for example), and relations with friends allcombine in any one person, along with genetic traits(inherited through one's parents), to create thatperson's "essential" character. It is hardly deniablethat "character" as such exists. And there is un-doubtedly a certain correlation between one's ownintellectual ability and that of one's forebears.However, the argument as to whether intelligence isinherited or cultivated has been long and bitter. It is

131

The point: numbers and imagination

of course not possible to determine the "structure"of a child's intellectual capacity after the moment ofbirth. On the other hand, genetic and biologicalengineering is still at an early stage-and its use isalready raising fundamental ethical problems.

Thus we can only discuss the way things are atpresent, using the present to develop as far aspossible every individual's capacity for profoundmotivation (towards success, friendship, culturaland professional achievement, family life, and soon) and interest in life. For these constitute perso-nality. And personality is always changing. Certainaspects predominante at certain times, others atother times.

Human behaviour is predictable only to a limiteddegree. Very many people "make up for" pooreducation or what might be termed average intelli-gence by sheer force of will and dedication to theachieving of a fixed goal. Others perhaps solve theirproblems by "stepping beyond themselves," so tospeak, in a sort of creative act that sets everything ina new and more "workable" order.

Yet it is somewhat artificial to separatethe variousaspects of a person's being (whether one does soprofessionally or as an amateur). They are in facttoointimately linked-just as the history of a singleindividual is bound with that of the social group towhich he or she belongs, just as myth can stimulatescience, and just as the intellectual process ofanalysis and synthesis goes hand in hand withcreativity.

Thus it is that the number stories in the followingparagraphs involve myth, imagination, and culturaland economic facts of history.

The intelligent crow

Medieval man had much greater contact with anim-als than we have today. Hunting had its own laws,and of course there were no guns, so the survival ofwhole species was not endangered. The castles offeudal lords would also have been full of animals-the lower floors were often overrun with rats andmice, while crows and other birds nested in thetowers.

One day one of these lords decided to try and getrid of a particularly noisy crow that was nesting in hiscastle watchtower. On a number of occasions he hadattempted to creep up on it and take it by surprise,but the wretched creature's hearing was too sensi-tive; it heard him coming and flew to a nearby tree,from where it could safely observe the proceedings.Once the man had left, it simply returned to itsformer place. For the lord, it was no longer just aquestion of disposing of an irritating bird pest: it had

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become a battle of wits. He thus thought up a planthat would test the crow's counting ability. He senttwo men up the tower. The crow immediately flewoff to its safe perch in the tree. Then one of the menleft the tower, in full view of the bird. It was notto bedeceived, however, and only returned to its nestonce the second man had also left. The lord thentried with three men, two leaving together first.Again, the crow was not taken in. Wishing to discov-er how far the crow could count, the lord then sentfourmen, but once again with negative results. Onlywhen five men went into the tower and four cameout did the bird lose count and so finally fell into thetrap.

From this story it emerges that crows are intelli-gent birds, with, however, only a limited ability tocount !

Counting: a human faculty!

One mode of behaviour we can unhesitatingly calltypically human is the use of language. All animalsseem to have some form of communication-signs,postures, and sounds-that represent messages. In

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mankind, however, language has evolved such avariety and multiplicity of meanings, and a capacityfor conveying such subtle nuances, that it is unlikeany other form of communication. But there is onelanguage in particular that man uses and has de-veloped to a supreme degree, clearly distinguishingthe human species from the other animals: that ofnumbers. Whether true or not, the anecdote of thecrow is based on the observation that certain anim-als have some sense of numbers. There are animalsthat clearly possess elementary perception of con-crete quantities. Yet this is very different from anability to count. Rather, it is an ability to distinguishbetween groups of objects of the same nature. Inmany species of animal, for instance, mothers be-come agitated if one or two of their young aremissing.

The tools of counting

It would appear that no human community, how-ever primitive, has been without some basic form ofcounting.

The "tools" used in counting are numbers, sym-bols representing abstract concepts, in that they canstand for any object whatever without having anydirect relation to it. Different civilizations in thecourse of history have used different systems andcodes to represent numbers and numerical calcula-tions.

Even the most primitive peoples now alive, withthe most crude counting systems, have words fornumerical concepts such as "one" and "two," then"two and one" to mean three, and "two and two" tomean four: numbers above four being simply"many."

Numerical bases and systems

Linked with the concept of number is that of systemof enumeration, which denotes all the symbols(necessarily a finite quantity) used to enlarge therange of use of those same symbols. What we callArabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are merelythe symbols relating to the system of enumerationthat the West has used for centuries. Other cultures,both past and present, had and have different sys-tems. Numbers are thus simple sequences of sym-bols arranged according to the rules of a certainsystem. The base of a system of enumeration indi-cates how many units of a certain order are neededto make one unit of the next higher order. The baseof our own counting system is decimal. That is, wechange to a higher order every ten figures: hence,starting from ten (11, 12, 13 . .. 17, 18, 19, 20), we

come to the next higher order (in this case, 20) afterten units (20 being the tenth figure after 10, which isthe first unit of a higher order). Generally the basisfor the counting systems that employ our figures isdenoted by 0.

If, for instance, we had taken six as base, wewould then switch to a higher order every sixfigures. In such a system, the numbers would go 0,1, 2, 3,4,5, and 6would be symbolically representedas 10 (11 representing 7, and so on).

Let us now end with a little problem: To whichnumber in the decimal system would 20 correspondif we knew that it was in base 6-for example, 20)6) =

?(10)'One simply needs some familiarity with natural

numbers and perhaps a certain aptitude for solvingthis kind of problem. Thus:

(2 x 61) + (0 x 60) x 12

The number 20 in base 6 therefore corresponds to12 in the decimal system. This resultcan be symboli-cally represented as 20(6, = 12no

How many hands?

Our counting system-the decimal, possessing tenfigures-is neither the only nor the best possiblesystem. The philosopher Aristotle observed manycenturies ago that the reasoning behind his state-ments lay in the fact that man has ten fingers dividedbetween two hands and finds it convenient to countin tens. For this same reason numerical systemsbased on twenty were not uncommon: it was notdifficult to include toes with fingers, to create foursets of five!

It is told how one day a team of anthropologistswishing to make a close study of an ancient tribe in aremote corner of the world encountered a bizarremode of arithmetical reasoning. Written on a largedried leaf they found the following calculations:

1) 140 - 323 - 1013;2) 2223 + 4123 = 11401.

Immediately they pondered this problem: ifcounting systems developed historically from thefact that man has ten fingers, distributed betweentwo hands, how many hands would the members ofthis obscure tribe have had to have to result in thiscounting system?

The answer can be obtained by identifying exactlywhat that system was. First, note that there are nofigures higher than 4; then 3 plus 3, carrying 1, canapply only in a system in which the move to the

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higher order occurs every 5 units-that is, in asystem of base 5. We must conclude, then, that theinhabitants of that far part of the earth had only onehand!

It would in fact appear that numerical systems inbase 5 were the most widespread. Clear proof of thiscan be found in certain cultures in which the wordsfor "hand" and for "five" are identical. In ancientPersian, to say "hand" one would use the wordpentcha, corresponding etymologically with theGreek penti (Latin quinque), meaning five-equivalent also to the Sanscrit panQa. Greek also hasa word pampaxein, which in the classical era wasused to mean "count"-the strict etymologicalmeaning being "to count in fives": "quindrise," onemight now say. The South American Tamanacostribe had a quinary counting system. Their word forfive also meant "a whole hand"; for higher num-bers, they would say "one on the other hand," "twoon the other hand" (six, seven -. ), and so forth.

We know, too, that the ancient Aztecs divided themonths into weeks of five days. And many othersimilar examples could be cited.

The planet of the one-handed

Life on earth had long been impossible. As theforests and grassland were killed off by acid rain and

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the seas became irreversibly polluted, more andmore humans travelled to other planets to seek anew existence.

One such group voyaged to the planet of theone-handed (so-called because prior exploration,mainly by satellite, had revealed that its inhabitantshad only one hand). Having landed the spaceshipwithout any problems, the captain and crew, then allthe passengers, cautiously disembarked and beganto explore. Soon their attention was attracted bystrange inscriptions carved on a rockface (seefigurebelow).

Various features, the arrangement of the sym-bols, and the results themselves indicated thatthesemade up a mathematical sum. Earlier surveys hadestablished that the planet's numbering system fol-lowed roughly the same rules as our own, and that itwas based on the number of fingers on the inhabi-tants' single hands.

The problem can be expressed thus: Supposingthat the numbers in the example below do not startat zero, how many fingers did each inhabitant of thiscurious planet have on their one hand?

To obtain the answer, it is necessary to find thebase of their numbering system. Remembering howwe have defined any such base, this should not behard. The base of any numerical system tells us howmany units of a certain order are needed to bring usto the next higher order. Let us examine the exam-ple carefully and work it out normally, as though itwere written in our own system. Two figures addedtogether** suffice to take us to a higher order,which is indicated as *. This means that the systemmust be ternary, in base 3.

Thus the planet's inhabitants have just one hand,with three fingers. Having identified the numericalsystem, we should now be able to tell what theleft-hand figures should come to in our own decimalsystem. The only possible solution is

12 + 12= 101.

And that "translated" into decimal is

5 + 5 = 10

Three times one-five = five-one\W../ /I , .

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"If three times one-five equals five-one, how muchis six times one-three?"

The formulation of the problem makes it seemmore complex than it actually is. However, it doesconceal one interesting feature. The way the num-ber 15 is represented (as one-five) should tell us that

134

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we are not dealing with the decimal system but arein another base, which we must discover if we are toobtain the correct answer in our system. If we read15 as "fifteen," we would be confusing it with thedecimal base; in fact, "15" merely tells us that,reading from right to left, there are five units of thefirst order and one of the second. To identify thebase of unknown numbers, there are various empir-ical methods of procedure (note, for instance, that in3 x 15 = 51 the highestfigure is 5), butthe safest istoresort to an algebraic equation. Representing theunknown base as B,we can establish:

(3 x BO) x (1 x B1 + 5 x b0)(3 x 1) x (B + 5 x 1)3 x (B + 5)3(B ± 5)3B + 155B 3B28= 14 B=14/2

= (5 x B1 + 1 x B6)=51± + 1=5B + 1= 563+ 1= 5B + 1= 15- 1= 7

The base of this system is thus 7.Having discovered this, we can then answer the

original problem, thrown up at the start: What is sixtimes one-three?

It is advisable to do this problem as it stands,without translating it into decimal, working it outand then translating it back into base 7. Let usrationally try to work in the same numerical system,even though this is different from ours (the Latinword ratio, from which "rational" stems, in factmeans both reason and, specifically, "counting"and "calculation"). To do this we need only remem-ber that every seven units we need to pass to thenext order above:

13 x6

114

Thus we can write:

13(7) x 6(7) = 114(7)

We can checkthis result bytransposing the wholeoperation into decimal and ensuring that all is cor-rect:

13(7 - 1 x 71) + (3 x 70) = 7 + 3 = 10(10)6(7) = (6 x 70) - 6 x 1 - 6(lo)114(7) = (1 x 72) + (1 x 7') + (4 x 70) == 49 + 7 + 4 = 6011i,

In effect:

10()np x 611o0 = 60(1o)

The "Black Cat" Society

The Black Cat Society had been implicated in someserious crimes. Its members were mostly ex-convicts who sought obscure compensation fortheir past in secret rites and ceremonies and myste-rious practices that in the end led back to the samecrimes as before. The police had decided the orga-nization should be outlawed. However, the crimesstill continued. It thus seemed probable that thesociety was still operating clandestinely. A messagewas received that they would be meeting in a cellaron a certain night, and the police duly descended onthe place that evening-to find it deserted! Every-one had vanished. Nevertheless there were signsthat they had been there shortly before. Some oddmathematical scribblings on a blackboard caughtthe attention of the police inspector:

16 x 253 = 5104

He examined this for a while, then exclaimed,"The Black Cat Society uses a different base !"

What base did the society use for their mathema-tical calculations?

Let us proceed by trial and error. Note first thatthere are no figures higher than 6: thus the basemust necessarily be higher (7, 8, 9 . ).

So let us begin by imagining that the base was 7;this would mean that when multiplying, a new orderis reached after every seven units:

253 x16

22442530

5104

Luck is with us: the base indeed is 7.By way of example, let us go over the way we

worked: six times three is eighteen, but since we arein base 7 we have to base ourselves on the firstmultiple of seven before eighteen, namely "four-teen"; this being four less than eighteen, we writefour and carry two. Continuing on, we come to sixtimes five = thirty, plus two carried = thirty-two.Here again, we need to return to the first multiple ofseven-which is twenty-eight. This is four short ofthirty-two, so again we write four, and this timecarry four. Finally six times two = twelve, but six-teen = two times seven plus two, so we need towrite "two-two," and so on.

Try out some other calculations using differentnumerical bases.

135


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left-hand little finger for "one," then the ring fingerfor "two," and so on, until we reach the right-handthumb (see figure B left) or the right-hand littlefinger (see figure C). Also of interest is the systemused by Arabs in North Africa-starting with all tenfingers folded and extending the left- or right-handindex finger for "one" (see figure D left).

It is true to say that digital calculation, which stillsurvives today in areas of Europe and Russia, datesback to earliest times.

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"Mathematics gives a V sign"This remark is attributed to the celebrated Italianphysicist G. Battista Venturi (1746-1822). He wasreferring to the similarity between certain formsof numerical representation with fingers and theinsulting gesture (the suggestive V sign), which issti ll used in various mutations today!

The rules of "indigitation" (reckoning by thefingers) were widely followed all around theMediterranean, especially in the Middle Ages andduring the Renaissance. The habit of counting onone's fingers had become so instinctive that digitalrepresentation came to be used even in music (seeillustration on page 138).

One name worth mentioning at this point is LucaPacioli (1450?-?1520), a mathematician who stu-died in Venice and then became a Franciscan friar.With his friend Leonardo da Vinci, he worked in theservice of Ludovico Maria Sforza in Milan. In 1494 hepublished in Venice his Summa de aritmetica,geometria, proportioni et proportionalita-the firstgeneral treatise on practical arithmetic and algebrato appear in print. In this book he illustrates one ofthe commonest forms of finger counting (see illus-tration opposite).

The practise of counting on one's fingers con-tinued so long that for centuries arithmetic wasthought of in terms of fingercounting. Clear signs ofthis can be seen in paintings and sculptures; andcertain passages in old books and manuscripts onlymake sense when one remembers the conventionsof "indigitation," the act of reckoning by the fingers.

"Happy is he indeed who can count his years onhis right hand," declared the satirist Juvenal (c.55-130 AD.). And as we can see from the illustrationfrom Pacioli's book (opposite), the years that werecounted on the right hand started at one hundred.The poet was therefore saying, "You're happy if youlive to over a hundred!"

Important dates and events are sometimes refer-red to in medieval texts in such a way that theywould be indecipherable if one did not know thecode of finger counting that was being used. Still inuse in the medieval schools as an arithmetical text-book was the Liber de loque/a pergestum digitorum(Book on the Language of Finger Gestures) by theVenerable Bede.

Hand calculations

It was during the Middle Ages and the Renaissanceespecially that systems of doing calculations onone's fingers became widespread-probably be-cause only a few people had abacuses or multiplica-

Opposite page: a system formaking calculations on one'sfingers, from a book onmathematics by Luca Pacioli,published in Venice in 1494.People continued to count on

theirfingers in later centuries, aswe discover, for instance, fromthe TheatrumArithmetico-Geometricum byJacob Leupold, published inGermany in 1727.

tion tables beyond 5 x 5. One very simple system,known as the "old rule," employed the concept ofthe complement to ten: if n is a number, its comple-ment is 10 - n. Then, if we want to multiply, say, 8 x9, we start by finding the complements (2 for 8, 1 for9); next, subtract either complement from the num-ber of which it is not the complement (hence 8 - 1 =7, or 9 - 2 = 7). This tells us the "10"s digit in thefinal result-namely, 7. The product of the twocomplements (2 x 1 = 2) then gives the final digit:placed in the correct order, 72 indeed emerges asthe result of 8 x 9.

141

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This same principle was applied to calculationsworked out on fingers. Keeping our example of 8 x9: each finger is assigned a numberfrom 6to 10 (seefigure below); to multiply 8 by 9, one finger called-8" has to be placed against one called "9."

It can be seen that the complement of 8 is repre-sented bythetwo top fingersof the left hand (above

/70 ) 8 x 9 = 72

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This method of multiplying on fingers can beapplied to all the "10"s above 10 itself, though withslight modification. Suppose, for example, we wishto multiply 13 x 15: the fingers are then numbered11 to 15 (see figure below), and finger "13" will beplaced against finger "15." We then multiply theeight fingers below (representing the last digits) by10: (5 + 3) x 10 = 80. Taking the lower fingers again,we multiply the two hands (5 x 3 = 15), and thisproduct is then added to the previous total 1 5 + 80= 95). And the final step is to add the constant, 100.

Thus:8x 10=80 /

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2 x 1 =2 70+2=72

the two that are touching), and that of 9 by the singlefinger (referring to thumbs as "fingers"!) on theright hand. The number of the "touching" finger onone hand, less the complement represented by theupper fingers on the other totals 7: this is the -1 O"sdigit. The product of the upper fingers, namely 2 (2x 1) gives the units: 7 and 2 are 72. The figuresbelow show multiplications of 6 x 6 and 7 x 10.

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100 + 95 = 195

The disadvantage of this method is that it is tooslow, however, it has considerable value as ateaching method. First, children are entertained bydiscovering new ways of producing the same re-sults in arithmetic. Second, the device of usingcomplements to ten is a concrete example of thealgebraic multiplication of binomials. Glancing backto the example of 8 x 9, we could express the sameprocess in terms of the relation of each figure to5-(5 + 3) and (5 + 4)-and multiplying them in adifferent way:

5+35+4

25 + 1520 + 12

25 + 35 + 12 =72

Finding out how certain calculations were madeand problems of arithmetic solved in the past is not

/L - mere idle curiosity. Sometimes examining methodsthat have been dropped in favour of faster and moreefficient processes brings into relief or reveals newand interesting properties of numbers and relationsbetween them that all too often pass unnoticed.

142

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What is XLVIII by CCLXXXVIII?

Most of us are familiar with the ancient Romannumerals, although we no longer use them to calcu-late with. In fact, they remained in use throughoutthe West up until the fourteenth and fifteenth cen-turies, when they were replaced by the quicker andmore economical Indo-Arabic system. (And itshould be pointed out that even then the-adoption ofthis latter system by Western Europe was hinderedby a general prejudice against anything derivingfrom the world of Islam.)

To bring home the great advantages of the Arabicsystem, let us see how long it takes to multiplyXLVIII by CCLXXXVIII in comparison with 48 to 288.No documents or accounts illustrating how theancient Romans coped survive. It must have beenquite complicated!

One way of proceeding might be as set out in thefigure below:

Roman numerals with matches

Ancient Romans had no matchsticks: to light a fire,they used a flint (lapis ignarius or pyrites), whichproduced sparks when struck with either anotherstone or a piece of iron. These sparkswould fall ontotinder (igniarium)-some easily flammable mate-rial, usually dry wood. It is not easy to imagine howmuch time this rather crude system of creating fivetook. How much effort must have been wasted indamp or rainy weather. Doubtless no one everdreamed of a day when a flame on the end of a littlestick of wood was a perfectly normal and easilyattainable thing. Ironically, then, these same "mira-culous" little bits of wood are now ideal for us to useto represent ancient Roman numerals.

Below are some examples of Roman numbers.However, the sums are not quite correct. By movinga single match in each case, is it possible to obtaincorrect mathematical expressions?

F: XLVIII X CCLXXXVIII

XLxC =MMMMXLxC =MMMMXLXL = MMXLxX = C C C CXLxX = C C C CXLxX = C C C CXLxV = C CXLxIII = C X X

VxC = C C C C CVxC = C C C C CVxL = C C LVXX = LVxX = LVxX = LVxV = X X VVxIII = X V

IIIxC = C C CI II1Ix C = C C CIIIXCI= CCC!Ilixi = C L

IIIxX = XX X1IIxX = X X XI IlxX = X X XI IxV = X V1ixill = I X

OI

Th soluto . n

The solutions

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are as follow:

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MMMMMMMMMMCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCLLLLLX XXXXXXXXXXXXXXXXIV= M MMMDCCCXXIV

143

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A system for the human brain

As we saw when we tried to multiply 48 x 288, eventhe simplest calculations are laborious when usingRoman numerals. As time went on, it became in-creasingly uneconomical to retain them, and whenin the thirteenth century the decimal-based Indo-Arabic system was introduced to the West byLeonardo Pisano (better known as Fibonacci), theRoman system was slowly discarded. The successof the new system was due entirely to its greatsimplicity: any number could be expressed in termsof the ten figures O, 1,2,3,4,5,6,7,8,9. The positionof each figure in any given number was what mat-tered: the quantity represented by each digit wasdetermined by its position in the number as conven-tionally written down. For instance, taking the num-ber 2222, the 2 second on the right represents thetens, the third from the right the hundreds, thefourth (the first 2 on the left) the thousands, and soon. The criterion here, therefore, is one of position,whereas in the Roman system it is mostly a matterof addition-that is, each symbol always has thesame value regardless of its position: a symbol ismerely added to the previous one when it is of lowervalue, or subtracted from the following symbolwhen it is of a higher value. Where symbols arerepeated they are simply added together, but nosymbol is repeated more than three times.

Despite early opposition the new decimal-baseIndo-Arabic system, was so clearly superior that iteventually won the day. Its superiority lay not in itshaving ten figures, however, but in the advantagesof a positional system over an additional system.

The "farmer's system of multiplication"

In the long history of mathematics many differentways of doing calculations have been evolved, andwe never learn the less common ones. An exampleof such a case is the so-called farmer's system ofmultiplication, which in fact does have considerableinterest and works according to its own logic. See ifyou can identify the underlying principle in theexample below. Let us suppose we are to multiply14 x 24.

14731

244896

192

336

14 x24 =

56280

336

If we multiply the same figures in the normal way(see right-hand calculation), we can check that theresult is indeed correct. How does the "farmer'ssystem" work?

A look at the left-hand figures will show that the"14" column numbers are just divisions by two,forgetting the remainders (2 into 14 goes exactlyseven times, into 7, three times, and into 3, once:and here the division by two stops), while those inthe "24" column are a progressive multiplication bytwo (2 x 24 = 48; 2 x 48 = 96; 2 x 96 = 192).

The total of 336 is made up of the sum of all thenumbers opposite odd numbers in the "14" col-umn: 48 + 96 + 192 336. Note, too, that the sameanswer would be given if the factors were reversed:

2412631

142856

112224

336

We deliberately took two even numbers as anexample, but in fact the same method can be ap-plied equally to odd numbers.

This curious way of multiplying offers the oneadvantage of enabling us to forget about remain-ders. On the other hand, it is somewhat cumber-some and slow. Hence it is rarely used, and themethod normally taught at school proves quickerand more reliable. The "farmer's system" is in-teresting as a relic of an ancient use of the binarysystem, using only two figures.

Let us now look more closely at the chief featuresof this.

Only two symbols

Initially, the binary system of numbering, in whichany number can be expressed by means of only twosymbols, "O" and "1 ," appears the simplest. Yet it isimportant here to consider what we mean by "sim-ple." Although it does only have two symbols, inpractice it works less neatly and quickly than othersystems.

Let us examine some examples of binarymathematics, for it is always useful to have someacquaintance with it. (Incidentally, the word "bin-ary" comes from the Latin bis, meaning "twice.")Take the number 101c2,, which would read "one-naught-one" (in this sort of exercise it is advisable toindicate the base) in order not to confuse it with thenormal decimal "one hundred and one." This num-

144


ber tells us that-reading from right to left-there isone unit of the first order, no units of the second, andone of the third. With only a few, and simple, cal-culations, it is possible to move from the binarynumber to the corresponding decimal, or a numberin any other base.

For example, to find the equivalent in the decimalsystem of the binary number 11001(2), we need onlymake a quick calculation:

11001(2) = (1 x 24) + (1 x 23) + (O x 22) + (O x 21) +(1 x 2°) == 16 + 8-+0 + 0 - 1 25

One could thus write: 11001(2) = 25(10)It does not matter which numerical system one

uses, the actual mathematics is not affected. Thisindicates that the rules of arithmetic are indepen-dent of whatever particular base may be used.

Though there is a variety of bases, then, it may besaid that they are all just different ways of symboliz-ing and presenting the same "argument." Numbersand the activity of counting must always obey thesame laws, regardless of whetherthe person count-ing is a top boffin or a wild Amazonian Indian !

Is there a reason?

Is there a reason why, whatever the base, if onemultiplies a number by 10 (which, remember, weread in decimal as "ten," but which in binary is"one-naught," and in yet other numerical systemswill be different again), the result will always be thesimple addition of a naught?

Consider. Adding a naught in a "positional" sys-tem means passing to the next higher order: andthis process is expressed by moving all the otherfigures one place to the left.

Anyone with any familiarity with the abacusknowsthisfrom physical action: foron an abacustomultiply by ten involves moving everything onespace to the left and leaving the right-hand endspace empty, to indicate naught.

The hidden binary principle

What is the principle behind the curious method ofhandling numbers, which previously we referred toas the "farmer's system"?

Having now given some explanations of the for-mal features of the binary system, we are in aposition to identify that principle. Let us take twonumbers-not too high, say, 35 and 11-and multi-plythem as already illustrated: dividethe number atthe top of the left-hand column (35 in the table

following) by two, disregarding the remainder, andat the same time multiply that at the top of theright-hand column (11) by two. Then cross out thelines that consist of even numbers in the left-handcolumn and add up the remaining numbers in theright-hand column. Note that it does not matterwhich number is on the top of the right-hand col-umns. The important thing is that the lines crossedout are those that correspond with the even num-bers on the left.

110001

35 x 1117 228 444 882 1761 352

385

Let us now convert 35 into binary (in other words,do the reverse of what we did earlier).

35 21 1 2

18 20 2

0220 1

Thus 35(1o) = 10001 1(2).Now place these binary figures beside the column

headed "36," with the first figure at the top-andobserve how the naughts correspond to the cros-sed-outeven numbers. Similarly, the final answer tothe multiplication, it can be seen, was reached bytotalling the numbers of the right-hand column cor-responding tothe binary "-1s.

By now it should be fairly easy to see the principleunderlying the "farmer's system" of multiplication.When the binary version of the number at the top ofthe left-hand column is written vertically beside it,the right-hand-column numbers corresponding tothe binary naughts are crossed out. (Incidentally, itis perhaps worth recalling that any number multi-plied by naught equals naught.)

A system for a computer "brain"

In order to understand what numbers really are, it isuseful to compare different numerical systems. Thesymbols by which we represent numbers, andwhich form the different bases, are purely conven-tional. Various factors may result in one systembeing more practical than another, but there is one

145


in particularthat makes a vital difference: "position-al" systems prove far more effective than those thatare essentially just a chain of additions. "Positional"systems are all in fact equally good; and aswe heardfrom Aristotle earlier, the adoption of a ten-scalesystem as the norm is really due to the way our ownbodies are made. However, the binary system canalso in its own way become "quite complex," be-cause it alone is applicable in modern digital com-puter work. Most of us have at least a rough idea ofwhat a computer is: a fairly intricate arrangement ofvarious basic components. These components (cir-cuits and elements that can be used inside thecomputer) are such as can exist in two states: in anygiven circuit, current may pass or not pass (open/closed); an electronic tube may be either alive ordead, a certain material may be magnetized in onedirection or in the opposite direction, and so on. Itcan be helpful to think of these two states as "O" (forthe "off" state-when current does not pass) and"1" (for the opposite state). It is in general true thatthe binary system is positional, but nobody wouldever dream of doing normal calculations in thatmode; it would take far too long. We should realize,though, that what is a limitation for us is an advan-tagefor an electronic machine (such machinesoper-ate atthe speed of electric current-which for every-day purposes we can think of as instantaneous).

How to count in binary on your fingers

Human fingers are eminently suitable for countingwith "O" and "1"-curved for the former, perhaps,and outstretched for the latter. In this way one cancount from 1 to 1111111111(5) corresponding to10231o). Start by folding all yourfingers, thus form-ing two fists, with the back of your hands upwards.For "1" extend the little finger on your right hand;for "2" (written 10(2)), fold back the little finger andextend the ring finger of the right hand; for "3"extend both the little and the ring fingers ... and soon.

0 1 1 1 0 1 0 0

How could one represent 0111110100(2)? The left-hand figure below suggests a way.

With a little practice it is possible to do binarycalculations. Whether we do them on our fingers oron paper, however, they are still slow, crude, andtedious. Such drawbacks cease to matter, though,when a machine like a computer is involved-for aswe have already seen, they work at the speed atwhich electric current travels.

A game of strategy in binary

There is a game called Nim (probably Chinese inorigin), which is based on a binary concept. It is fortwo players, and the rules are as follows: Piles ofmatches are placed on a table, and each player inturn removes as many matches as he or she wishesfrom any one pile-the whole pile if so desired. Thewinner is the player who removes the last match.

Try this game out, and you will find it is not assimple as it sounds, especially in the early moves, asknowing how many matches to take away each turn,in order to be the last to remove one, requires somecunning.

Let us imagine we have four piles, one with 7matches, another with 5, the third with 3, and thelast with just 1 match.

Say we kindly agree to let the reader start: heremoves 2 matches from the pile of 5:

146

1 (2)If

0 S

11 (2)

* 0 0 0 *

11(2)

111(2)

It is now our turn, and we take 6from the pile of 7,thus leaving only 1 match in that pile. After the firstround, the table is therefore, 1, 3, 3, and 1.

1(2)

* S 0

* 0 * if . * 0 1(2)

What can the reader do? Does he yield victory?Nim has a secret inbuilt binary strategy that can

be used right from the start. Letting the reader beginwas not such a polite or kind thing afterall: infact, ifwe had had to start, itwould have been hard forustowin. What is this hidden strategy?

Let us transcribe the numbers of the four piles (7,5, 3, and 1) into binary:

1(1o)3(1o)5(1 of7(10)

1 (2)l 1 (2)

101(2)

111(2)

In the first column, from left to right, there are twounits, in the second, again two, and in thethird, four.When this situation arises it is impossible for the

11(2) player who starts to beat a knowledgeable player.All that is needed is an even number of units in eachcolumn. And our game had this from the outset (seecalculation below left); thus, in letting the readerstart, we were actually giving ourselves the victory.Whatever our reader did, he or she was bound to

* * 11(2) create an odd number of units in some column.Knowing this, let us look more closely at what

happens when the reader removes 2 matches fromthe pile of 5 (see calculation below, right): thereremain piles of 1, 3, 3, and 7, which in binary wouldbe 1(2), 1112), 11)2), and 111 (2).

Let us then suppose that the reader removes oneof the two piles of 3 matches. We then take thesecond pile of 3. The following diagram illustratesthe state of play at this stage.


1(2)

1 (2)l 1 (2)

10102)

111)2)

224)10)

147

1 2M11(2)1 1 12i1 1(2)11142,

1 34( 0

0 0


There are therefore two columns with an oddnumber of units. Our move (taking 6 matches fromthe pile of 7) restored each column to an evennumber of units.

Finally, we have also removed the other pile of 3:

1(2]

1(2)

11(2)

1 (2)

20(o)11(2)

1(2) The end came when only two piles of one matcheach remained, and itwasthe reader'sturn to move.

24o1o)

The reader's next move (taking away one of thepiles of 3) inevitably once again created an oddnumber in either one of the two columns, or both:

1 (2)

Another problem

Would the opening player win if the piles consistedof 10, 4, 7, and 2 matches respectively?

First, change these numbers into binary, thentotal the units in each column (in decimal):

10(10) =4(10) =

7010) =2)1o) =

1010i2i

100(2)111(2)

10(2)

1231

This gives us two columns with an odd number of11(2) units. So the opening player can win, as long as he

makes no mistakes during the game.Is it possible for the reader's first move to resu It in

an even number of units in each column? (Sugges-tion: try removing 9 matches from the first pile....)

Logic and mathematics:falseltrue-naught/one

We have already said that arithmetic is essentially alanguage-a convention of symbols with no par-

1(2)

13 (1o)

148


Computers use binary logic, asthe mass of electronic circuits ofwhich they are constructed existonly in two modes: either theyallow current to pass, or they donot. These circuits can thus bereferred to as logical circuits, orlogicalgates. There are six basictypes, shown here on the rightwith the symbols conventionallyused to denote them. As can beseen, each has one or two inputelements (A, BI and an outputlL-ll, and in the explanatory table,"1 " represents a passage ofelectric current and "0" ablocked current. For our presentpurposes we have just shown afew examples of logical gates,but readers who wish to learnmore should consult thebibliography at the end of thisbook for books on mathematicallogic. In an AND gate(corresponding to mathematicaladdition and a conjunctionbetween propositions), theremust be two input impulses 11,1ito produce an output of (11;otherwise the output will be (10.In an OR gate on the other hand,only one input impulse isneeded to produce an output ofI1): 1, 0; or 0,1 . A NOTgatesimply reverses the input, and sothe logic of the gates develops.As the explanatory table shows,1 and D are the "building bricks"of computer logic.

ticular concrete referents: the natural number 3, forinstance, can mean three apples, eggs, or evenideas. Arithmetic is thus a code of rules and indica-tions for setting these symbols in relation with eachother (addition, subtraction, multiplication, and divi-sion), while the significance we find in the results isessentially that which we attribute to them. There isthus nothing to stop us from thinking of the numberI1"V as "true," and of naught as "false." Now, thesetwo figures are the symbols of binary arithmetic:hence we are now ready to transform logic into aprocess of logical calculation.

Modern computer language is also based on thedigits "O" and I1 ." Thus the underlying logic of thisnew technology remains purely mathematical.

Appendix of games with numericalsystems

In line with the rather discursive style in which wedealt with the origins of numbers and arithmetic,

here is a series of problems presented as games, inwhich we can all try out our facility with numbers-or rather, our ability to master the language ofnumbers. There is one thing to bear in mind, how-ever: these numbers are not given in a decimalbase. Thus you will have to vary your numerical"points of reference" and your normal sense of"home territory" in simple calculations, and freeyourself of habits of mathematical reasoning thatare less than absolute. Sometimes stepping outsideour usual acquired mental habits and trying to docalculations by methods with which we are notfamiliarcan be not only entertaining but also helpfulin giving us a better mastery of mathematicalreasoning. It enables us to see numbers, and therelations between them, in a new light.

Some of the tests relate to concepts discussed inthis section and are similar to the examples usedas illustrations. Others require more individual"input." Even these, however, can easily beworked out with some concentration and straightthinking.

149

The point; numbers and imagination

oHow does the series continue?

111101 111

0Using the "farmer's system" of multiplication, work out the following:

a) 37 x 9 b) 28 X 13

WhtiO h isn ubr ern nmn

What is the missing number, bearing in mindthat the squares have an operative purpose?

,1 / 110 1

.4L

A.+

/ 101,e/e

0DSupply the missing number.

1

a)

11011

I'n d, ,...

110

b) i

. 110110 I 10010

.C

150

1001 7

c) 49 X 7

11

1001

1001 A

I

i

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1

9 1


S m ....... missing n ..

|Supply the missing number. Insertthe missing

I IHo d the series - -c.iu e.-. i


-. .. -.. .. . ...

n8)

Whwhi

L.-

ich of the following numbers are even, andich odd?

a) 110(2)

b) 1011M2)

c) 2012(3)

d) 1021(3)

e) 2013,)

f) 112(4)

g) 42(5)

h) 103(5)

- - ..-. Ii-.- ..... .............. .. . . .151

151

numbers. |

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Tests 1-8 on pages 150-1: answers and explanations

1) 1011. As we can see, in this first exercise the numbers areexpressed in binary form, and each is the sum of the previousone plus 10: thus 11 = 1 + 10,101 = 11 + 10. Since

11 + hence 001= 111 +10 10

101 1001.Thus, to establish the number that continues the series, we

need simply add IO to 1001: 1001 + 10= 1011; the answer,therefore, is 1011.

At this point we might, for interest's sake, work out whatthiscorresponds to in decimal. Remember that 10 in thebinary system equals 2 in the decimal system, while 1 equalsone factor in both systems. Thus the number that continuesthe series is 11: 1 (+2), 3 (+2), 5 (+2), 7 (+2), 9 (+2), 11.

2) Answer 333, 364, 343.

a) 37 x189421

: 91 83672

144288

b)28x 1314 267 523 1041 208

364

c) 49 x2412631

333 343

3) 100. A careful examination of a will show that the "head"figure 111 represents 11 + 100. The two flaglike squaresemanating from a indicate an addition. In b, however, if thenumber in the "head" (1) is the result of some calculationbetween the two numbers in the flags, we can hazard that thelower number has been subtracted from the higher, since I isless than both 110 and 101. Therefore:

110 -101

It is therefore probable that in c,too, the relative positionsof the flags indicate a subtraction 1 101 - 1001 = 100),Hence the missing number in c is 100. It is not difficult to seethat here, too, we are working in binary and so can establishthat 1101)2) = 131o1) and 1001)2) =S 91o), and hence 13)1a) - 90)o- 4n0) = 100)2)-

4) 1. In square a the series goes 1 x 11 1, then l 1 x 11 = 1001since

11 x11 =

1 111 -

1001andthenl001 x 11 = 11011; inbitisnottoohardtosee

that a similar principle has been followed, with simply anextra naught. The missing number is thus 10. We can checkthis by switching all the numbers into decimal: thus in a 1)2)1(10), 1112) = 3(10),1001)2) - 9)10), 11011(2) 271o0); while in b,however, 10)2) = 2)1o), 110)2) = 6(lo), 10010)2) = 18(o1), 1 101 10)2)

= 54t101.

5) 112 orO. Firstwe must identify the numerical system-which,since the highest figure is 2, we suspect is ternary. The series

proceeds by the addition of 2 at each stage. Thus: 2(3) + 2)3) -

11)3) and 22)3) + 2(3) - 101(3). Hence 110)3) + 2(3) 112(3). Inthe decimal system this corresponds to 2,4, 6, 8, 12, 14; thusthe final number is 14 and the first number (before 2) is zero.

6) 102. It will be noticed first of all that the highest figure amongthese numbers is 4; we are thus involved with a systemwhose base is at least higher than 4. Already in the firststep-from 1 1 to 14-we see that 3 units have been added.Now let us reflect: in what numerical system would we, bythe addition of 3, end upwith this result, 33 + 3 = 417 Clearly,if 3 plus 3 equals 6 and is written carrying 1 (which, added tothe 3, gives the 4 in 41), we can only be in a quinary system (inbase 5). Thus to supplythe number that continues the series,we need only calculate 44ro + 3) 5)-giving the result 102(5).Transcribed in the decimal system, the series would go 6, 9,12, 15, 18a 21, 24, 27.

71 26 and 51. Here let us first note how the series develops:11-15-22... .While 11 to 15 leaves a difference of4 units, 15to 22 is more. So in what system could 15 + 4 - 22? A shortcalculation (5 + 4 = 9- 2 = 7) will tell us that we are in base 7.Let us then confirm this discovery by looking at the followingnumbers: 33(7) + 417) = 40)7) and this coincides; 40)7) + 4)7) =

44(7); thus the two missing numbers are given by 22)7) + 4)7) =26)7) and 44)7) + 4)7) = 51(7).

The reader may transpose this into decimal if he or she sowishes.

8) We might answer this problem by transcribing all thenumbers into decimal:a) 110)21 ) 1 x 22 + 1 x 21 + 0 X 20 = 4 + 2 = 6)10 );b) 1011(2) -1 x 23 + 1 x 21 -rl x 20= 8+3 = 11)1o0.

This much can be said regarding the binary system: if thelast figure of the number in question is a 1, then it must be anodd number; and if a naught, then the number is even.c) 2012) 3)=2x33+0x3

2+1 X3

1 +2x3 0°54+3+2-

d) 102113) = 1 x 33 + 0x 32 2+ 2x 31 t+ 1 x 30 = 27 + 734t(o)-

Note how in ethe number ended with an even number butcame out odd, while the reverse happened in d.e) 2013) 4 >)2x4 3 +0x42 +1 x4 1 +3x 1=64+4 +3=71(10);f) 112)4)= I x42 + 1 x41 +2 x4 0 = 16+4+2=22)1or.

In e the numberwas odd in base 4 and also came out odd indecimal; and the same also occurred in f, Let us examine thenumbers in base 5.g) 42) 5 ) = 4 x 51 + 2 x 50 = 20 + 2 = 2211o(.h) 103) 5 ) -4 x 52 + 0 x 5' + 3 x 5° = 29 + 3 = 32)10 o)

The final figure of the quinary number in g is even, and sotoo is the corresponding figure in decimal; whereas the lastfigure in h is odd, but the decimal equivalent is even. As thereader may have gathered, the last figure is not a reliablecriterion for establishing whether numbers written in anumerical system with an odd-number base are even or odd.

Is it possible to find a criterion for telling whether a numberexpressed in any base is even or odd? We shall attempt toformulate one: if the base is given by an even number, then anumber is identifiable as even if the unitary digit is even, andas odd if the unitary digit is odd. However, if the base is givenby an odd number, then the number in question is even (orodd) if the sum of the values of each digit, counted in decimal,is (respectively) even or odd.

152

Pick your own gamesAnd still it is not finished How well / know it, the whole book is like that! /have wasted a lot of time on it: a perfect contradiction remains mysterious

to wise and foolish alike. This art, friend is both old and newMephistopheles' reflections n Goethe's Faust

A brilliant solution

Some years ago, Omar Fadhami, the emir of a smallstate in the Persian Gulf which until the oil boom hadbeen poor and little known, but which had sincebecome rich and great in world esteem, paid a visitto a certain European country. Naturally he wasaccompanied by a retinue of wives, advisers, minis-ters, children, and servants.

One day he decided to give his court presents, andat the head of a kind of procession he went into thesmartest and biggest store in the capital. The cour-tiers swarmed around the laden counters.

The emir was very satisfied, until he noticed thatthe children were squabbling overtoys, sweets, andpicture books. An equable man, however, he did notthink it appropriate to intervene. But he began to beannoyed when he observed that his wives and theladies of his retinue were behaving in a similarfashion, fighting over lengths of material, makeupand perfumes. Solima, usually so sweet-natured, hitout at young Iris, who had "robbed" her of a littlevanity case. Things were hardly much better amongthe servants: arguments became louder, faces moreflushed. Omar, pensive as ever, reflected on theirrationality of the situation. Everybody was choos-ing in a hurry, without bothering to think whetherwhat they were becoming enraged over was reallyeither necessary or useful. A moment of reflectionwould quieten everyone down.

At this point he made a decision. Clapping hishands three times, he obtained sudden silence, as ifby magic. Then he turned to the store manager andsaid: "Pack everything for me, and send it to me. Athome, in the quiet of the palace, everyone will seewhat they really want."

The emir's order was speedily carried out by thedelighted manager and the affair was widely publi-cized by the media. Everyone was so happy toreceive everything in the store because now Solima,Iris, the children, and all the courtiers had time tosort things out.

And it is in the same spirit that we offer thefollowing "pick" of games-for each reader tochoose from, to classify, and to evaluate as he or shewills.

Geometrical figures with matches

Two fundamental concepts of geometry are thepoint and the straight line. Upon these-which aregiven, rather than defined by means of otherconcepts-the whole structure of the disciplinerests.

What, in geometrical terms, does a match remindus of? Despite the slight bulge of the sulfur cap at theend, it can be seen as a short straight line; or better,as one of an infinite number of possible embodi-ments of the abstract notion of a "section of a

153

Pick your own games

fig. 7 &1

I

4

iS

An4.

fig. 2.1

.

straight line." This perhaps explains why, ever sincethey were invented, matches have been used forcreating geometrical figures to "kill time." Theirunvarying size and shape, rather than being a draw-back, in fact proves a spur to devising ever moreunusual and curious geometrical patterns. There is,too, a less obvious advantage-though it is for allthat no less potentially fruitful: since each match isidentical, they can be used as units of measure-mentsforthefiguresformedfrom them. This meansthat in the games that follow, mathematical con-cepts and calculations are necessarily also involved.

If, for example, we have twelve matches, we canform various types of polygon all with sides repre-senting a whole number. A square is a polygon withfour equal sides, and twelve is exactly divisible byfour-three times, as we all know. So let us takethree matches at a time and make a square withsides of three units (using the concept of matchesrather than inches, centimeters, and the like). Theresult will be a square divisible into nine smallersquares with one-match-long sides (figure 1).

Taking this figure and modifying it slightly, we canform (figure 2) a cross made up of five smallersquares. In other words, four of the small squaresout of the total of nine in the first figure wereremoved, leaving five.

Let us now consider the following problem: Is itpossible to construct a polygon consisting of foursmaller squares, using all twelve matches?

One solution might be the following. First, make aright-angled triangle, as in figure 3.

The surface of this triangle consists of six squareunits: 4 x 3 . 2 = 6! Thus the problem now is tomodify this six-unit figure to achieve one of fourunits. The solution is shown in figure 4.

fig. 3

./) is* /A.

' ./.' I

-2?A t" ....V - h 4 ¢

fig. 4

a

154

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BACKGAMMONMany games developed fromthe classical "royal table"-agame known to the Romans astabula, which was playedthroughout the Middle Ages andinto more recent times in Italy.One of the best known of thesedevelopments is the Englishgame of backgammon whichuses the same characteristicboard, and the rules of whichrelate quite closely to theoriginal form of "royal table."The name "backgammon"comes from "back game"-for,as we shall see, a single man{piece), known as a "Slot,"resting on a point, and hit by hisopponent, has to go back to the"Bar" and start all over again(two or more Blots may be hit inone play).Today backgammon is verypopular asagameall overtheworld. A game for twoplayers-although there is aversion, chouetfe, for up to fiveplayers or more-it has adistinctive board (see figure a),

with 12 wedge-shaped points ofone colour and 12 of a differentcolour. There are 15 white orlight-coloured men and 15 red ordark ones. The board is dividedinto four tables, each playerhaving an "inner table" and"outertable." Each table has 6points, those in the innertableare numbered from 1 to 6, thoseon the outer from 7 to 12. Eachplayer has 2 six-face dicenumbering 1-6; there is alsoanother die for double scoring,also six-faced, numbering 2. 4, 8,16, 32, and 64, usually usedwhen the game is played forhigh stakes.

* The men are distributed on theboard as shown in figure b(overleaf). Starting from thislayout, which may atfirst seemodd but which in fact is designedto make the game quicker andmore interesting, white movesfrom his opponent's inner tableto the far side of the board,across to his own outer tableinto his inner table. Red movesin the opposite direction. Thedividing space of the two tables

(inner and outer) is called the"Bar." Each player moves hismen according to the throw ofthe two dice into his inner table,and when all are there, hethrows or bears them off (startsto throw off men from pointscorresponding to dice thrown).The first player to remove all hismen wins.

* To decide which player startsthe game, a single die is cast thehigherthrow deciding the issue;ties are rethrown. For eachmove, both dice are thrown. Thecount begins on the point next tothe one on which the man isresting. Numbers on both dicemust be played if possible. Twomen are moved according to thescore of each die (one score perdie), or a single man can movebased on the score of one die,then continue on the score of thesecond die, but the two numberscannot be added and played asone move. In the event of adouble score (4 + 41, eachcounts twice, so the number ofmoves also doubles. The movesmay be effected by the same

man Ifourtimes), or by two,three, even four.If, having thrown the two dice, aplayer can only use one of thetwo scores, it is the higher scorethat must be used-that is, solong as such a move is possible;otherwise the lower score mustbe used. In all other casesplayers must use both scores. Inother words, a player must playone or other score.

* Other basic rules of the gameare as follows:A point with two or more men onit is blocked against anopponent, although it may belumped, and when a playercannot move because of blockedpoints, he loses the move.There is no limit to the numberof men either player may haveon one point.A Blot, hit by his opponent andsent to the Bar, must reenter inhis opponent's inner table andtravel all around the board to hisown inner table before moremen can be moved or taken off.Thus a man "sent off" can onlycome beck into play when one or

155

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ri

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other of the die scores puts it ona free point in the opponentsinnertable (a point that does nothave two or more enemy menon it, in which case it would beblocked). If the enemy point hasone man on it, it is a Blot andgoes to the Bar in turn. Thereturned man (or any other onthe board) is then movedaccording to the score of thesecond die, but only providedthe player does not have anyother men at the Bar.

* As mentioned, the final stagesof the game consist of theremoving of men from one'sinner table. This can onlyhappen when a//the men still in

play are assembled together inone's inner table. Two kinds ofmoves are possible:

11 One can removeone ortwomen from the board, dependingon the throw of the dice, forexample, if one of the numbersthrown is higher than thehighest-numbered point onwhich a man is resting, the nexthighest can be removed (forexample, if one die scores 6, andthe highest man is on 5, it can beremoved from the board.) Onecan, if one wishes, take the lowerscore fi rst (for example, in theevent of a 5 + 6 score, one couldremove the single man on 5,then that-or one, if there are

several-on 4, to make up the 6score).

21 On the other hand, one mightwish to reduce the number ofmen on the higher points ofone's inner table (S and 6), or toturn a Blot into a "house" (twoor more men on a single point).This is most important whenone's opponent has a man at theBar-for on reentering thegame, he could force an enemyBlot back to the Bar in turn, fromwhere it would have to goaround the whole course again.As long as every point in one'sinner table has a "house, it isimpossible for an opponent tomove from the Bar.

t The skill in this absorbinggame lies in impeding theopponent's progress as much aspossible, hitting Blots andbuilding up "houses," whichmake it difficult or impossible forthe other player to proceed or toreenter from the Bar. At thispoint it should also bementioned that if one managesto build "houses" (to group atleast two) on points 7 and5-so-tslled key points-one'sopponent becomes greatlyhandicapped (and even more so,the more houses one forms onthe opponent's outer table).What requires most skill isgauging how safe or otherwise itis to leave one's own menexposed as Blots. In fact, thereare manuals that listexhaustivelythe variouspossible permutations of themoves during a game, giving theprobabilities for and againstcreating a Blot in any situation.

* There are three types ofvictory: a) single win, whenone's opponent has also begunthe process of removing his menfrom the board, having alreadyremoved at least one; bigammon, when the opponenthas not yet removed any of hismen; and cl backgammon, whenthe opponent still has one ormore men in the other player'sterritory or atthe Bar. In theevent of a single win, the winnertakes the straight stake; with agammon, double the stake; andwith a backgammon, triple thestake.

* The single die (alreadymentioned) for the doublingversion of the game is only usedwhen the game is played formoney (and thus a stake hasbeen fixed). A player who feelshe or she has the upper hand atany point can set this die to 2,thereby doubling the stake; if theother player at any time duringthe course of play thinks he hasthe upper hand, he can setthedie to 4 (double the alreadydoubled stake). This cancontinue up to 24 times thestake-though alternating eachtime between the two players(thus, A, 2; B 4; A, 8; 8, 16; andso on). A player can refuse adouble at any point: in whichcase, he pays the stake alreadyreached, and a new game isbegun.

156

-v -e11

PUFF-ALEA ORTABULAPuffWhen a game becomesestablished in the "heritage" of asociety, an indication of itsvitality is the way it fragments.so to speak, into all sorts ofvariants. This happened, forinstance, with the ancient andvery widely played game ofbackgammon (see pp. 155-1571.One quite common version ofbackgammon, played in Englandand Germany, is generallyknown as puff. Anyone familiarwith the rulesof backgammonwill easily graspthe rules of thisvariant. It is always played bytwo people, with 15 pieces each:one set white (or some otherlight colour), the other black (orany dark colour); two dice areneeded. Basically the same asfor backgammon, the board, onright, has spaces. marked withcapital letters, corresponding tothe arrows. Puff is a less flexiblegame than backgammon. Thisdoes not make it any slower,however: games can sometimesbe short, but they can equally belong drawn out.Puff is related closely to othergames besides backgammon:for instance, to the Frenchtric-trac, which acquired acertain notoriety after it wasplayed by Niccolo Machiavelli(1469-15271 during his enforcedleisure in San Casciano; and toan extremely ancient game,known in Roman times as aleaand very widespread throughoutmedieval Europe. Puff hasbasically the same rules as thislatter game.

Ales or tabula"Alea sive tabula" ("the dicegame or the game board"): thisreference comes from lateRoman times, concerning agame that has rightly been seenas the ancestor of a number ofgames using a board and dice.Board games were alreadycommon in Greece and the Eastbefore the Romans conqueredthere and began to adopt boththe culture and the everydaycustoms and activities theyencountered.It is not known when the scoreobtained from thrown dice wasfirst used to determine themoves of objects on a board.What we do know, though, isthat by classical times alesliterally. "a die") also meant aboard game played with dice.Referring to this game, thefamous historian Suetonius170-140A.D.), author of thepopular Lives of the Caesars,

S Q

i OUTER TABLEM

w V ~U TRp 0 N M7x

INNER TABLE

L.A

used the term alea and nothingelse.In the hurly-burly of life inRepublican Rome, there werepeople who earned their livingas professional dice players,called aleatores. By Cicero's dayit was considered a somewhatdisreputable profession,although by all accounts it wasmostly old men who playedgames of this kind, since theyrequired little physical strength.Times must have changedconsiderably, then, by the age ofthe Emperor Claudius (45-54A .D.-who, Suetonius tells us,was so fond of alea that he had itfitted into his carriage, to playduring long journeys. Claudiuswas indeed so expert at thegame that Istill according toSuetonius) he was able to write atreatise on it. Unfortunately thisbook, which would have been ofno small interest, has been lost.The game spread throughout theempire and is in fact played eventoday in the area around theAegean, under the name thvil.Anyone who saw the filmTopkapi (a successful film of the1960s, starring Melina Mercouriand Maximilian Schell) willprobably remember the scene inwhich two people are seenplaying tavli. We have no directdetailed account of the rules oftabula or alea. But from thescraps of information we dohave, we are able to more or lessreconstruct the game, filling inthe gaps with rules that seempragmatically necessary.

I Basic facts. There were twoplayers and three dice (whiletoday, puff is played with twoonly>. Each player had 15 pieces,one set black, the other white,and these were arranged oneboard that, though it may nothave had 24 wedge-shapedmarkings, would have had 12 -12- 24 spaces of some kind,similar to backgammon.

B IC D

* The dice. The ancient Romansused two sorts of dice: tesserae,which had six sides, marked 1,11,III, IV, V, and VI; and ta/i, whichwere oblong in shape (usuallymade of joint bones, rounded ateither end, and marked only onthe other four sides. On one sidewas a dot, called unio (in Latin,"one", an ace known as "thedog," canis; on the opposite sidewere six dots (senio - "six"); onthe remaining two sides, threeand four, tern/c and quaternio.During the game, four taliandthree tesserse were used. Andfrom this we can guess fairlysafely that alea was played withtesserae rather than tali.Normally, the dice were shakeninside a small tower-shaped box,known as the fritilubs or turricula,and thrown onto the board. Theluckiest throw was called Venus,/actus venereus, or basilicus andconsisted of 3 sixes (if tesserae)orthe score of the taliwhen eachlanded with a different face up.The unluckiest throw, known asjactuspessimus or damnosus, oralso cans or canicula, was threeaces (with tesserae) or when allthe ta/ifell the same way up. Theother throws were worth whatthe numbers added up to, If oneof the ta/i landed end up, it was"cock dice," and the throw wasrepeated. When throwing dice,the custom was to express awish, say the name of one's truelove, and the like.

' How to play. Alea was a sort ofrace in which each player startedin tu rn at space A and proceededanticlockwise, according to thedice throw, towards X, obeyingcertain rules. The winner was thefirst to remove the last of his orher pieces from space X. Thescore for each throw would beworked out roughly as follows:

all scores from any dice throwwere valid;- apiece could be moved asmany spaces as the total of the

E F G H

points scored by the three dice;- twop/eces could be moved inone go, one as many spaces asthe total of two dice scores andthe other the number of spacesof the third die score;- or similarly, threepieces couldbe moved, according to thescores, respectively, of eachindividual die.

* Important rules.1 (If one player has two or morepieces on a single space, theother cannot use that space.2) If the fall of the dice lands aplayer's piece onto a spacealready occupied by oneopponent piece, this latter maybe removed and put on thecentral bar. This means that hisopponent must bring that pieceback into the game, starting at Aagain, before moving any otherpieces.31 No player may go beyondspace L until all his or her piecesare in play.4) No piece can be taken from Xuntil all that player's pieces havepassed R.51 All points scored must beused, as we have described: it isnot permitted to use part of thescore of one die to move a piece.Points that Cannot be used (forexample, because of spacesoccupied by enemy pieces) arelost.

Another keen player of a/ea, sotradition has it, was theBy2antine Emperor Zeno1474-491 A.D.), The colour plate,opposite page, reproduces thestate of play during one game:Zeno was playing the darkpieces: his next score was 2, 5,and 6.

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fig. 5

fig. 8

160

Adding three matches ...

1) Figure 5 is of an equilateral triangle formed bythree matches: by adding another three, it is possi-ble to form three triangles that, taken together withthe first, make four in all! How!

The answer is illustrated in figure 6.11) Figure 7is also of an equilateral triangle-thoughthis time each side consists of two matches. If threeare added, how many triangles will there be?

Normally the following answer is offered: Theextra matches are laid inside the triangle, so as toform four small triangles; the answer thereforewould be four. In fact, though, there are five, sincethe arrangement of the four is such as to form a fifth.A complete answer would thus be: By adding threematches, onewould makefive in all-four small andone large (figure 8).

fig. 6

fig. 7

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Removing three matches ...Four squares together form a single larger square(see figure 9); with three fewer matches, we canmake three diamonds! How?

The solution is shown below (figure 10).

fig. 9

p.

Let's play with squaresWith twenty-two matches we should be able todevise all kinds of quite complex geometricalfigures. But let us take one of the very simplest, thecommon square. How many squares with sides of

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one match length can be made with twenty-twomatches? After various combinations, we haveended up with a maximum of eight (figure 17).However, we shall not be surprised to hear that ourmaximum has been beaten: we are very willing torecognize our limitations. By constructing two 3-dimensional cubes with twenty matchsticks we canobtain eleven squares.

Other games with squares can also be tried. For

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fig. 12 fig. 13

instance, what is the most number of matches wecan remove and still be left with four squares? Let us fig. 14proceed methodically. If we take away six, sixteenremnain~-mrore than enough for four squares (see .

figure 12).Even if seven or eight matches are removed, there

are still easily enough for four squares (figures 73and 74 show this quite clearly).

With nine or ten matches less, we are left withfewer options; however, it is still possible to formfour squares with sides of one match length (figures75 and 76).

Abandoning any restriction on the length of thesides of the squares, what is the minimum numberof matches needed to make four squares?

Figure 77 illustrates our solution, using only sixmatches.

The number of squares can increase to infinity,but obviously the number of matches needed tosubdivide the available space will similarly increase.

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Pick your own games

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What are the coins in Peter's pocket?"I have two coins in my pocket, totalling 30 cents,"Petertold his friend. "Butone of them is nota nickel.What are the two coins?"

The friend answered that he must have two coinsof 15 cents, in that case!

"Don't be stupid!" Peter laughed.The friend's answer was in fact along completely

the wrong lines-though perhaps manyof us wouldhave thought similarly. Let us look more closely atthe trick question.

Some problems are simple enough in them-selves, yet become problematic because of the waythey are expressed. And this is a case in point. Let usreread it carefully and analyze the language trap inwhich Peter successfully caught his friend. First ofall, we are told, "I have two coins in my pocket,totalling 30 cents"; then comes the second piece ofinformation, ("But one of them is not a nickel").

Wrong answers are often common in such prob-lems because we are not really listening to thatnegative bit of information: here we are informedthat one of the two coins is not a nickel-it does notsay that neither is!

So a correct answer would be: The coins in Peter'spocket are a quarter and a nickel. As we were told,one of them was not a nickel: and that was thequarter.

How many horses has the farmer got?

A farmer owns ten cows, three horses, and twentysheep. How many horses does he own, if we alsocall cows horses? The quick answer would, ofcourse, be thirteen (the three horses + the ten cows,which we are calling horses). However, thirteen isincorrect.

Here we are dealing with the nature of language:for even though we have agreed to call a cow"'horse," that does not actually make it a horse.

Language is only a system of symbols that we useto designate things, but the system itself is quitearbitrary. If we call a horse "an ox," the creaturedoes not suddenly become an ox. Hence the answerto the question is the straight fact: the farmer hasthree horses, not thirteen at all.

We take language so much for granted that wesometimes confuse the object and the word signify-ing it and therefore fail to distinguish between twodifferent levels of reality.

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A square and a triangle

Sometimes the game is to visualize a certaingeometrical figure without adding or removing anymatches but simply altering the position of one ortwo, in order to create a very different shape.

For instance, the six matches in figure 18 make afairly conventional matchstick house shape: now,can you make four triangles just by moving threematches?

Figure 19 shows how this can be done by placingover the triangular house roof another triangle, thepoints of which rest on the midpoints of the originaltriangle's sides.

fig. 18

The ocean liner

World-famous buildings, churches, towers, monu-ments, old sailing boats and ships, galleys, modernvessels of all kinds are frequently taken as subjectsfor matchstick models.

Figure 20 gives us in outline a representation ofthe prow of a big oceangoing liner, seen from thefront. It is made up from nine matches.

Now, by moving just five of these matches, canyou create a pattern of five triangles?

The solution is given in the diagram in the middleof the opposite page (figure 21).

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How old was Livy?The historian Livy was born in Padua in 59 B.C. andspent most of his life in the capital, Rome. However,as age crept upon him he returned to his belovednative city, where he died in 17 A.D.

A Latin teacher familiar with the life and works ofLivy asked his class the following question:

"How old was Livy when he died?"At once the answer came back, "Seventy-six.""Wrong," the teacher replied. "He was seventy-

five." Why seventy-five? The pupil who hadanswered had added 59 + 17 = 76, without allowing

for the fact that the year naught (0) does not exist.Hence he should have calculated: 59 + 17 - 1 = 75.

"Trick questions" like this rely on never havingconsidered the nonexistence of a year naught!Thus, when we need to calculate the age of some-body born before the birth of Christ, who died after-wards, in an anno Domini, we tend mistakenly justto add the two numbers. Normally, such questionswould be phrased a little more cunningly.Let us, for instance, imagine that Gaius was born

on the ides of April in 25 BC. and died on the sameday, the ides of April, in the year 25 A.D. How old washe at his death?

We would be tempted to answer fifty. Yet thatwould be wrong. He would of course have beenforty-nine, since the year 0 cannot be included.

To understand this better, try to envisage thelifespan of an individual as a straight line dividedinto short stretches: from 3 B.C. to 3 A.D., for example,

the first year of life goes from 3 to 2, the second from2 to 1 B.C., then the third-since there is no year0-from I B.C. to 1 A.D., the fourth from 1 A.D. to 2 A.D.,and the fifth finally from 2 to 3 A.D. Hence one yearseems to have disappeared, but in fact the actualnumber of years lived was five and not six.

Tony's socksTony was somewhat lackadaisical by nature andmore than once went out with his girlfriend wearingodd socks. In the rush of going out, he would fail tonotice that they were of different colours. This parti-cularly irritated Prudence, his fiancee, who was astickler for neatness and smart clothes.

Although it had not in fact been the cause of toomuch disagreement, Tony had nevertheless re-solved to pay more attention to these little matters.As he was getting ready to go to a party that even-ing, there was a power cut, and he found he had toput on his socks in the dark (typically, he had nocandles in the house and trying to find matches inthe chaos in which he lived was hopeless). So he satdown and thought about it. In the drawer he knewthere were ten brown socks and ten light gray ones.How many, then, should he take to be sure of havingtwo of the same colour?

Answer: three.There were two possibilities: either all three

would be the same colour, or two would be thesame and one different, In either case he would besure of having a pair of the same colour.

... and Prudence's gloves

Meanwhile, Prudence had been aboutto lookfor hergloves as the power cut occurred. The thought ofarriving at the party with odd gloves was not to becountenanced! In her drawer she had five pairs ofblack ones (ten gloves in all) and five pairs of white(ten white gloves in all).

Prudence's problem was rather harder thanTony's, for half of the gloves she had to choose fromwere right-handed and the other half left-handed.After a moment's thought, she took out elevengloves, confident that among them there would beat least one pair-both a right-hand and a left-handglove of the same colour.

Now, had she really found a solution? Should shehave taken either more or fewer?

Answer: She had indeed taken the minimumnecessary. If she had taken one fewer, she mighthave found herself with ten right-hand gloves, fiveblack and five white.

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Pick your own games

The mill

Most of the energy used by the ancient Greeks andRomans was human- or animal-powered. Waterenergy (the principle of water turning a wheel by

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falling onto jutting-out paddles) was known, butslaves were so easy to come by that they were usedeven for such crude tasks as raising water andgrinding corn.

In the Middle Ages, when slavery as it had beenknown in antiquity no longer existed, water millswere builtthroughout Europe, replacing the manualmachines used by the Romans. For centuries thestructure of these mills remained the same,although they did change in size and came to bebuilt in different materials and used for a widervariety of purposes. Ever better and strong gearingsystems enabled the simple rotating motion of thepaddled wheels to drive many different kinds ofmachine, from hammers to looms, so that the word"mill" itself came to mean a factory (at least incertain areas).

The very existence of Holland is largely due to thewindmill: with tireless application, they won landfrom the sea by digging drainage dikes, throughwhich stagnant water was pushed out to sea bymeans of energy harnessed by windmills.

The bridge to the island

Figure 22 represents an island in the sea. Yearsback, the Dutch built a windmill on it.

It makes an ideal site, for a wind is always blowingoff the nearby sea. Unfortunately, the one roadlinking it with the mainland has been destroyed byrough seas, and the mill is now cut off.

Is it possible, with two matches, to build a tempor-ary bridge to reestablish the link with the mainland?One complication is that if we align two matchesperpendicularly, we shall find they are not longenough to reach both sides. Figure 23 offers aningenious solution.

A wheel with paddles

The most basic element of any mill is the wheel,turned by some form of energy (animal, hydraulic,or wind), which is linked to a gearing mechanism inorder to turn other wheels, for various purposes, notjust the grinding of corn. In the eighteenth century,for instance, machines built on this principlepumped water out of mines, thus enabling shafts tobe dug deeper. In water mills, the principal wheel isfitted with paddles, ontowhich thewaterfalls, there-by pushing it around.

Take the sixteen matches used in figure 22 toseparate terra firma from the sea, and see if you canconstructc" a water wheel with the necessary pad-dles. Figure 24 shows how this can be done.

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Now for a rather unusual game: set out four match-es in a cross, as in figure 25.

Is it possible to create a square by moving just onematch?

fig. 25

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Claustrophobia

Patricia was an oversensitive girl who was not al-ways able to control her fears and anxieties and seeher own.weaknesses in perspective. Among otherthings, she became agitated when she found herselfin closed spaces: to put it bluntly, she was a bitclaustrophobic. In certain situations she had thefeeling of being trapped and unable to escape. Forher work she often had to travel by train to visit afirm in a nearby town. Shortly before this particularoccasion, a bomb had exploded as a train was goingthrough a tunnel. Now just outside the stationwhere Patricia took hertrain there was a long tunnel.And even before getting on board, she could feelmounting terror. She feared she would not be ableto bear it once she was in the train.

What could she do? She could at least make sureshe was in the quickest carriage, so as to spend asshort a time as possible in the tunnel.

So she asked a guard if there was a carriage thatwould take herthrough thetunnel more quicklythanthe others. The guard said yes, there was one.Which one was it?

Answer: The guard recommended that she sit inthe back carriage; sincethetunnel came immediate-ly outside the station, and as the train would still beaccelerating, the tail carriage would thus traverse itfaster than the others.

fig. 26

Initially, the reader will probably try to think of thematches as forming the sides of a square-in part,no doubt, because this was the case in previousgames. Here, however, the solution is in itself a bitcrude, but it does stretch one's imaginative abilities,forcing them to think of the problem in a slightlydifferent way.

The only answer is to lower the bottom matchuntil a little square is formed by the ends of all fourmatches (see figure 26).

167

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Games with clocks

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(see answers onp. 175)

168

MAGIC SQUARESAre you feeling depressed, evendownright melancholy? See ifthe following game helps raiseyour spirits. Figures a, b, and care of magic squares; the sum ofthe numbers in the boxes isalways 15, whether added alonghorizontal, vertical, or diagonallines. The reader can check thispersonally. In all three examples,a, b, and c, the same "magic"workal Figure dis a square ofthe same sort, but two of thenumbers have been swapped, sothat in some lines and somecolumns the answer is no longer15. The object of the game istodetect the misplaced numbersand put them back in the rightboxes to re-create a "magic"square. A careful examination offigure dwill showthat bymoving the 6 from the topleft-hand box and swapping itwith the bottom right-hand 4, wecan make a square with thesame "magic" as a, b, and c.Perhaps you are still feelingmelancholy, despite this littleexercise? Never mind. We weresimply suggesting a remedyadvocated by sixteenth-centuryartists and scientists to stimulatethe human intellectual faculties.

Their strange allureMagic squares are very ancientgeometrical and arithmeticalfigures. It would appear thattheChinese were the firsttodiscover them: the lu shu (figurec), to which particular religiousand mystic significance wasattached, is recorded as ea rly asthe fourth century B.C. Similarsquares were known very earlyon in India, whence the Arabsbrought them to the West, to bemade known to Christianthinkers in the fourteenthcentury by a Greek monk calledMoscopulos. The peculiarmathematical properties offig. a

fig. d

6 9 2

3 5 7

8 1 4

these geometrical figures caughtpeople's imaginations, so muchso that mysterious magicpowers were attributed tothem-hence their name. Atvarioustimes they have been asource of artistic inspiration,equally fascinating both to thelearned and to the ignorant.Worth mentioning from the timeof the Renaissance are themathematician CorneliusAgrippa (1486-1535), who hadan unflagging passion for magicsquares, and the German painterAlbrecht Diirer (who featuredone in his famous printMelancholia, 15141. Morerecently, too, men'simaginations have been stirredby the mysterious relationshipsof numbers in magic squares.Benjamin Franklin was onlytoohappy to be able to devotehimself to them, and his workscontain complex examplestogether with comments. In thepresent century, the Americanarchitect Claude Fayette

fig. b

fig. e

10 5 6

3 7 11

8 94

Bragdon has based some of hisbuildings on geometrical modelsinspired by magic circles.Superstitious belief in theirmagic properties still survives: itis said that during the war inCambodia, local women drewsuch squares on their scarves forprotection against bombs; andin certain areas of the East it isstill common to find them madeof bone or wood and worn asamulets. This aura of magic andpower is not so surprising onceone becomes aware of theirgeometrical and arithmeticalproperties. Let us now lookbriefly at these.

Definition andpropertiesA magic square is anarrangement of whole positivenumbers, without repetition, in aregular pattern of adjacentsquares, such that each line(from left to right, or vice versa),each column (from top tobottom, or vice versa), and bothdiagonals all add up to the same

fig. c

figure. The figure n determinesthe order (or base, or module) ofthe magic square.Figures a, b, and c, are all of thethird order, because each sideconsists of three squares withthree numbers. Figure e is alsoof the third order: what isdifferent is the total for each line,column, and diagonal. This isnow 21 rather than 15. Lookingat the colour plate on page 171,what will the number on thecentral ball be? We are dealingwith a third-order magic squareand know from the completerows that the total to attain is 21:hence we simply subtract 19 + 5)from 21 = 21 - 14 = 7. Themissing number, then, is 7.Squares can also be devised ofthe fourth and fifth-and evenhigher-orders. The first order isnot really worth considering-asimple square with one numberin itl It is curious, however, thatno second-order squares existbut perhaps if you try to makeone, you'll discover why.

169

2 7 6

9 5 1

4 3 8

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More complex squaresThe higher the order of thesquare, the more taxing, yet atthe same time more satisfying,the game. Try to make a magicsquare of the fourth order withwhole numbers from I to 16,such that the total for each rowof four figures (horizontal,vertical, and diagonal) is 34.Figure f is one suggestion;figure g is another.Some squares have otherremarkable properties. In figureg, for instance, the required 34can be found not only by addingthe numbers in the outsidesquares each side l1 + 12 + 8 +13 = 34), but also by taking thefour corner numbers 1 + 16 + 4fig. i

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+ 13 = 341, or again the center 4squares: 6 + 7 + 10 + 11 = 34.Figure h is a fourth-order squarewith similar properties, whilefigure is a square of the fifthorder. (Note: The brokendiagonals also total 65-23 A 15+ 6 + 19 + 2 - 65; 1 + 14 + 22 +10 + 18 = 65.) Then there areother yet more complicated andextraordinary squares, the"satanic" and the "cabalistic,"which require considerablemental effort to understand.

Some variantsMany variants have evolvedfrom magic squares. Here aretwo, which can be played asgames with friends.

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1) Draw seven circles as in figurejand number them from 1 to7such that the total of every rowof three circles linked by straightlines is 12 {the numbers printedhere are the answer).2) Make a pattern as in figure k

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THE 14-15PUZZLEShort historyNot many years ago, one onlyhad to board a bus to see someyoungster engrossed in aRubik's cube (a "magic" cubenamed after its inventor), orindeed even adults, similarlyengrossed, twisting and turningthe little coloured cubes of whichthe larger cube consisted.Legends grew up around thisexceptional toy, and nationaland internationalchampionships were held.However, the craze wasrelatively short lived andseemed to disappear like a merefashion. Such is the way ofthings Only a few decadespreviously, in the fifties, a closeforebear of Rubik's cube-the"14-15 puzzle"-enjoyed asimilar brief burst of popularity.As the Hungarian inventor of thecube himself admits, this gamecan be considered atwo-dimensional ancestor ofthelater three-dimensional game.It was invented by the AmericanSam Loyd around 1870, and fastbecame a craze in the UnitedStates. A number of peopleconsider Samuel Loyd thegreatest inventor of games andpastimes ever to have lived.Much of what he invented can befound in the Cyclopedia ofPuzzles, edited and published byhis son, Samuel Loyd, Jr., in1914.fig. a

How to playThe game is played with asquare panel containing fifteensmall movable square sectionsnumbered 1-15 whence thename), which occupy the wholearea of the panel except for theequivalent of one extra (asixteenth) small square Iseefigure a).Given this space, the smallsquares can be shifted around,one space at a time, as on achessboard. Norm asty one startswith them at random (see figurebl and tries to arrange them inSome predetermined pattern bymeans of methodicalintermediate moves.The pattern might very well bethat in figure a, readinghorizontally.The little squares can only bemoved up or down orhorizontally. It may at first sightappear an easy game: however,one can find onself with, forinstance, a beautifularrangement as in figure a-butwith the 14 and 15 the wrongway around.

Some examplesBoth children and adults find thisgame compelling, especiallywhen the arrangement to beachieved is more complex thanthat illustrated in figure. Forinstance, one could decide on avertical ordering of the squares(figure ci, with the numbersreading from top to bottom.Perhaps more difficult againwould be a diagonal order

{figure d, next page), startingfrom one in the bottom left-handcorner.If any reader feels so inclined, itmight also be worth trying aspiral arrangement, as shown inthe colour plate opposite, withthe numbering starting from one

ofthe central squares andproceeding outward to 15 in aspiral direction.

fig. b

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Some variantsIf instead of 15 we take 31squares, we get the rectangulargame shown in figure e: thiscertainly offers greaterpossibilities, but at the sametime it is more complicated.Fifteen squares afford a total of

fig. d

some 1,300 billion possiblearrangements; and obviouslywith 31 this number will be evengreater.By replacing numbers withletters, we have an alphabeticalversion of this same gamefigure /), which in turn offers a

variety of possibilities.One could, say, make the nameof a friend out of the letters, froma higgledy-piggledy startingpoint. Or one could make wordsor short phrases and then makeanagrams. After the initial crazeforthegameof"14-15," it fell

out of vogue and is now onlyplayed by a few enthusiasts.(Bobby Fischer, the former worldchess champion, could solve thepuzzle blindfolded.)

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How do you make out in mathematics?(Five progressive easy problems)

1) Of thirty-six eggs in a basket, four in every dozenare bad. How many good eggs are there?

2) Peter spent half of what his mother had given himon chips, then with half of what he had spent onchips he bought some sweets. He was then left withthirty cents change. How much did he spend onchips?

3) The number y is as much higher than twenty-oneas it is lower than thirty-seven. So what is y?

4) If a, b, and c are numbers, totalling d, is it true thatd less a equals b plus c?

5) If a, b, and d are numbers, and d is the differencebetween a and b, what would be the necessarycondition for d plus a to equal b?

Answers and comments on the clock games

1) c goes with f: since a shows 3:00, b shows 6:00, and c shows9:00, the logical sequel is 12:00.

2) e. (Here it is not a question of looking at hours and minutes,but atthe geometrical relation of the two hands; the secondpair in each set of hands has been turned forty-five degreesclockwise.)

3) Now what we are faced with is a quite different kind ofproblem: to guess the time in e we must first identify theprinciple behind the times shown in a, b, c, and d: a shows9:15, b 7:20, c 5:30, and d3:45. Note that the hour lessens bytwo each time, while the minutes increase in multiples of five(5,10,15, 20 . .. .Thus we need simplytake 2 from 3 and add20 to 45 (giving us one hour and five minutes, or five past thehour): the time in e should therefore be 1:05.

4) This exercise is like the previous one but has the addedcomplication of the seconds:

a 8:10:00b 9:00:05c 10:50:10d 11:40:15

It is easy to use that the hours increase by one each time,while the minutes decrease by ten, and the seconds increaseby five. Thus we just have to do a simple calculation:

23 + 1 = 24 (hours)40 - 10 = 30 (minutes)15 + 5 = 20 Iseconds).

Therefore e will read 12:30:20.

Answers to the mathematical games

1 Twenty-four eggs. Since thirty-six eggs make three dozen, iffour eggs in every dozen are bad, the other eight will be good.Hence there are 3 x 8 = 24 good eggs.

2) Sixty cents. Anyone at all familiar with basic algebra willeasily solve this problem. Whatwe are required to do is workout how much money Peter spent on chips. To denotethissum we shall use the symbol x. Now we know that what hespent on sweets was half of the total he spent on chips, x/2.This left him with thirty cents: in otherwords, if we subtractx12 from x, we are left with 30 cents. We are now in a positionto express this in algebraic form:

x - x/2 = 30

2x-x -=30

x/2 = 30

x = 30 x 2

x = 60.

3) Twenty-nine. A correct answer could perhaps be given byintuition, but a simple algebraic calculation will be almost asquick:

37 - 21=2 x

16/2 = x

8 x

Thus the number is 21 + 8 = 29.

4) Yes. For the answer here, we need to do a calculation withletters: a + b + c = d. Now we also know that in an equation,if the same amount is added or subtracted on both sides, thevalue will not change.

Thus:

a+b+c-a=d-a

which gives us:

b+c= d - a

6) dplus a would equal bon condition that is greater than a.To see why this is so, we have only to do a simple calculation:

dear b Cd b -a.

Some comment ought to be made on these last twoexercises. If we had used numbers instead of letters, theywould probably have been quicker and easier to answer.Expressed as they were, they required more mental effort: forit is harder to calculate with letters, which can have any value,than with numbers. Stepping from numbers to letters(arithmetic to algebra) involves accepting a higher degree ofgeneralization and abstraction. In that stepwe see anexample of how mathematics develops.

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Pick your own games

The prince's legacy

Many years ago there was an Indian prince who hadruled wisely and peacefully, administering his terri-tories and seeing that his eight children (six boys,two girls) were given the best possible upbringing.

Finally, sensing that his life was nearing its end, hedecided to retire to one of his estates-a sort ofparadise on earth, with sea all around it (see figure27).

He decided at the same time to keep the palaceand garden, which were at the heart of the estate, for

fig. 27

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himself, and to divide the surrounding country andforest equally between his six sons.

How might he have done this while still keepingthe central square intact?

One possible solution is given in figure28Hardly had the poor prince sorted all this out

when his wife, the princess, demanded fair treat-ment for her two daughters, claiming equal rights ofinheritance forthem. The prince loved his daughtersso he set about dividing up the estate into eightequal portions. Perhaps the reader can help.

Figure 29 (next page) offers a solution.

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The trapezoid

Figure 30 is easily recognizable as an isoscelestriangle, the base formed of three matches, and theother two equal sides of two each. By moving thematches around, is it possible to create a trapezoid-a four-sided figure with only two parallel sides?

Figure 31 suggests how.

earth. The comet even aroused a great deal of popu-lar interest among those not normally much con-cerned with astronomy. By mid 1985, the cometwas visible through most sorts of telescope usedby amateurs for about a year. Visible with the nakedeye in Britain around the end of 1985, it was to beseen clearly in southern Europe (forty degrees lati-tude) in the early months of 1986.

Ffi ... -. 1

fig. 31

1985: rendezvous with Halley's comet

In 1682 the English astronomer Edmund Halley(1656-1742), a personal friend of Newton's, dis-covered the periodic nature of the comet that nowbears his name. By a series of very precise calcula-tions, he established that its orbit took exactlyseventy-six years. Thus the last time it was seenpassing was in 1986, and the next appearance willbe in 2062. Halley's comet is thus visible only everyseventy-six years, and many of us will not be alive tosee its next appearance. For the astronomers of thepresent generation, therefore, 1986 saw a uniqueoccurrence which was not to be missed. Hence thegreat interest in the comet recently and the attemptsto achieve a "rendezvous" with the comet in spaceas it passed some tens of millions of miles from the

Giotto's Comet

Comets were often portrayed in medieval paintingas stars with five points, similar to that representedwith ten ordinary matches in figure 32.

There is one celebrated exception, however: thatincluded by the Florentine painter Giotto in thefresco Adoration of the Magi in the ScrovegniChapel in Padua. Here the head of the comet is nolonger the traditional star shape, but a sphericalbody emitting rays of light.

Giotto, a master painter of the fourteenth century,was indeed a great innovator. But a comet starhardly seems worthy of such a definite iconogra-phical break with tradition. How are we to explain it?- The most likely explanation is that in 1301, thevery year when Giotto was decorating the

179

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Scrovegni Chapel, Halley's comet appeared in theheavens, clearly visible to the naked eye. Un-doubtedly the painter was among those who saw it,a nd he was struck by the fact that it was not so muchstarlike as spherical, diffusing light.

The crater

It is true to say that in all ages there has been fear ofthe catastrophic effects of the possible collision of acomet with the earth. A German astrologer pre-dicted a disastrous collision between our planet and

fig. 33

fig. 35

fig. 34

a comet on June 13, 1857, and in France this led todisplays of collective panic. Perhaps even theappearance of Halley's comet in 1986 is seen bysome people as a "rendezvous" with the end of ourworld.

With eight matches it is possible to create severalgeometrical figures (see figures 33, 34, 35, 36).

Using just eight matches, can you make a geo-metrical figure, consisting of four triangles and twosquares, to represent the crater that would appearwere Halley's comet to crash into the earth?

A suggestion is given in the bottom right-handfigure on this page (figure 37).

fig. 36

fig. 37

180

Select bibliography

General titles on gamesFor all kinds of information, curiosities, and games (not onlymathematical and logical) and their variants, it is worthconsulting Martin Gardner, an American writer on pastimes andmathematical questions, who up until 1983 contributed asection on mathematical games in the "Scientific American."The following books are of particular interest:Gardner, M. Mathematical Circus. New York, 1979 and London,

1981Mathematical Puzzles and Diversions, New York, 1963 and

London, 1961MathematicalMagic Show, New York and London, 1977The Magic Numbers of Doctor Matrix, Buffalo, N.Y., 1980N.B. Not all the games in these books are easy to grasp: some

presuppose a mastery of mathematical and geometrical mattersthat perhaps some readers may lack.)

Other worksArnold, A. The World Book of Children's Games, New York, 1972Baslini, F.l 1"solitaire," Florence, 1970Dossena, G. Giochi da tavolo, Milan, 1984Girault, R. J. Traitd du jeu du Go, Paris, 1972Grunfeld, F. V. (ed.l. Games of the World, New York, 1975

Games of logicLogic has been treated very much in layman's terms in thisbook, without the use of technical "jargon" of symbols, and theanswers are on the whole presented through processes ofintuition and reasoning. Underlying this approach was a desireto introduce readers unfamiliar with this way of thinking to thebasic language and methodology of science. Although ourgames, stories, and puzzles were arranged in increasing order ofdifficulty, as well as according to their own logical development,

they do not claim to be a self-contained introduction to theproblems of logic. Our aim was primarily to arouse interest andstimulate curiosity, and encourage further reading. The mainsources for the games of logic are:Copi, l. Introduction to Logic, 6th edn. New York, 1982Johnson, D. A., Glenn, W. H., & Norton, M. S. Logic and

Reasoning, Bologna, 1978Smullyan, R. The lady orthe Tiger?, New York, 1983

What is the Name of this Book? The Riddles of Dracula andother Logical Puzzles, Englewood Cliffs, N.J., 1978 andLondon, 1981

Further readingDalla Chiara Scabia, M. L., Logica, Milan, 1981Mendelson, E. Introduction to MathematicalLogic, New York

1976Tarski, A. Introduction to Logic, New York, 1954Wesley, S. C. Elementary Logic, Bologna, 1974

Mathematical stories and gamesAlthough brief, the chapter dealing with mathematical storiesand games constitutes a sort of history of numbers (withparticular attention to the binary system) and a concise andreadable account of the origins of arithmetic. Sources of thegames are:Alerne, Jean-Pierre. Jeux de l'espritet divertissements, Paris,

1981Berloquin, P. Geometric Games, London, 1980Boyer, C. B. History of Mathematics, New York and London,

1968if rah, G. Histoire universelle des chiffres, Paris, 1981Johnson, D. A., & Glenn, W. H. Exploring Mathematics on Your

Own, New York, 1972

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Further readingCourant, R.,& Robbins, H. What is Mathematics?, London, 1941Freudenthal, H. Mathematics Observed, New York and London,

1967Waismann, F. Introduction to Mathematical Thinking: The

Formation of Concepts in Modern Mathematics, New York,1951 and London 1952

Intelligence puzzles and testsThe tests, verbal, visual, mathematical and otherwise, weretaken primarily from:Berloquin, P. Testez votre intelligence, Paris, 1974Bernard, W., & Leopold, G. Test Yourself, Philadelphia and New

York, 1962 and London, 1964Eysenck, H. J. Know Your Own 1... HaTmondsworth, 1962

Intelligence, creativity and personalityFor a discussion of problems relating to intelligence, creativityand personality, it is worth referring to the following books onpsychology:Anastasi, A. Psychological Testing, London, 1961Eysenck, H. J., & Kamin, L. Race, Intelligence andEducation,

Hounslow, 1971

HarrA, R., & Secord, P. F. The Explanation of Social Behaviour,Oxford, 1976

Hilgard, E. R., Atkinson, R. C., & Atkinson, R. L. Introduction toPsychology, New York and London, 1964

Games with matchesNeumOller, Anders. TOndstickor- Konst och ek, Stockholm,

1983

Jealousy testThe test on jealousy (pp. 114-116) is by Pino Gilioli, whose bookPersonality Games appeared in the same series (New York andLondon, 1986)

Other books in the seriesAgostini, F. Math andLogic Games, New York, 1983

Mathematical and Logical Games, London, 1983De Carlo, N. A. Psychological Games, New York and London,

1983Gilioli, P. Personality Games, New York and London, 1986

182

Index of games

Adding three matches, 98Alea, 158-9Alma, 78Alquerque, 34-5Analogy game, The, 121Animals and matches, 98Antinomy of the liar, The, 82Appendix of games with numerical systems, 149-52Asa Ito, 67At the Assembly of the Wise: all truth tellers or all liars,

75-6At the "Full Moon" inn, 125

Backgammon, 155-7Bagh-bandi, 36-7"Black Cat" Society, The, 135Blind associations, 122Bridge to the island, The, 166

Capacity for synthesis, see synthesisChinese checkers, 78-9Clock of Ctesibius, 63Crater, The, 180

Definitions, 27-8Derrah, 126Dwarf, The, 13

Enigma. 62Eyes of the mind, The, 71

Fanorona, 34"Farmer's" system of multiplication, 144-5

Figures in motion, 39-40First exercise, A, 99Four balls, five coins, six matches, 98Fox and geese, 67

Game of strategy in binary, 146-8Games with clocks, 168Games with numbers or figures, 43Games with shapes, 32-3, 38Geometrical figures with matches, 153-4Giotto's comet, 179-80Go, 107-12

Hand calculations, 141-2Hero's machine, 63Hidden square, The, 169Hidden trick, The, 120How is it possible? 70How is your mechanical skill? 105-6How many are we in our family? 70How many hands? 133-4How many hares have the hunters bagged? 69How many horses hasthe farmer got? 163How many members? 76-7How old is Peter? 70How old is Livy? 165How's your memory? 43

In the world of opposites, 73-7,82Intelligent crow, The, 132

Jealousy, 114-17

183

Index of games

Kolowis Awithlaknannai, 34

Let's exercise our intelligence, 13-15, 20-3Let's gauge our intelligence, 52-9Liar from antiquity, A, 82Logical riddle, A, 71Love story, A, 122-3

Macabre game, A, 123Magic squares, 169-71Making up a story, 95Mancala, 128-30Meeting with the prime minister, The, 75Mill, The, 166Mixed bunch, A, 84-5Mogol Putt'han, 37

Nim, see Game of strategy in binaryNine Men's Morris, 80-1Numbers, Games with, 29-31

Ocean liner, The, 164Oedipus and the sphinx, 60-2Officers and Sepoys, 67One, two, three ... ten: on your fingers! 137Opposites, see In the world of opposites

Palindromes, 28-9Peralikatuma, 34Ping-Pong balls, 72-3Planet of the one-handed, The, 134Prince's legacy, The, 176-8Problem of logical deduction, A, 87-8Prudence's gloves, 165Puff, 159Puzzle, The 14-15,172-3

Question and answer, The game of, 27

Reasoning with figures, 88Re-creative games, 100Removing three matches, 161Rimau-rimau, 37Roman numerals with matches. 143

Same-letter game, The, 123-4Seega,126-7Senet, 16-17Shrook! 74Singing fountain of Hero, 63Solitaire, 64-5Square, a circle and ... a child!, 96Square and a triangle, A, 164Squares, Let's play with, 161 3Story of the exclamation mark, The, 95Success, The game of, 101Synthesis, Capacity for, 102-4

Tabula, 158-9Technical aptitude, 105Three times one-five = five-one, 134-5Tic-tac-toe, 81Tony's socks, 165Topological games, 41-2Tough case for Inspector Bill, A, 86-7Trapezoid, The, 179True and false, The game of, 88Two buffoons, 74-5Two roads, The,73-4

Ur, The royal game of, 19

Wari,128-30Weights and scales, 71-2What are the coins in Peter's pocket? 163What colour are their clothes? 84What is it? 118- 9What is itthat ... .? 68-9What is XLVIII by CCLXXXVIII? 143Wheel with paddles, A, 166-7Who is it? 117-18Who is the guard? 85-6Who said ... .?121-2Word chains, 28Words, Playing with, 24-5- First games with, 26-7

Yes, no,120

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With more than 100 brain teasers, board games, riddles, puzzles, and gamesinvolving memory, logic, words, and numbers, Intelligence Games tests your judg-ment and creativity and measures your ability to compete in the power game onthe job and in choosing a partner, friends, hobbies, and practical moves.

By putting verbal, visual, mathematical, and logical forms of intelligence in adifferent light, this lively and entertaining book offers you an opportunity toreflect about yourself and the way your mind works. Here is a selection of thebest-known games from all over the world, such as Solitaire, Backgammon, andChinese Checkers, as well as hundreds of lesser-known games such as The RoyalGame of Ur, Fox and Geese, Shrookl, Go, Nine Men's Morris, Tony's Socks, BinaryStrategy, Magic Squares, Jealousy, Claustrophobia, and many more.

As informative as it is fun, this treasure chest for the mind will help you developa better understanding of intelligence, decision-making, and the way in whichyou think.

About the AuthorsWith a degree in experimental psychology, Franco Agostini teaches classics

in Verona, Italy, and contributes a regular column to the Italian computer maga-zine Zerouno.

Nicola Alberto De Carlo edits Scienza e Cultura and teaches at the Universityof Padua.

A Fireside Book o 14 1295 U gPublished by Simon & Schuster, Inc.New York $12.95 Cover design by Karen Katz

71295 0-671-63201-9

I

0 -671 -63201 -9 1171295

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