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Mathematics and Cultures Across theChessboard: The Wheat and
ChessboardProblem
Alberto Bardi
In number did outmillion the accountReduplicate upon the
chequer’d board(Dante Alighieri, Divine Comedy,Paradise 28, 93;
transl. Henry Francis Cary)
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 2Mathematics and the Invention of Chess . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 3Mathematics and the Origins of Chess . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5Geometric Progressions and Chess . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5Arabic Sources on the Computation 264 − 1 . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Greek
Sources on the Computation 264 − 1 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 12Western Sources
on the Computation 264 − 1 . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 15Number Theory . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 19Summary . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
20Cross-References . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 21References . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 21
Abstract
This chapter is an introduction to the wheat and chessboard
problem and theinterplay between chess and mathematics in the
several authors and culturalcontexts that have inherited, faced,
and modified this problem, ranging from
A. Bardi (�)Polonsky Academy for Advanced Study in the
Humanities and Social Sciences, Van LeerJerusalem Institute,
Jerusalem, Israele-mail: [email protected]
© Springer Nature Switzerland AG 2019B. Sriraman (ed.), Handbook
of the Mathematics of the Arts and
Sciences,https://doi.org/10.1007/978-3-319-70658-0_82-1
1
http://crossmark.crossref.org/dialog/?doi=10.1007/978-3-319-70658-0_82-1&domain=pdfmailto:[email protected]://doi.org/10.1007/978-3-319-70658-0_82-1
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2 A. Bardi
Antiquity to the Renaissance, considering a selection of Arabic,
Persian, Greek,Latin, Italian, Romance, and Germanic sources.
KeywordsChess · Chessboard · Geometric progression of 2 ·
Mathematics · Wheat andchessboard problem
Introduction
Walking through Union Square or Washington Square in New York
City, one willfind several street chess players. They sit by round
tables set up with chessboards,ready to play against passersby for
just a few dollars. Such recreational and socialactivity represents
only one aspect of the wide range of features and
capabilitiesencompassed by the chessboard. The chessboard is indeed
an attractive andfascinating object: not only is it the basic
material field of an internationally famousgame of strategy and
intelligence, known worldwide under the name chess, withofficial
rules and an official federation (FIDE 2000); but it has also
inspired tasks,puzzles, and mathematical challenges. For instance,
the legend of the origins ofthe game of chess is related to a
calculation on the chessboard, which is knownto mathematicians to
this day as the wheat and chessboard problem. The legendhas been
transmitted through several redactions, in different languages, and
fromdifferent cultural traditions. At this introductory stage, it
is helpful to remindourselves of the standard features of this
legend. The ruler of India discovered thegame of chess at his court
and was fascinated by this original idea and the multitudeof
possible combinations for placing the pieces on the board. He
learned that theinventor of this game was at his court, so he
summoned him and congratulated himon his wonderful idea. As reward,
the ruler promised the inventor to grant any wishhe might express.
The inventor’s wish was for a quantity of wheat grains to becounted
on a chessboard: one grain for the first field of the board, two
grains forthe second, and so on, so that each field contained twice
as many grains as the onepreceding it. The ruler ordered his men to
grant the inventor his wish. The followingday, the court
mathematicians told their lord that such wish could not be
fulfilled,for it would require more wheat than the world could
supply. The quantity of grainsrequested by the inventor of the
chess is equal to
1 + 2 + 22 + · · · + 263 = 264 − 1 = 18446744073709551615.
At first glance, the game of chess might appear to us as a mere
street game,but upon closer inspection, it proves to be more than
simply a recreational activity:the unexpected conclusion of the
tale about the invention of chess shows the greatmathematical
possibilities hidden in the game. For instance, particular
mathematicalproperties of the chessboard lead to its so-called
magic squares. A magic square oforder n is a squared table of n × n
dimensions, which contain the integers from 1 to
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
3
n2. The numbers contained in the table are positioned in such a
manner that the sumof the numbers of each row, each column, and the
two major diagonals is equal toone other. The fascination of this
mathematical object went beyond pure mathemat-ics. For example, the
German Renaissance mensch Albrecht Dürer was so impressedby the
magic squares that he reproduced one in his famous engraving
Melancholy.(We use “mensch” to avoid sexist connotations, as
suggested by Sriraman 2009, 75.)By means of experimenting with a
chessboard and chess pieces, first-rate modernmathematicians such
as Luca Pacioli (c. 1445–1517), Leonhard Euler (1707–1783),and Carl
Friedrich Gauss (1777–1855) formulated solutions to problems of
recre-ational mathematics related to the chessboard. Pacioli
composed a treatise on chessproblems at the beginning of the
sixteenth century, entitled De Ludo Schacorum(“On the Game of
Chess”). The Knight’s Tour Problem, which deals with the searchfor
a knight’s tour in chessboards of different sizes, was of
particular interest toLeonhard Euler: he provided remarkable
solutions to this for an 8 × 8 chessboardin 1759 (Watkins 2004,
3–8). Carl Friedrich Gauss worked brilliantly on the so-called
8-queens problem (Watkins 2004, 164–169), which focuses on how to
placeeight queens on an 8 × 8 chessboard so that none of them
attacks any of the others,elaborating a new arithmetical solution.
In the twentieth century, the game of chesswas a source of
inspiration for new branches of mathematics, namely, set theory
andgame theory (Zermelo 1913; Von Neumann and Morgenstern 1944).
For instance,chess is the field of application for the theory of
so-called zero-sum two-persongames (Von Neumann and Morgenstern
1944, 124–125), and it sparks discussionson finiteness and the
applicability of the minimax theorem (Ewerhart 2002).
These are only a few examples to indicate how mathematics is
intertwinedwith the chessboard and the game of chess. For a more
in-depth consideration ofthese interconnections, one might consult
Gik (1986), Petković (1997), and Watkins(2004). Moreover, the
influence of chess is not confined to the mathematics itinspires,
for its long history involves strategical, moral, civic, poetic,
and literaryissues, several aspects of which are outlined
masterfully in the groundbreaking workon the history of chess by
Harold J. Murray (1913, but see also earlier works on thehistory of
chess, to which Murray is indebted: Hyde 1694; Van der Linde
1874,1881).
The current chapter is an introduction to the entanglements
between chess andmathematics in relation to the wheat and
chessboard problem. Due to the recentdiscovery of unpublished
sources on the wheat and chessboard problem, it is
worthreconsidering its main cultural and mathematical aspects,
their developments, andtheir diachronic interplay.
Mathematics and the Invention of Chess
The history of chess – to make a very long story short – is
believed to have its originin India, between the third and the
sixth century BCE, during the Gupta Empire.From India it was
introduced to Persia, during the Sassanid Empire, and then
entered
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4 A. Bardi
the Islamic world after the Islamic conquest of Persia. In the
seventh century CE, itpossibly spread to China too. It was
transmitted to the West through exchanges withthe Arabs, who had
settled in Northern Africa, Spain, Southern France, Sicily,
andSouthern Italy before the tenth century.
Mathematics and chess are connected in the earliest sources on
the history of thegame. As far as we can reconstruct from the
extant written sources, the legend of theinvention of chess indeed
provides a version of the wheat and chessboard problemwithin its
narrative (Wiedemann 1908; Ruska 1916; Wieber 1972, 88–102;
Tropfke1980, 630–632).
The ninth-century Arab historian Ibn Wad. ı̄h. provides one of
the mostancient written accounts of the tale of the wheat and
chessboard problem withinthe wider context of a story about the
legend of the invention of chess. The storygoes as follows
(Wiedemann 1908, 43–44; Ruska 1916, 280–281; Tropfke 1980,631). An
Indian philosopher named Qaflān developed a game (chess) which
wouldfunction as a war simulation and protected the soul from
participating concretelyin cruelty and bloodshed. The queen
summoned him and praised his wisdom andinvention and promised him
any reward he wished. Qaflān asked for a quantity ofcorn grains to
be counted on a chessboard: one grain for the first field of the
board,two grains for the second, and so on, so that each field
contained twice as manygrains as the one preceding it. The grain
stores of that reign were not enough togrant Qaflān’s wish, but he
said he did not need such a quantity, adding that for himjust a
small piece of lawn was enough. The queen then asked him about the
numberof the grain corns he wished. He therefore explained the
computation: the first rowhas 255, the second one 32768, the third
8388608, the fourth 2147483648, the fifth549755813888, the sixth
140737488355323, the seventh 36028797018963968, andthe eighth
9223372036854975808. The total of the corn grains on the chessboard
isequal to 18446744073709551615.
The Arab historian Al-Khāzinı̄ (about 1130) also provides this
tale in his Bookof the Balance of Wisdom (Khanikoff 1858) but
ascribes the problem to an Indianphilosopher called S. is.s.a ben
Dāhir (Wiedemann 1908, 45–54; Wieber 1972, 96;Tropfke 1980, 631).
Instead of grains of corn, his version deals with the doubling
ofpieces of gold. A tale similar to Al-Khāzinı̄’s was transmitted
by the thirteenth-century Arabic historian Ibn Khallikān (Ruska
1916, 276) in his biographicaldictionary of Arab scholars (Slane
1961). Another Arab historian, (about946), reports a similar tale
and agrees on the Indian origins of chess (Ruska 1916,280; Tropfke
1980, 631).
The renowned Arab scholar Al-Bı̄rūnı̄ (973–1048) mentions the
wheat andchessboard problem in his Chronology of Ancient Nations.
His account concernsonly the solution, that is, the computation of
the sum of the grains on the 64 squaresof the chessboard and the
properties of such computation; he does not provide anytale on the
invention of chess (Schau 1879, 134–136). This constitutes the
mostancient source which provides an account of the sum of the
grains on the chessboard,but it omits the tale about the invention
of chess.
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
5
Mathematics and the Origins of Chess
Given the well-documented link between chess and the wheat and
chessboardproblem, exploring the history of this problem could shed
new light on the origins ofthe game of chess. The widespread
distribution of the wheat and chessboard problemmakes it difficult
to trace with certainty its origins and subsequent
dissemination.Nevertheless, Arabic sources on this problem agree on
one point: the inventor ofchess was an Indian philosopher, and the
game to be played on a chessboard of 64squares was invented by
him.
It is very likely that this game reached the Arabic-speaking
world through Persianintermediaries (Murray 1913, 207–210). A
Persian source, however, ascribes theinvention of chess to the
Persian king Khosrow I Anushiruwān (501–579). This textis written
in Pahlavi (a variant of Middle Persian), and the history goes as
follows:the Indian king Devasārm sent an embassy to the king
Khosrow Anushiruwān withthe game of chess, which the Persian king
had invented, asking for an explanationof the game. The Indian
ambassadors learnt that this was to be understood as awargame and
in contrast to the chance-governed game nard, which was dependenton
astrology (specifically on combinations of planetary positions and
zodiacal signs)(Ruska 1916, 280). In this instance, chess and
mathematics represent order andcivilization versus chance and
chaos. However, the Persian source does not mentionany version of
the wheat and chessboard problem.
Since all the Arabic sources that refer to the invention and the
chessboardproblem agree on the Indian origins of chess, it is
therefore likely that the wheatand chessboard problem – in the form
whereby the solution must be the sum of aunity redoubled 64 times
on a chessboard – shares the same origin. Moreover, allsources
agree on the mathematics involved in the solution to the problem,
which is,in more technical terms, the sum of a geometric
progression of reason 2.
Geometric Progressions and Chess
Geometric progressions, that is, sequences of numbers where each
term after thefirst is found by multiplying the previous one by a
non-zero number (properly calledcommon ratio), were known about and
used extensively in problem-solving beforethe invention of the
chessboard and the game of chess. Applications of
geometricprogressions were already an object of recreational
mathematics in Babylonian andEgyptian mathematics in Antiquity. The
most ancient versions of the wheat andchessboard problem can be
traced back to before the ninth century and are thereforemore
ancient than Al-Bı̄rūnı̄’s and versions, which constitute, as
hasbeen suggested, the earliest testimony about the wheat and
chessboard problem ona 64-squared chessboard (Ruska 1916, 282;
Høyrup 1994, 96–99). The differencebetween this version and the
most ancient versions is twofold: the latter give aredoubling that
differs from the 64 times and are not included as part of legendson
the invention of chess. The most ancient account of the wheat and
chessboard
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6 A. Bardi
problem is a text on Babylonian mathematics from the eighteenth
century BCE.This version deals with corn grains which are to be
doubled 30 times (Soubeyran1984).
As for geometric progressions unrelated to the wheat and
chessboard problem,their use is attested by written sources that go
back to ages and contexts moreancient than the invention of chess.
For instance, the renowned Rhind papyrus,found in Egypt and
originating from around 1700 BCE, is a document redacted inhieratic
script which provides a collection of mathematical problems: a
geometricprogression with common ratio 7 (7 + 72 + 73 + 74 + 75) is
part of the contentof the papyrus (Newman 1952, 27; Boyer and
Merzbach 1991, 11–16). A fragmentof a Greek papyrus of unknown
provenance, originating from the third to the fourthcentury BCE,
contains an application of a geometric progression of ratio 2 with
thenumber 5 as the initial term, which should be doubled 30 times.
The sum of theprogression constitutes the solution to a currency
conversion problem of ancientcoinages, that is, for drachmae and
talents (Boyaval 1971, 165–168). Treatiseson geometric progressions
and their properties, such as Euclid’s Elements BookIX (esp. Prop.
35 and 36), redacted between the fourth and the third centuriesBCE,
can be traced back to Antiquity. These topics would be reprised in
Thābitibn Qurra’s renowned treatise On the Determination of
Amicable Numbers (tenthcentury) (Rashed 2015, 337, 399–410).
As for our wheat and chessboard problem, this takes place on an
8 × 8chessboard and must be solved by computing the sum of the
geometric progression1 + 2 + 22 + . . . + 263, which is equal to
264 − 1. The method of solving the wheatand chessboard problem is
usually provided in the extant sources by explanationsin textual
form or by means of displaying the geometric progression in a
column. Ifthe tale of the invention of chess is provided, the
computation always follows that.The column displays the number of a
field of the squares and the correspondingresults of the
reduplication. Some partial results of the progression are
usuallyconverted into different units of measurement in order to
make the computationeasier, so that whoever is computing has to
deal with quantities smaller than theywould otherwise be. We will
see some examples below. Some sources also providerules to
demonstrate the validity of the computations alongside
considerations of theproperties of the progression.
While Arabic and Western sources (in Latin and in Romance and
Germaniclanguages) were extensively examined in the scholarship on
the history of the wheatand chessboard problem (Wiedemann 1908;
Ruska 1916; Wieber 1972; Tropfke1980, 630–633; Sesiano 2014,
138–141), only recently have examinations of Greekmanuscripts
brought to light new evidence on the dissemination of the sum of
thegeometric progression to solve the wheat and chessboard problem.
(This scholarlygap was pointed out long ago: “There is one branch
of the later Greek literature,fairly circumscribed in extent, which
might possibly give us some reference tochess earlier in date than
any I have cited. The mathematical problem known as‘the doubling of
the squares of the chessboard’ may have been known to the
laterGreek mathematicians, as we find it included in the oldest
Western mediaeval MSS.on mathematics. The Greek MSS. have not so
far been examined for this purpose”
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
7
(Murray 1913, 167).) In what follows, we provide sections on
Arabic, Western, andGreek sources on the sum of the geometric
progression to solve our problem, that is,on the computation of 264
− 1. The sections on Arabic and Western sources offer
anintroductory account on this topic, displaying significant
sources alongside a briefcommentary. The section on Greek sources,
given the lack of scholarship on thistopic, constitutes the most
up-to-date account concerning the wheat and chessboardproblem in
Greek literature.
Arabic Sources on the Computation 264 − 1Several Arabic sources
present the wheat and chessboard problem within books ofalgebra.
The most ancient example of our problem in this genre of
mathematicalliterature is in a lost book of the mathematician and
polymath Al-Khwārizmı̄ (780–850 CE). We learn this from a book of
algebra by the mathematician Abū Kāmil(ninth century), who
mentions Al-Khwārizmı̄ as one of his sources (Rashed 2012,724–726;
Sesiano 2014, 139). Later on, this problem is contained in many
books ofalgebra, such as that by Al-Kashı̄ (mid-fifteenth century)
(Luckey 1951, 26–27, 61).
The simplest computational method to solve the wheat and
chessboard problemis the one mentioned in the legend of (see
above). It consists in the sumof the quantities that the reader
obtains at each final square (i.e., on the extreme rightcorner) of
each row. Therefore:
The first row has 255The second 32768The third 8388608The fourth
2147483648The fifth 549755813888The sixth 140737488355323The
seventh 36028797018963968The eighth 9223372036854975808
The total of the grains on the chessboard is equal to
18446744073709551615 (i.e.,264 − 1).
Among the several primary sources for this geometric progression
(of which thereader can find a comprehensive overview in Wieber
1972, 103–119), the Arabicmanuscript Orientalis A 1343, preserved
in the Gotha Research Library, deservesa special mention, for it is
entirely devoted to the mathematics of the wheat andchessboard
problem (Pertsch 1878–1892, 3: 14–15; Wiedemann 1908, 54–58).
In the following, we provide a sample of a computation of a
geometric progres-sion related to the wheat and chessboard problem
from the manuscript Orientalis A1343. The quantities in the third
column are not provided in the manuscript, but wehave added them to
allow the read to become acquainted with the raw results of
thereduplications.
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8 A. Bardi
Square Unit of measurement Quantity
1 1 h. abba 12 2 h. abba 23 4 h. abba 44 8 h. abba 85 16 h. abba
166 32 h. abba 327 64 h. abba 648 128 h. abba 1289 256 h. abba
25610 512 h. abba 51211 1024 h. abba 102412 2048 h. abba 204813
4096 h. abba 409614 8192 h. abba 819215 16384 h. abba 1638416 32768
h. abba = 1 qadah. 3276817 2 qadah. 6553618 4 qadah. 13107219 8
qadah. 26214420 16 qadah. = 1 waiba 52428821 2 waiba 104857622 4
waiba 209715223 8 waiba = 1 irdabb + 2 waiba 419430424 2 irdabb + 4
waiba 838860825 5 irdabb + 2 waiba 1677721626 10 irdabb + 4 waiba
3355443227 21 irdabb + 2 waiba 6710886428 42 irdabb + 4 waiba
13421772829 85 irdabb + 2 waiba 26843545630 170 irdabb + 4 waiba
53687091231 341 irdabb + 2 waiba 107374182432 682 irdabb + 4 waiba
214748364833 1365 irdabb + 2 waiba 429496729634 2730 irdabb + 4
waiba 858993459235 5461 irdabb + 2 waiba 1717986918436 10722 irdabb
+ 4 waiba 3435973836837 21845 irdabb + 2 waiba 6871947673638 43670
irdabb + 4 waiba 13743895347239 87381 irdabb + 2 waiba
27487790694440 164761 irdabb + 4 waiba = 1 šūna 54975581388841 2
šūna 109951162777642 4 šūna 219902325555243 8 šūna
439804651110444 16 šūna 8796093022208
(continued)
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
9
45 32 šūna 1759218604441646 64 šūna 3518437208883247 128 šūna
7036874417766448 256 šūna 14073748835532849 512 šūna
28147497671065650 1024 šūna = 1 madı̄na 57294995342131251 2
madı̄na 114589990684262452 4 madı̄na 229179981368524853 8 madı̄na
458359962737049654 16 madı̄na 916719925474099255 32 madı̄na
1833439850948198456 64 madı̄na 3666879701896396857 128 madı̄na
7333759403792793658 256 madı̄na 14667518807585587259 512 madı̄na
29335037615171174460 1024 madı̄na 58670075230342348861 2048 madı̄na
117340150460684697662 4096 madı̄na 234680300921369395263 8192
madı̄na 469360601482738790464 16384 madı̄na 9387212036854775808
The conversions in the second column are intended to facilitate
the computations.It is Ibn Khallikān who claims to have learnt
such methods from a mathematicianfrom Alexandria in Egypt (Wieber
1972, 106). A brief explanation of this is thatthe first sequence
of reduplications occurs from the 1st to the 16th square, wherewe
obtain 32768 h. abba (grain of corn), which is equal to 1 qadah.
(cup); from the16th to the 20th, we obtain 16 qadah. , which is
equal to 1 waiba (dry measure of20 units/mass of wood); from there
to the 40th square, we have irdabb (buschel)and šūna (store), so
that 164761 irdabb + 4 waiba, that is, 174762 + 2/3 irdabb =
1šūna. From the 40th square to the 50th, we have 1024 šūna, which
corresponds to 1madı̄na (town). By redoubling until the 64th house,
we obtain 16384 madı̄na, whichcorresponds to 263 =
9223372036854775808.
The manuscript Orientalis A 1343 contains further geometric
progressionsrelated to the reduplications on the chessboard. In the
Arabic versions of thecomputation 264 – 1, it is common to read
conversion methods which are designed tomake the computations
easier. There are many variants in the units of measurement(Wieber
1972, 103–119).
It is worth reporting Al-Bı̄rūnı̄’s chapter on the wheat and
chessboard problem,for it constitutes the most extensive source on
the mathematical properties ofthis problem in Arabic literature.
Al-Bı̄rūnı̄ takes for granted that the reader isacquainted with
the tale on reduplication of grains on the chessboard and
providestwo rules for it. The first rule is:
The square of the number of a check x of the 64 checks of the
chessboard is equal to thenumber of that check the distance of
which from the check x is equal to the distance of thecheck x from
the first check.
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10 A. Bardi
For example: take the square of the number of the 5th check,
i.e. the square of 16(162) = 256, which is the number belonging to
the 9th check. Now, the distance of the 9thcheck from the 5th is
equal to the distance of the 5th check from the first one.
(TranslationSchau 1879, 134–136)
Al-Bı̄rūnı̄’s second rule on the doubling of the chessboard
is:
The number of a check x minus 1 is equal to the sum total of the
numbers of all the precedingchecks. Example: The number of the 6th
check is 32. And 32 − 1 is 31, which is equal tothe sum of the
numbers of all the preceding checks, i.e. of 1 + 2 + 4 + 8 + 16
(=31). If wetake the square of the square of the square of 16,
multiplied by itself [ . . . ], this is identicalwith taking the
square of the number of the 33rd check, by which operation the
number ofthe 65th check is to be found. If you diminish that number
by 1, you get the sum of thenumbers of all the checks of the
chessboard. The number of the 33rd check is equal to thesquare of
the number of the 17th check. The number of the 17th check is equal
to the squareof the number of the 9th check. The number of the 9th
check is equal to the square of thenumber of the 5th check. And
this (i.e. the number of the 5th check) is the abovementionednumber
16. (Translation Schau 1879, 134–136)
After the second rule, we find further considerations of the
problem and ofgeometric progressions, taken from another work by
Al-Bı̄rūnı̄, entitled “Book ofCiphers.” The text is
self-explanatory:
I shall explain the method of the calculation of the chess
problem, that the reader mayget accustomed to apply it. But first
we must premise that you should know, that in aprogression of
powers of 2 the single numbers are distant from each other
according toa similar ratio. (Lacuna?) If the number of the
reduplications, i.e. the number of the singlemembers of a
progression is an even one, it has two middle numbers. But if the
number ofthe reduplications is an odd one, the progression has only
one middle number.
The multiplication of the two ends by each other is equal to the
multiplication of thetwo middle numbers. (In case there is only one
middle number, its square is equal to themultiplication of the two
end numbers.) This is one thing you must know beforehand. Theother
is this: −
If we want to know the sum total of any progression of powers of
2, we take the doubleof the largest, i.e. the last number, and
subtract therefrom the smallest, i.e. the first number.The
remainder is the sum total of these reduplications (i.e. of this
progression).
Now, after having established this, if we add to the checks of
the chessboard one check, a65th one, then it is evident that the
number which belongs to this 65th check, in consequenceof the
reduplications of powers of 2, beginning with 1, is equal to the
sum of the numbersof all the checks of the chessboard minus the 1st
check, which is the number 1, the firstmember of the progression.
If, therefore, 1 is subtracted from this sum, the remainder is
thesum of the numbers of all the checks of the chessboard.
If, now, we consider the 65th check and the 1st as the two ends
of a progression, theirmedium is the 33rd check, the first
medium.
Between the checks 33 and 1, the check 17 is the medium, the
second medium.Between the checks 17 and 1, the check 9 is the
medium, the third medium.Between the checks 9 and 1, the check 5 is
the medium, the fourth medium.Between the checks 5 and 1, the check
3 is the medium, the fifth medium.Between the checks 3 and 1, the
check 2 is the medium, the sixth medium, to which
belongs the number 2.Taking the square of 2 (22), we get a sum
which is a product of the multiplication of the
number of the 1st check by that of the 3rd check (1 × 4 = 22).
The number of the 1st checkis 1. This product, then, is the fifth
medium, the number of the 3rd check, i.e. 4.
The square of 4 is 16, which is the fourth medium of the 5th
check.
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
11
The square of 16 is 256, which is the third medium in the 9th
check.The square of 256 is 65536, which is the second medium in the
17th check.The square of 65536 is 4294967296, which is the first
medium in the 33rd check.The square of 4294967296 is
18446744073709551616.If we subtract from this sum 1, i.e. the
number of the first check, the remainder is the
sum of the numbers of all the checks of the chessboard. I mean
that the number which at thebeginning of this digression we have
used as an example (of the threefold mode of numeralrotation).
(Translation Schau 1879, 135–136)
Given the immense quantity of the final result of the sum of the
grains, Al-Bı̄rūnı̄provides a method to better grasp the number
18446744073709551615 by means ofdividing it by dry measurements
based on the Arabic coinage, the dirham (A dirhamcoin is made of
silver (Miles 2012)):
The immensity of this number cannot be fixed except by dividing
it by 10000.Thereby it is changed into Bidar (sums of 10000
dirhams).The Bidar are divided by 8. Thereby they are changed into
(loads).The are divided by 10000. Thereby the mules, that carry
them, are formed into
(herds), each of them consisting of 10000.The are divided by
1000, that, as it were, they (the herds) might graze on the
bordersof Wâdîs, 1000 kids on the border of each Wâdî.The Wâdîs are
divided by 10000, that, as it were, 10000 mountains should rise out
of eachWâdî.In this way, by dint of frequently dividing, you find
the number of those mountains to be2305. But these are (numerical)
notions that the earth does not contain.(Translation Schau 1879,
136)
Further Arab authors were aware that the final sum was too big
to be easilycomprehended and therefore developed methods to allow
the reader to visualizethis quantity (Wieber 1972, 116–119). For
instance, Al-Khāzinı̄ expounds a methodusing conversions of units
of measurement into dirhams, in order to distribute264 − 1
throughout the surface of the Earth by means of coins. This can be
explainedbriefly as follows. Given the diameter of the Earth of
6490 + 10/11 miles and thecircumference of the Earth of 20400
miles, the surface of the Earth correspondsto the multiplication
6490 + 10/11 miles × 20400 miles, which gives a result of132416400
miles2. This number corresponds to 2118662400000000 cubits2. If
wedivide 264 − 1 dirhams by 2118662400000000 cubits2, we obtain a
result of 8708dirhams pro cubit2.
The weight of a cubit of silver corresponds to circa 420867
dirhams pro cubit3. Ifwe take a cube with edge length of 1 cubit
(which consists of dirhams), we have 75dirhams + a fraction, that
is, the length of a cubit in dirhams3. The square of this is5625.
If we divide 8708 dirhams by 5625, we obtain 1/48 cubit, that is,
the quantityof silver that represents the dirhams stretched out
along the surface of the Earth. Ifwe distribute this throughout the
inhabited part of the Earth, the quantity is equal to1/12 cubit
(Wiedemann 1908, 53–54; Wieber 1972, 117).
Later on, the tenth-century Arab astrologer and mathematician
Al-Qabisiemploys a very similar method to distribute 264 − 1 along
the surface of theEarth in dirham coins. His result of the
computation of the surface of the Earth incubits2 is the same as
Al-Khāzinı̄’s, that is, 2118662400000000. The procedures
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12 A. Bardi
after this passage are slightly different. According to his
computations, a cubit2 isequal to 500 dirhams; therefore the
surface of the Earth is equal to the multiplicationof
2118662400000000 × 500, that is, 1059331200000000000 dirhams. The
total ofthe reduplication of the squares of the chessboard is about
17 + 2/5 times higherthis sum (Sesiano 2014, 148–149).
Greek Sources on the Computation 264 − 1Alongside Greek sources
on geometric progression, such as Euclid’s Elements IXand later
mathematical literature in Greek, such as collections of
mathematicalproblems originating from the Byzantine era (Hunger and
Vogel 1963; Vogel 1968;Chalkou 2006; Deschauer 2014), we have only
to look to the Greek scientifictradition of the fifteenth century
to find applications of geometric progressions thathelp to solve
the wheat and chessboard problem. Up until now, only one
Greeksource has been detected (Heiberg 1899, 168–169), but we have
also found twoadditional sources (more below). These sources do not
contain any tales aboutthe wheat and chessboard problem or the
invention of chess, but simply providea geometric progression of
reason 2, which deals, as usual, with a unity (withoutfurther
specification of unit of measurement) to be doubled 64 times.
It is likely that the geometric progression on the squares of
the chessboard wasbrought into Greek mathematics through Arabic or
Persian intermediaries. A lexicalinvestigation supports this
interpretation. In fact, the Greek word for chessboardis zatrikion,
which stems from the Sanskrit catur-aṅga (“four members”) and
wasmediated into Greek through shatranij, an occurrence detectable
both in Persian andArabic literature. Greek historiographical works
agree in assigning Eastern (Persianor Assyrian) origins to the game
of chess (Murray 1913, 162–163).
The Greek source on the doubling of the chessboard that we
already knowabout is provided in the fifteenth-century manuscript
Vindobonensis phil. gr. 65(henceforth V), which contains a
selection of mathematical problems and wasredacted at around 1436
(Hunger 1961, 182–184). The progression is arranged intotwo columns
and is entitled “doubling of the chessboard.” The Greek numerals
formα to θ, supplemented with a sign for the “zero,” and are used
as we use the numeralsfrom 1 to 9, supplemented with 0; this
results in a positional notation for numbers.This is evidence of
the influence of Arabic or Indian notation. Such a system
wasalready known in Byzantium before the fifteenth century, for
Indian numerals werethe object of a mathematical treatise by
Maximos Planudes, a Byzantine scholaractive in the half of the
thirteenth century (Allard 1981).
We found another geometric progression of the wheat and
chessboard problemvery similar to V in the fifteenth-century Greek
manuscript Ambrosianus I 112 sup.(henceforth A), preserved at the
Ambrosiana Library in Milan (Martini and Bassi1906, 562–564). In
the following, we provide a translation of the progression intoour
notation according to the version in the manuscript V.
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
13
Doubling of the chessboard
1 1 29 2684354562 2 30 5368709123 4 31 10737418244 8 32
21474836485 16 33 42949672966 32 34 85899345927 64 35 171798691848
128 36 343597383689 256 37 6871947673610 512 38 13743895347211 1024
39 27487790694412 2048 40 54975581388813 4096 41 109951162777614
8192 42 219902325555215 16384 43 439804651110416 32768 44
879609302220817 65536 45 1759218604441618 131072 46
3518437208883219 262144 47 7036874417766420 524288 48
14073748835532821 1048576 49 28147497671065622 2097152 50
57294995342131223 4194304 51 114589990684262424 8388608 52
229179981368524825 16777216 53 458359962737049626 33554432 54
916719925474099227 67108864 55 1833439850948198428 134217728 56
36668797018963968
57 7333759403792793658 14667518807585587259 29335037615171174460
58670075230342348861 117340150460684697662 234680300921369395263
469360601482738790464 9387212036854775808
At row 50, both A and V share an error, that is, 249 =
572949953421312,when it should be 562949953421312. This is the
cause of all the subsequenterrors, so that 263 becomes
9387212036854775808 and not as it should be,i.e.,
9223372036854775808. On this account, both scribes made a mistake
inmultiplying 248 × 2. Alternatively, it may be that the error was
already presentin the manuscript from which they copied. In
contrast to the Arabic sources which
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14 A. Bardi
provide progressions in columns, no conversion between units of
measurement isundertaken.
Both manuscripts provide two texts about the properties of the
geometricprogression.
Demonstration of the doubled numbers for each houseSeek the
doubling of a house you wish; from this subtract 1; the remaining
number is
the sum of the above numbers of the doubled houses; in this way,
the first house doubledup to the eighth becomes 128; subtract 1,
127 remains; and so this is the whole sum of thedoubling of the
first house until the seventh; 127; in fact, the sum of 1 and 2 and
4 and 8and 16 and 32 and 64 becomes 127; the same happens also for
the others. (My translation:cfr. A, f. IVr; V, f. 147r)
It is equivalent to this: 1 + 2 + 22 + 23 + 24 + 25 + 26 = 127 =
27 − 1.This rule can be applied to all the numbers provided in the
two columnsof the doubling. Therefore, in general terms, we could
write in this way,1 + 2 + 22 + 23 + 24 + 25 + . . . + 2n = 2n + 1 −
1.
There is also another demonstration for the group of five
houses, namely the fifth, thetenth, the fifteenth, the twentieth,
and the houses which follow and which are sought for5, namely 16,
which lays beside the fifth house; multiply this by 32 and the
multiplicationof these is 512; this is the doubling of the tenth
house, 512; multiply this by 32 and themultiplication of these is
16384; this is the doubling of the fifteenth house, 16384;
multiplythis by 32 and the multiplication of these is 524288; this
is the doubling of the twentiethhouse, 524288; the same also
happens for the other [numbers] that are sought for groups offive.
(My translation; cfr. A, f. IVr; V, f. 147r)
The second text deals with the following property about the
fifth house (24) ofthe progression: by multiplying the fifth house
by the very next house (sixth), onegets the number corresponding to
the tenth house; by multiplying the tenth houseby the sixth house,
one gets the fifteenth house; and so on. This is equivalent to24 ×
25 = 29, 29 × 25 = 214, 214 × 25 = 219, etc. The multiplication
property with25 remains valid in all the terms of the geometric
progression. In general terms, wecan write: 2n × 25 = 2n + 5.
We have found an unpublished geometric progression of 2 on the
chessboard inthe fifteenth-century Greek manuscript Ambrosianus E
80 sup. (Martini and Bassi1906, 329–331). In contrast to the Greek
and the Arabic geometric progressionsmentioned above, this example
contains an unusual trait: it arranges the geometricprogression
into an 8 × 8 chessboard and, similarly to the Arabic progressions
incolumns, employs conversions between units of measurement.
On the one hand, the current progression provides no sum of the
terms of theprogression, in exactly the same way as the
progressions in columns mentionedabove. On the other hand, in this
instance, the Greek numerals are employed in theirstandard
notation, and no further text is provided after the display.
In the following, we provide the progression on the chessboard
as it appears inthe manuscript Ambrosianus E 80 sup. Letters and
numbers in bold are added tofacilitate the commentary.
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
15
Zatrikion
A B C D E F G H
1 [?] 1 2 4 8 16 32 64 1282 256 512 1024 2048 4096 8092 [?]
16384 23 4 8 16 32 [?] 64 [?] 2 4 84 16 32 64 128 256 512 1024
20485 4096 8192 [?] 16384 [?] 2 4 8 16 326 64 128 256 512 1024 2048
4096 81927 16384 32768 65536 131072 262144 524188 1048576 20971528
18619392 37238784 74477568 148955136 297910272 595820544
There is an error at cell F2, for it should be 8192, not 8092.
This is the cause ofall the subsequent mistakes in the other cells,
especially in the rows 7 and 8.
This geometric progression of 2 employs a conversion system
similar to thesamples seen in the Arabic sources, but the
abbreviations of the Greek language,which are located in the places
where we have inserted question marks, have notyet been solved by
expert Greek philologists and paleographers. New light willbe shed
on this by solving those abbreviations. The abbreviated nouns have
thesame function as the units of measurement encountered in the
Arabic geometricprogressions in columns, which is to make
computations easier by means of dealingwith numbers smaller than
the raw results.
As usual, the number 1 (A1) is the initial term of the
progression with thecommon ratio 2. The progression continues till
215 (G2). From here, the partialresult is converted into a
different unit of measurement, and the progressioncontinues with
the same ratio up to cell E3, where we find 43. Then the
partialresult is converted into a different unit of measurement. At
cell C5, we find anotherconversion, and then the progression
continues until square F8.
Western Sources on the Computation 264 − 1The wheat and
chessboard problem is discussed not only in the Middle East andthe
Eastern Mediterranean but also in the medieval and premodern West,
both incollections of recreational mathematics and treatises on the
properties of numbers.
The famous collection of mathematical problems entitled
Propositiones adacuendos juvenes (Problems to Sharpen the Young)
constitutes the earliest Westernsource and is ascribed to the
English polymath Alcuin of York (ca. 800) (Folkerts1978). This work
is also the earliest collection of mathematical problems inLatin
literature. Problem 13 (Proposition 13) contains a version of the
wheat andchessboard problem in a similar fashion to the Arabic
version provided in a bookof algebra by Abū Kāmil (Høyrup 1986).
Instead of wheat or grain or gold, it dealswith the doubling of
soldiers of the army of a king (Folkerts 1978, 51–52). The
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16 A. Bardi
reduplication occurs 30 times. The following is a modern
translation of Alcuin’sProposition 13:
A king ordered his servant to collect an army from 30 villages
as follows: He should bringback from each successive village as
many men as he had taken there. The servant went tothe first
village alone; he went with one other man to the second village; he
went with threeother men to the third village. How many men were
collected from the 30 villages?
Solution.After the first village there are 2 men in the army,
there are 4 after the second village,
there are 8 after the third village, . . . . So after the nth
village there will be 2n men collectedfor the army. After the 30th
village there will be 230 = 1073741824 men in the army.(Translation
O’Connor and Robertson 2012)
As for the wheat and chessboard problem in the version with 64
reduplications,it has also survived in the West. Sources attest to
its progression after Alcuin; wefind it in the late Middle Ages and
in the Renaissance. An example of the standardwheat and chessboard
problem is provided by the Italian mathematician LeonardoFibonacci
(c.1170–c.1250) in his renowned Liber Abaci (1202 CE)
(Boncompagni1857, 309–318; Sigler 2002, 435–445). The Liber Abaci,
redacted in Latin, containsseveral problems concerning geometric
progressions, sometimes accompanied bytales, which constitute an
example of the application of the method; for instance, astory
about a financial dispute (on the sum of debts between two parties)
providesthe application of a geometric progression on a chessboard
to show the usefulness ofsuch a computational method in a
problem-solving context. The coinage mentionedby Fibonacci is the
denarius (denaro), a silver coin originating in the
third-centuryRome (Metcalf 2012, 300):
A certain man gave one denaro [denarius] at interest so that in
five years he must receivedouble the denari, and in another five he
must have double two of the denari, and thusforever from 5 to 5
years the capital and interest are doubled; it is sought how many
denarifrom this one denaro he must have in 100 years; you divide
the 100 years by the 5; thequotient will be 20; therefore the
denaro is doubled twenty times. Whence 20 places ofthe chessboard
carry a similarity; therefore if we shall double the denaro twenty
times weshall have the amount to which the denaro increases in 100
years; or in another way youdouble the denaro; there will be 2
which is the number to which the denaro grows in thefirst five
years; and the 2 again you multiply by itself; there will be 4
which is the numberto which the denaro increases in the second five
years; this 4 you again multiply by itself;there will be 16 which
is the amount to which the denaro increases in four times five
years;this 16 you double; there will be 32 for the amount after
five times five years; this 32 youmultiply by itself; there will be
1024 for the amount after ten times five years; this 1024you
multiply by itself yielding 1048576 denari for the amount after
twenty times five years,namely the 100 years; this is 4369 pounds
and 16 denari. The same method works for a manwho sold 20 pairs of
hides; from the first he had 1 denaro; for the second 2, for the
third4, and thus forever doubling up to the last pair; the sum is
the aforesaid amount minus 1denaro. (Translation Sigler 2002,
437–438)
Similarly to Arabic mathematicians, Fibonacci provides methods
to help thereader visualize the immense quantity of the result of
264 − 1 (Sesiano 2014, 149–150). The first method describes how to
fill a cash box with coins. The coinageemployed by Fibonacci in
this text is the bezant (bizantius aureus), a gold coin
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
17
that originated in Byzantium (Grierson 1991, 1, 287).
Fibonacci’s text is self-explanatory:
From the sum of two rows of the chessboard 65536 is summed,
namely from 16 places, andfrom these one coffer [aura] is filled,
and then in order this coffer is doubled, and thus weshall have in
the seventeenth place, namely in the first place of the third row
two coffers;in the second place of the same row there are 4
coffers, in the third 8, in the fourth 16, inthe fifth 32, in the
sixth 64, in the seventh 128, in the last place of the same row
256. In thefirst place of the fourth row 512. In the second place
1024, in the third 2048, in the fourth4096, in the fifth 8192, in
the sixth 16384, in the seventh 32768, and in the last place
youwill have 65536 coffers; from this if we shall fill one house
[domus], then we shall have inthe first place of the fifth row 2
houses. In the second 4, in the third 8, and thus doubling inorder
we shall have in the last place of the sixth row 65536 houses. From
these if one shallmake one city [civitas], and the remaining places
we continue doubling, then we shall havein the last place of the
chessboard 65536 cities; therefore the sum of all the numbers onthe
chessboard reaches 65536 cities; each city has 65536 houses, and in
each house thereare 65536 coffers, and in each coffer there are
65536 bezants; because of the abovesaiddemonstration one must have
in one coffer 1 bezant less (translation Sigler 2002, 435–436).
Later on, Fibonacci explains the sum by means of ships that are
to be filled withgrains of corn, eventually claiming “that the
number of ships is effectively infiniteand uncountable is here
easily observed”:
And you will wish to double beginning with one grain of corn on
the first place, and youwish to know how many ships are needed to
carry the corn if each ship will carry 500Pisan modia, each of
which is 24 sestari, each of which weighs 140 pounds, each of
whichweighs 12 ounces, and each ounce weighs 25 pennyweights; each
pennyweight weighs 6carobs; each carob weights 4 grains of corn;
all of these are disposed under a fraction inorder thus: 1/4 0/6
0/25 0/12 0/140 0/500, that is 18446744073709551615 and is the sum
ofthe grains of corn on the chessboard, you divide by the above
written parts that are underthe fraction, and whatever will remain
over the 4 that is at the head of the fraction will begrains, and
whatever will remain over the 6 will be carobs, and whatever is
over the 25will be pennyweights, and whatever is over the 12 will
be ounces, and whatever is over the140 will be pounds, and whatever
is over the 24 will be sestari, and whatever will remainover the
500 will be modia; truly the integer which remains after the
division will be thenumber of ships to be loaded, as here is shown:
3/4 3/6 0/25 6/12 115/140 13/24 123/5001725 028 445; that the
number of ships is effectively infinite and uncountable is here
easilyobserved. And you note that the 500 modia of each ship are
seagoing modia, namely 16000Roman modia, or 8000 Syrian modia, or
4000 Sicilian salme. Truly in the second of thedoubling of the
chessboard squares, namely when any place in the sequence of places
isproposed to be the sum of all the preceding doubles, one can find
them in two ways; thefirst is indeed from place to place up to the
last number. The second truly is the way inwhich you take the 1
which is proposed for the first place, and you add it to the 2 that
is putin the second place; there will be 3 which you multiply by
itself; there will be 9 which isthe number of the sum of the first
and second places, namely three. For example, if in thefirst place
1 is put, in the second two, and in the third 6, namely double the
sum of the twopreceding places, the sum of them will be 9, as we
said before; this 9, if it is multiplied byitself makes 81 which
number is the sum of the first place and double the two
followingplaces, namely 5 places. For example, if at the first
place 1 is put, at the second 2, at thethird 6, at the fourth 18,
at the fifth 54, undoubtedly they add to 81; if you multiply this81
by itself, then it makes 6561; this number is the sum of the first
place and double thenn following places, namely 9 places. For
example, the number at the first place is 1, thesecond 2, the third
6, the fourth 18, the fifth 54, the sixth 162, the seventh 486, the
eighth
-
18 A. Bardi
1458, the ninth 4374; all added together they make 6561, which
6561 you multiply by itselfmaking 43046721; this above written
number disposed is the sum of the doubles of thefirst place and
double eight following places, namely 17 places. Whence if you
multiplythe 43046721 by itself, then 1853020188851841 will result
for the sum of the doubles ofthe first place and double 16 places,
namely 33 places; this number multiplied by itselfyields
3433683820292512484657849089281 for the doubles of the entire
chessboard andone place more; this place is double the entire
chessboard; therefore it must be that a thirdpart of the above
written number is the sum of the doubles of all the chessboard
squares;therefore this number divided by 3 yields for the sum of
all the doubles on the chessboardsquares 1144 561 273 430 837 494
885 949 696 427. (Translation Sigler 2002, 436–437)
Fibonacci’s account is the most comprehensive source on the
wheat and chess-board problem in Latin literature.
A few centuries later, the Italian mathematician Luca Pacioli
wrote his mas-terpiece Summa de arithmetica in Italian, which
contains another account of thewheat and chessboard problem
(Pacioli 1494). (Full title: Summa de arithmetica,geometria,
proportioni et proportionalita, that is, Summary of arithmetic,
geometry,proportions and proportionality.) Pacioli provides two
methods for doubling grainsof corn on a chessboard. Here is an
excerpt from Pacioli’s chapter on the wheat andchessboard
problem:
To double a grain of corn as many times as there are white
houses and black ones on thechessboard, which are in sum 64, can be
understood in two ways. The other way means thatonly the following
house doubles the preceding house and nothing more. In this way it
doesnot grow up as much as the previous way. Now, in order to
double a small grain in this way,be aware that if you add the first
term to the sum of 4 houses, and you multiply the sum foritself,
and from that product you subtract the first term you added, what
remains is equal tothe sum of the doubling of 8 houses, for the
doubling of 4 houses is 1 2 4 8, whose sumis 15. I say that if you
add the first term, which is 1, it makes 16. Multiply this by
itself, itmakes 256; subtract the first term, it remains 255. So
much will be the sum of the housesmultiplied in the way mentioned.
(My translation; cf. Pacioli 1494, f. 43 nr. 28)
Pacioli’s methods would be reprised in another important
mathematical treatiseredacted in Italian, namely, General trattato
di numeri et misure (General treatiseon numbers and measures;
printed in 1556) by the Italian mathematician NiccolòTartaglia
(c.1499–1557). His works contains a chapter devoted to geometric
pro-gressions. We learn from his text that Tartaglia considers
Pacioli an authority on thewheat and chessboard problem, for he
explicitly refers to him in methods to solveit: “according to the
rules of the friar Luca” (cf. Tartaglia 1556, book 1, f. 16r).
Further discussions of the wheat and chessboard problem are
provided in Latinand in Romance or Germanic languages, for
instance, in the so-called AlgorismusRatisbonensis, a
fifteenth-century collection of problems in Latin and
German,redacted by a monk (Frederick) in the Benedictine monastery
of Regensburg (Folk-erts 1971, 71; Sesiano 2014, 9 n.12), and in
the so-called Algorismus Columbia, afourteenth-century collection
of problems similar to Fibonacci’s Liber Abaci (Vogel1977).
The Western sources that include the wheat and chessboard
problem often have afinancial or trading focus, which sits
alongside the mathematical nature of the workthat provides the
context for the problem. This reflects Italy’s economic growth
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
19
between the thirteenth and sixteenth centuries. The works of
Fibonacci, Pacioli, andTartaglia were the most advanced accounts of
their age in computational techniques.These were useful not only
for Italian scholars but also for merchants and traders,whose
activities were indeed flourishing at the time when Fibonacci,
Pacioli, andTartaglia redacted the treatises.
In contrast to Arabic sources, Western sources never present
tales about theinvention of chess and give no hints as to the
origins of the problem; they simplydeal with applications of the
geometric progression on the squares of the chessboard,accompanied
by methods and considerations about the properties of the
progressionitself.
Moreover, the Western sources provide evidence of the
dissemination of ourproblem in non-mathematical literature. For
instance, the renowned Dante Alighieri(1265–1321) alludes to the
wheat and chessboard problem in a verse of his Paradise(see the
verses quoted above in the epigraph to this chapter). This
demonstrates thatour problem was being discussed among learned
Italians as early as the thirteenthcentury.
Number Theory
The mathematical possibilities triggered by the wheat and
chessboard problemare not confined to the Renaissance; indeed, they
embrace a modern branch ofmathematics known to date as number
theory. By computing 264 − 1, as wellas any 2n − 1 of the geometric
progression of the chessboard, we obtain a so-called Mersenne
number. This name derives from Marin Mersenne (1588–1648), aFrench
polymath who devoted some of his scholarship to studying the
propertiesof numbers. Mersenne numbers are equal to a power of 2
minus 1. Given aninteger n, we will obtain a number Mn that is
equal to 2n − 1. Therefore,Mn = 2n − 1. Following Mersenne’s
puzzles, mathematicians have been challengedto find criteria to
determine which integer n we must use so that Mn = 2n − 1results in
a prime number. To obtain such special numbers, the integer n
mustbe one of the following: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89,
107, 127, 521, 607,1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941,
11213, 19937, 21701, 23209,44497, 86243, 132049, 216091, etc. Such
numbers are called Mersenne primes.Scholars have been using
computers to find large Mersenne primes. Such a pursuithas given
rise to significant collectives of mathematicians and number
amateurs,thus extending this pursuit beyond the boundaries of
academic mathematics intopopular culture. For instance, the Great
Internet Mersenne Prime Search (GIMPS –see the official website:
www.mersenne.org) comprises an international team ofvolunteers
developing software to search for Mersenne prime numbers. The
pursuitof Mersenne primes is still ongoing. To date, the largest
known Mersenne primeis 282589933 − 1, which is the 51st known
Mersenne prime. This discovery wasmade by Patrick Laroche and
officially claimed on December 21, 2018
(https://www.mersenne.org/primes/?press=M82589933). Moreover, the
properties of suchnumbers gave rise to new mathematical problems,
some of which remain unsolved.
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{}orghttps://www.mersenne.org/primes/?press=M82589933https://www.mersenne.org/primes/?press=M82589933
-
20 A. Bardi
For instance, mathematicians are currently discussing the
possibility of the existenceof an infinite set of Mersenne numbers
(see Mersenne number. Encyclopediaof Mathematics. Available through
http://www.encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=36008
– accessed 4/9/2019).
Summary
The history of the wheat and chessboard problem proves to be a
fil rouge throughcenturies and civilizations, leaving traces in
several cultural contexts, crossing lan-guages and literary genres.
Its earliest source stems from Babylonian mathematics ina version
where the reduplication occurs 30 times and is not related to a
chessboard.From that, the version featuring a unity to be redoubled
64 times on a chessboardwas developed, and this proves to be the
most widespread version (in Arabic, Greek,and Western sources). The
first reference to the latter version traces back to the
ninthcentury and is transmitted by Arabic sources. This is
precisely the age in which wehave evidence of the relationship
between our problem and the game of chess andits invention. On this
account, evidence of intrinsic connections between chess
andmathematics can be traced back to the ninth century.
According to Arabic sources, the sum of the geometric
progression 264 − 1constitutes the answer of an Indian philosopher
who was invited by a queen or aking to express a personal wish, in
reward for having invented the game of chess. Itis likely that this
tale, like most ancient legends, was transmitted orally in the
firstinstance; as such, it is reasonable to surmise that the
mathematical possibilities ofchess would have been known to Indian
or Arabic mathematicians before the taleabout the invention of
chess was written down.
The Arabic and Latin literature on the wheat and chessboard
problem present awide range of things to be redoubled. The tale
about the invention of chess featuresgrains of corn or wheat, or
pieces of gold, various coinages, houses, cities, soldiers,and an
extensive variety of dry measurements. These units of measurement
are usedby mathematicians to make the computations easier.
The transmission of the wheat and chessboard problem cuts across
literarygenres. The earliest sources are historiographical works
which present it throughtales about the invention of chess, where
chess is considered as a war simulator orstrategical game at the
disposal of kings and courtiers. Later on, it is transmittedin
books of algebra, a mathematical genre typical of Arabic
literature. In Westernsources, it is extant in chapters of
recreational mathematics or general treatises oncomputations and
the properties of numbers. Notably, Luca Pacioli’s account onthe
wheat and chessboard problem is inserted in a book of mathematical
problemswhich are likely meant to serve as stimulation for
Renaissance financial accountants(bookkeepers). It is not
accidental that Pacioli was the first scholar to systematize
thetechnique of double-entry bookkeeping (Fischer 2000, 303–304).
On this account,the history of the wheat and chessboard problem
mirrors the cultural contexts inwhich it is employed.
http://www.encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=36008http://www.encyclopediaofmath.org/index.php?title=Mersenne_number&oldid=36008
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Mathematics and Cultures Across the Chessboard: The Wheat. . .
21
All of this proves that the wheat and chessboard problem has a
great degree ofversatility. From a perspective of tale
transmission, the most widespread versionof our problem was born
through a combination of the invention of chess with atale that
allows for the application of geometric progression. Later on, the
secondcomponent become popular among Arabic mathematicians, and the
invention of thegame of chess was left to one side. The application
of the geometric progressionproved to be a field of
experimentation: Arabic and Western mathematicians haveprovided not
only extensive considerations of the properties of the progression
buthave also explored how to best visualize the geometric
progression and its finalresult. Apart from explanations in textual
form, the most common representationalformat is the column with the
number of the chessboard square alongside thecorresponding result
of the reduplication. In this instance, the Greek literatureproves
to be original inasmuch as it provides a representation of the
progressioninto an 8 × 8 chessboard. Moreover, the final result of
264 − 1 inspired Arabic andWestern mathematicians to develop
methods to visualize this huge quantity. In thiscase, one of the
methods consisted in distributing 264 − 1 throughout the surface
ofthe Earth, thus showing that only a tiny part of the total sum
was enough to cover thewhole globe. This method includes
measurements of the surface, the circumference,and the diameter of
the Earth, thus testifying to an interplay between the wheat
andchessboard problem and one of the mathematical problems that
puzzled thinkersfrom Antiquity: measuring the surface of the
Earth.
Cross-References
�Mathematics and Big Data�Mathematics and Economics/Social
Choice Theory� Puzzles and Mathematics
Acknowledgments A special thank to Sandro Caparrini for fruitful
discussions. For usefulsuggestions I am indebted to several
anonymous reviewers as well as Roshi Rashed, Jens Hoyrup,Stefan
Deschauer, Richard Kremer, and Reyhan Durmaz. This chapter was
completed duringa research-stay at Dumbarton Oaks Research Library
and Collection (Harvard University), aninstitution I do wish to
thank. I am grateful to the Polonsky Academy and the Van Leer
Institute aswell.
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Mathematics and Cultures Across the Chessboard: The Wheat and
Chessboard ProblemContentsIntroductionMathematics and the Invention
of ChessMathematics and the Origins of ChessGeometric Progressions
and ChessArabic Sources on the Computation 264 − 1Greek Sources on
the Computation 264 − 1Western Sources on the Computation 264 −
1Number TheorySummaryCross-ReferencesReferences