Chapter 2

M A K E A S Y S T E M AT I C L I S T 25

22Make a Systematic List

When you make a systematic list, you reveal the structure of a problem. Sometimes the list is all you need to solve it. Trainschedules are systematic lists that help travelers find informationeasily and quickly.

2722_KC_Johnson_Ch02 9/22/03 10:56 AM Page 25

LOOSE CHANGE

Leslie has 25¢ in her pocket but does not have a quarter. If you can tell her all possible combinations of coins she could have that add up to 25¢, she willgive you the 25¢. Solve this problem before continuing.

M

26 C H A P T E R 2

any people start solving this problem as follows: “Let’s see, we could have 5 nickels, or 2 dimes and 1 nickel. We might have

25 pennies. Oh yeah, we could have 10 pennies, 1 dime, and 1 nickel.Perhaps we could have . . . .” Solving the problem this way is extremelyinefficient. It could take a long time to figure out all the ways to make25¢, and you still might not be sure that you’d thought of all the ways.

A better way to solve the problem is to make a systematic list.A systematic list is just what its name says it is: a list generated throughsome kind of system. A system is any procedure that allows you to do something (like organize information) in a methodical way. Thesystem used in generating a systematic list should be understandableand clear so that the person making the list can verify its accuracy quickly.Additionally, another person should be able to understand the systemand verify the solution without too much effort.

Many systematic lists are in the form of a table whose columns arelabeled with the information given in a problem. The rows of the tableare used to indicate possible combinations. As you read the followingsolutions for the Loose Change problem, make your own systematiclist. Label the columns of the list Dimes, Nickels, and Pennies, and thenfill in the rows with combinations of coins that add up to 25¢.

Brooke started her list inthe first row of the Dimescolumn by showing themaximum number of dimesLeslie could have: two. In theNickels column, she showedthe maximum number ofnickels possible with twodimes: one. In the second row,she decreased the number of nickels by one because it’spossible to make 25¢ withoutusing nickels. She then filled in the Pennies column by showing how

Dimes Nickels Pennies

2 1 0

2 0 5

1 3 0

1 2 5

1 1 10

1 0 15

and so on


many pennies she had to add to her dimes and nickels to make 25¢.After finding all the ways to make 25¢ with two dimes, Brooke continuedfilling in her list with combinations that include only one dime. In thethird row, she showed the maximum number of nickels possible inone-dime combinations: three.As she did for the two-dimecombinations, she decreasedthe number of nickels by onein each row until she ran outof nickels.

Brooke’s completed list is shown at right. It includesall the possible zero-dimecombinations. Finish yourown list before reading on.

Brooke’s systematic list is not the only one that willsolve this problem. Heatherused a different system.Before you look at her entiresolution, which follows,cover all but the first threerows of the table at bottomright with a piece of paper.Look at the uncovered rowsto figure out her system,and then complete the listyourself.

Heather explained hersystem like this: “I startedwith the largest number of pennies, which was 25.Then I let the pennies godown by fives and filled inthe nickels and dimes tomake up the difference.”

Making systematic listsis a way to solve problemsby organizing information.In this chapter you’ll make

Dimes Nickels Pennies

2 1 0

2 0 5

1 3 0

1 2 5

1 1 10

1 0 15

0 5 0

0 4 5

0 3 10

0 2 15

0 1 20

0 0 25

Pennies Nickels Dimes

25 0 020 1 015 2 015 0 110 3 010 1 15 4 05 2 15 0 20 5 00 3 10 1 2



systematic lists to organize information in tables and charts. You willalso learn a little about using a special type of diagram called a treediagram. Many of the strategies you’ll explore later in this book involveorganizing information in some sort of table or chart, and you’ll learnother strategies that involve organizing information spatially.

Remember that there is often more than one correct approach tosolving a problem. This is often true with devising systematic lists.Many different systems can produce a solution to a given problem.When you solved the Loose Change problem, you may have used adifferent list than those that Brooke and Heather used. Any list is fineas long as you have a system that you understand and can useeffectively. If you find that your original system is too confusing, try a different system.

Just as you can use the same strategy, such as making a list, to solve a problem in different ways, you will also often find that you can usemore than one strategy to solve a given problem. In Chapter 1 you solvedthe Virtual Basketball League problem with a diagram. Solve theproblem again, but this time use a systematic list. Don’t refer back tothe diagram solution!

VIRTUAL BASKETBALL LEAGUE

Andrew and his friends have formed a fantasy basketball league in which eachteam will play three games against each of the other teams. There are seventeams: the (Texas A&M) Aggies, the (Purdue) Boilermakers, the (Alabama)Crimson Tide, the (Oregon) Ducks, the (Boston College) Eagles, the (AirForce) Falcons, and the (Florida) Gators. How many games will be played in all? Do this problem before reading on.

Michael is a basketball player, and he’s always interested in the matchups. In this problem there are seven teams, which Michaelquickly assembled into pairs of teams for games:

Aggies vs Crimson Tide Crimson Tide vs DucksBoilermakers vs Gators Gators vs AggiesFalcons vs Aggies Crimson Tide vs GatorsDucks vs Eagles Boilermakers vs Aggies

28 C H A P T E R 2

There is often morethan one correctapproach to solvinga problem.


Crimson Tide vs Gators Falcons vs EaglesEagles vs Boilermakers Ducks vs GatorsCrimson Tide vs Gators Crimson Tide vs AggiesEagles vs Ducks Ducks vs BoilermakersBoilermakers vs Eagles Gators vs Eagles

Is Michael’s list systematic? Are all possiblematchups represented? Does the list contain omissionsor duplications?

Instead of trying to verify the accuracy of Michael’snonsystematic list, look at the first two columns ofMonica’s systematic list, at right.

Monica represented each of the teams by the firstletter of its name. For example, AB represents amatchup between the Aggies and the Boilermakers.She started her list by showing the matchups betweenthe Aggies and the other six teams. In the second column of her list, sheshowed the matchups between the Boilermakers and the other teams.Note that she didn’t include the matchup between the Aggies and theBoilermakers because she’d already shown it in the first column.

She continued by listing, in order, the opposing teams for eachremaining matchup. The complete list is shown below.

AB BC CD DE EF FGAC BD CE DF EGAD BE CF DGAE BF CGAF BGAG

There are 21 different pairs of teams, and each pair played 3 gamesagainst each other. So to answer the question “How many games willbe played in all?” multiply 21 by 3. The answer is 63 games.

Now compare Monica’s solution to this problem’s diagram solutionsin Chapter 1. You can see that the diagram lines, which representgames, were drawn systematically so that they’d be easy to understandand follow. Diagrams are often systematic. Notice also that the diagramlines correspond exactly to the pairs in Monica’s list.


AB BC

AC BD

AD BE

AE BF

AF BG

AG

Don’t list the samecombination twice.

Look for patternswithin your list.


PENNY’S DIMES, PART 1

Nick’s daughter Penny has 25 dimes. She likes to arrange them into threepiles, putting an odd number of dimes into each pile. In how many wayscould she do this? Solve this problem before continuing.

Randy solved this problem bymaking a systematic list of thepossible combinations. He madethree columns for his list andcalled them Pile 1, Pile 2, and Pile 3. In the first row of the list,he indicated the first combinationof dimes. He put 1 dime in thefirst pile and 1 dime in the secondpile. This left 23 dimes for thethird pile. In the second row hestarted again with 1 dime in thefirst pile, then increased the second pile by 2 and decreased the thirdpile by 2. (Remember that each pile contains an odd number of dimes.)He continued in this way for a while, as shown above.

At this point in his list, Randy needed to decide whether or not 1, 13, 11 is a repeat of 1, 11, 13. In other words, is 13 in one pile and 11 in the other the same as 11 in one pile and 13 in the other? Randydecided that the piles were indistinguishable and therefore that thesetwo combinations were indeed thesame. He realized that crossing outrepeats would save him a lot ofwork and make his list a lot shorter.So he crossed out the row with 1, 13, and 11. The next combinationwould be 1, 15, 9, which is a repeatof 1, 9, 15. So he concluded thathe’d exhausted the combinationsfor 1 dime in the first pile.

Next he began findingcombinations that started with 3 dimes in the first pile. The firstcombination he wrote down was

30 C H A P T E R 2

Pile 1 Pile 2 Pile 3

1 1 23

1 3 21

1 5 19

1 7 17

1 9 15

1 11 13

1 13 11


1 1 23

1 3 21

1 5 19

1 7 17

1 9 15

1 11 13

1 13 11

3 1 21

3 3 19


3, 1, 21. He quickly crossed out this combination because he realizedthat 3, 1, 21 was a repeat of the second combination in the list, 1, 3, 21.So he started with 3, 3, 19. He continued listing combinations with 3 dimes in the first pile until he reached 3, 11, 11. He stopped at thiscombination because he knew the next combination would be 3, 13, 9,which again would be a repeat.

Randy then moved on to listing combinations with 5 dimes in the firstpile. To avoid repeating 5, 1, 19 and 5, 3, 17, he started his combinationswith 5, 5, 15. He realized that when he changed the number in the firstpile, he had to use that same number in the second pile to avoid repeatingan earlier arrangement. He also noticed that he began to get repetitiouscombinations after the number in the second pile became as high asthe number in the third pile. For example, when he reached 1, 13, 11,he had a repeat of 1, 11, 13. So here is the primary pattern present inthis list: When moving from the first pile to the second pile to the thirdpile, the numbers cannot decrease. The second pile must be equal to orgreater than the first pile, and the third pile must be equal to or greaterthan the second pile. This type of pattern can appear in manysystematic lists.

Randy continued with his list, using the pattern he’ddiscovered. When he began listingcombinations with 9 dimes in the first pile, his first combinationwas 9, 9, 7. At that point his listwas complete, because 9, 9, 7 is a repeat of 7, 9, 9.

Randy’s complete list is shownat right. There are 16 ways toform three piles of dimes.

You can solve this problemdifferently by experimenting withother systems—we encourage you to do so. One possible systemwould begin with 23 dimes in thefirst pile. You might also decide tosolve the problem again, but thistime assume that the three pilesare distinguishable, which leads to



1 1 23

1 3 21

1 5 19

1 7 17

1 9 15

1 11 13

3 3 19

3 5 17

3 7 15

3 9 13

3 11 11

5 5 15

5 7 13

5 9 11

7 7 11

7 9 9

Finding a primarypattern helps toshorten the list.


a much longer list that has 78 possibilities. You should make this list,too. You’ll have to modify the system that Randy used, because it willno longer be true that 1, 3, 21 is the same as 3, 1, 21.

FRISBIN

On a famous episode of Star Trek, Captain Kirk and the gang played a cardgame called Phisbin. This problem is about another game, called Frisbin. Theobject of Frisbin is to throw three Frisbees at three different-sized bins thatare set up on the ground about 20 feet away from the player. If a Frisbee landsin the largest bin, the player scores 1 point. If a Frisbee lands in the medium-sized bin, the player scores 5 points. If a Frisbee lands in the smallest bin,the player scores 10 points. Kirk McCoy is playing the game. If all three of hisFrisbees land in bins, how many different total scores can he make? Make asystematic list for this problem before reading on.

You can make two different types of systematic lists for this problem.An example of each follows.

Derrick set up a list with columns titled 10 Points, 5 Points, 1 Point,and Total. He began by indicating the maximum number of 10-pointthrows: 3. He continued by indicating the other possible 10-point throws:2, 1, and 0. In each row he adjusted the 5-point and 1-point throws so that three throws were always accounted for. After calculating allthe point totals, Derrick concluded that Kirk McCoy can make tendifferent total scores.

10 POINTS 5 POINTS 1 POINT TOTAL

3 0 0 302 1 0 252 0 1 211 2 0 201 1 1 161 0 2 120 3 0 150 2 1 110 1 2 70 0 3 3

32 C H A P T E R 2


Notice the system in the list. The 10 Points column starts with thehighest possible number of throws, then decreases by 1. The columnentry stays at each particular possible number of throws (3, then 2,then 1, and finally 0) as long as it can. The 5 Points column follows a similar process: It starts with the highest possible number of 5-pointthrows for each particular score and decreases by 1 each time. The 1 Point column makes up the difference in the scores.

Derrick made this list very quickly, and anyone seeing the list for the first time should immediately be able to follow the system. To helpensure that the system is evident, we have provided an explanation of it. In this course, when you write solutions to problems that you’llturn in to your instructor, you’ll be asked to also provide a writtenexplanation of your work. By explaining your work, you’ll not onlybecome a better problem solver, but you’ll also become proficient at explaining your reasoning, which is a very valuable skill.

Julian used a different method, shown next. He labeled each columnwith the number of the three possible throws. Then he wrote downthe points for each throw. Describe Julian’s system.

THROW 1 THROW 2 THROW 3 TOTAL

10 10 10 30

10 10 5 25

10 10 1 21

10 5 5 20

10 5 1 16

10 1 1 12

5 5 5 15

5 5 1 11

5 1 1 7

1 1 1 3

Julian started by letting the first throw earn 10 points. He then adjustedthe other two throws to include all possible point combinations. Whenhe ran out of combinations for which the first throw earned 10 points, hebegan listing combinations for which the first throw earned 5 points, then1 point. (Note that he didn’t repeat score possibilities by rearrangingthe points earned. In other words, 10, 10, 5 is the same as 10, 5, 10 or



5, 10, 10.) The point totals came out exactly the same as those inDerrick’s list, but Julian’s approach made it easier to add up the totalscores.

What system did Cali use in the list below? Before reading further,study her list to figure out her system.

Throw 1 Throw 2 Throw 3 Total

10 10 10 30

5 5 5 15

1 1 1 3

10 10 5 25

10 10 1 21

5 5 10 20

5 5 1 11

1 1 10 12

1 1 5 7

1 5 10 16

Cali started by listing those situations where all three throws landedin the same bin. Then she listed the situations where two throwslanded in the same bin. Finally she listed the one possibility where allthree landed in different bins.

AREA AND PERIMETER

A rectangle has an area measuring 120 square centimeters. Its length andwidth are whole numbers of centimeters. What are the possible combinationsof length and width? Which possibility gives the smallest perimeter? Work this problem before continuing.

Tuan explained his solution for this problem: “I read that the area of the rectanglewas 120 square centimeters. The first thing I did was to draw a picture of a rectangle.

34 C H A P T E R 2


“I had no idea whetherthis rectangle was long andskinny, or shaped like asquare. But I did know thatthe area was supposed tobe 120 square centimeters.So I made a list of whole-number pairs that could bemultiplied to get 120.

“I knew I was done atthis point because the nextpair of factors of 120 is 12and 10, which I’d alreadyused. A 12-by-10 rectangle is the same as a 10-by-12 rectangle turnedon its side, and I saw no need to list it twice. I also realized that neither7 nor 9 would work for the width, because they don’t divide evenlyinto 120.

“Now I had to find which possibility gives the smallest perimeter. I knew that the perimeter of a rectangle is the distance around therectangle, so I needed to add up the length and width. But this wouldonly give me half of the perimeter, so I would have to double the sumof the length and width. I added the Perimeter column to my chart.”

WIDTH LENGTH AREA PERIMETER

1 cm 120 cm 120 cm2 242 cm

2 cm 60 cm 120 cm2 124 cm

3 cm 40 cm 120 cm2 86 cm

4 cm 30 cm 120 cm2 68 cm

5 cm 24 cm 120 cm2 58 cm

6 cm 20 cm 120 cm2 52 cm

8 cm 15 cm 120 cm2 46 cm

10 cm 12 cm 120 cm2 44 cm

“Now I can see from my chart that the rectangle measuring 10centimeters by 12 centimeters (which does have an area of 120 cm2)gives the smallest perimeter of 44 cm.”


WIDTH LENGTH AREA

1 cm 120 cm 120 cm2

2 cm 60 cm 120 cm2

3 cm 40 cm 120 cm2

4 cm 30 cm 120 cm2

5 cm 24 cm 120 cm2

6 cm 20 cm 120 cm2

8 cm 15 cm 120 cm2

10 cm 12 cm 120 cm2


WHICH PAPERS SHOULD KRISTEN WRITE?

For her Shakespeare course, Kristen is to read all five of the following plays andchoose three of them to write papers about: Richard III, The Tempest, Macbeth,A Midsummer Night’s Dream, and Othello. How many different sets of threebooks can Kristen write papers about? Do the problem before continuing.

Li explained her systematic list, shownat right: “I decided to abbreviate thenames of the books so I wouldn’t haveto write out the whole names each time.I used R3, TT, Mac, AMND, and Oth.Then I just made a list. I made my list byletting R3 stay in front as long as it could,and rearranged the other four books intothe remaining two spots. Once I had all the combinations that include R3, I dropped it from the list. Then I usedTT in the first spot and listed all thecombinations that included it. Then Idropped TT, and finally I used Mac in thefirst spot. I listed the combination that included Mac, but by that timethere was only one more way to do it. There are ten ways altogether.”

Travis used a different systematic list to solve the problem: “I madecolumns for the different books, and then I checked off three in eachrow. There are ten ways.”

R3 TT Mac Oth AMND

1 x x x

2 x x x

3 x x x

4 x x x

5 x x x

6 x x x

7 x x x

8 x x x

9 x x x

10 x x x

36 C H A P T E R 2

R3 TT MacR3 TT OthR3 TT AMNDR3 Mac OthR3 Mac AMNDR3 Oth AMNDTT Mac OthTT Mac AMNDTT Oth AMND

Mac Oth AMND


A tree diagram is another type of systematic list and is used toorganize information spatially. A tree diagram’s name reflects the factthat it looks like the branches of a tree. (Note that tree diagrams will be discussed further in Chapter 17: Visualize Spatial Relationships.)

After Hosa solved the Which Papers Should Kristen Write? problem,he wondered how many different orders Kristen could write the papersin once she had chosen the books. Suppose she chose Richard III, TheTempest, and Macbeth. Hosa solved the problem with a tree diagram.

Hosa’s tree diagram of the different orders in which the papers could be written for R3, TT, and Mac is shown at right.

Hosa explained: “The first branch of the tree shows the paper written first. The secondbranch shows the paper written next. In thesecond branch I didn’t repeat the paper that was written in the first branch. Finally, the third branch shows the paper written last.”

You can solve some systematic list problemswith tree diagrams. However, sometimes a tree diagram would be too confusing or cumbersome. You will need to decide when a treediagram would be more useful than a standard systematic list.

Make a Systematic List

Making a systematic list is a great way to organize information. Yourfirst attempt at a list will probably not be the one you end up using.

• Start with a messy list or several lists you are willing to give up.They will help you think more carefully about your planning.

• When you make your list, be sure you thoroughly understandyour system.

• Continue to monitor your system. When you reach a logicalbreak point, think carefully about the next entry so the next partof your list will continue the patterns you established earlier.

• If the system doesn’t seem to be working, don’t be afraid to revise it or to start over.

R3

TT

Mac

TT — Mac

Mac — TT

R3 — Mac

Mac — R3

R3 — TT

TT — R3


Plan your list, butdon’t hesitate tochange your plan.


Enjoy this strategy. Solving problems with systematic lists can be a lot of fun.

General Tips for Problem Solvers

As you work through this book, one major challenge you will face ischoosing an appropriate strategy to solve a problem. Often you willfind that the best strategy is the first one you chose, but sometimesyou’ll have to experiment with different strategies to see which one is most effective. As you work on the problems, keep the followingthoughts in mind:

• Sometimes you’ll need to use two or more strategies to solve a problem.

• Being persistent as you try different strategies will often pay off.

• On the other hand, you’ll need to develop a sense of when to trysomething completely different. Take a risk!

The most important thing to know about problem solving is thatmost problems can be solved. As you solve increasingly difficultproblems, your confidence and your abilities will increase.

Problem Set ASolve each problem by making a systematic list.

1 . C A R D S A N D C O M I C S

Charmaign’s daughter has $6.00 she wants to spend on comic booksand superhero cards. Comic books cost 60¢ each, and deluxe packagesof superhero cards cost $1.20 each. List all the ways she can spend allof her money on comic books, superhero cards, or both.

2 . T E N N I S TO U R N A M E N T

Justin, Julie, Jamie, Matt, Ryan, and Roland are the six players in around-robin tennis tournament. Each player will play a set against eachof the other players. List all the sets that need to be played.

38 C H A P T E R 2

Don’t give up—try a different strategy.


3 . F R E E C O N C E RT T I C K E T S

Alexis, Blake, Chuck, and Dariah all called in to a radioshow to get free tickets to a concert. List all the possibleorders in which their callscould have been received.

4 . A PA RT M E N T H U N T I N G

A management company offers two payment plans for leasing anapartment for one year. Plan A is designed so that a tenant’s entry costis low, and Plan B is designed so that there are more gradual priceincreases:

PLAN A PLAN B

12-month lease 12-month lease

$400 first month $500 first month

$30 per month increase $15 per month increase each month each month

Which plan costs more for only the ninth month of tenancy?

Which plan costs more for the entire year?

5 . S TO R AG E S H E D S

Andre’s company manufactures rectangular storage sheds. The shedsare made with aluminum side panels that measure 8 feet, 10 feet, 12 feet, and 15 feet along the bottom edge. For example, one possibleshed measures 10 feet by 10 feet. Another possible shed measures 12 feet by 15 feet. List the measurements of all the possible sheds.



6 . M A K I N G C H A N G E

Ms. Rathman has lots of nickels, dimes, and quarters. In how manyways can she make change for 50¢?

7 . F I N I S H E D P RO D U C T

The product of two whole numbers is 360, and their sum is less than100. What are the possibilities for the two numbers?

8 . B A S K E T B A L L

Yolanda scored 10 points in a basketball game. She could have scoredwith one-point free throws, two-point field goals, or three-point fieldgoals. In how many different ways could she have scored her 10 points?

9 . T W E N T Y- F O U R

How many ways are there to add four positive even numbers to get a sum of 24?

1 0 . TA R G E T P R AC T I C E

In a target shooting game, Spencer had four arrows. He hit the targetwith all four shots. With each shot he could have scored 25 points, 10 points, 5 points, or 1 point. How many total scores are possible?

1 1 . TA N YA ’ S T E R R I F I C T- S H I RT S

Tanya is visiting New Orleans, and she wants to bring back T-shirts for all her friends. She’s found T-shirts she likes for $5, $10, and $15.She has budgeted $40 for the gifts. List all the ways Tanya can spend$40 or less on T-shirts.

1 2 . W R I T E YO U R OW N

Create your own systematic-list problem.

C L A S S I C P R O B L E M S

1 3 . A R C H E RY P U Z Z L E

A target shows the numbers 16, 17, 23, 24, 39, and 40. How manyarrows does it take to score exactly 100 on this target?

Adapted from Mathematical Puzzles of Sam Loyd, vol. 2, edited by Martin Gardner.

40 C H A P T E R 2


1 4 . W H I C H B A R R E L WA S L E F T ?

There are six barrels, containing 15 gallons, 8 gallons, 17 gallons, 13 gallons, 19 gallons, and 31 gallons. Each barrel contains either oil or vinegar. The oil sells for twice as much per gallon as the vinegar. A customer buys $14 worth of each, leaving one barrel. Which barrelwas left?

Adapted from Mathematical Puzzles of Sam Loyd, vol. 2, edited by Martin Gardner.



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Chapter 2

Documents

list dimes

pennies nickels dimes

different list

brookes systematic list

given problem

nickels column

loose change problem

dimes column