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MATH ANDLOGIC GAMES


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Franco Agostini

MATH ANDLOGIC GAMES

Facts On File Publications460 Park Avenue South

New York, N.Y. 10016


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CONTENTS

7 Introduction

9 Games with numbers

9 An historical note11 A first criosity12 Fibonacci numbers14 A curious calculating dev ce: the abacus20 The origins of algebra21 Games with algebra23 Odds and evens24 The successor of a number24 A shortcut in calculations25 How much money is in your pocket?25 How to guess a birth date26 Guessing age and size of shoes26 Where is the error?27 Positional notation of numbers28 One rotten apple can spoil the whole basket29 Ordinary language and mathematical language

33 Games with geometrical figures33 Geometry and optical illusions41 Games with matches42 Lo shu, an ancient Chinese figure42 Magic squares: their history and mathematical features44 More intricate magic squares47 Diabolic squares48 Magic stars51 More about squares51 An extraordinary surface52 The bridges of Konigsberg55 Elementary theory of graphs57 Save the goat and the cabbage58 The jealous husbands61 Interchanging knights

62 A wide range of applications63 Topology, or the geometry of distortion70 Topological labyrinths73 The Mobius ring76 Games with topological knots77 The four-colour theorem84 Rubik's cube

91 Paradoxes and antinomies91 The role of paradoxes in he development of mathematical thought


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92 Pythagoras and Pythagoreanism94 Geometrical representation of numbers94 A tragic Pythagorean paradox: the odd equals the even96 Unthinkable numbers97 Zeno of Elea98 Zeno's paradoxes

100 Theoretical significance and solution of Zeno's paradoxes101 The part equals the whole102 Sets: an antinomic concept103 A postman and barber in rouble104 Russell's paradox105 A great game: mathematical logic105 A special chessboard105 What is a logical argument?106 Logic and ordinary language107 An ingenious idea of Leibniz108 Logic: the science of correct reasoning112 Logical variables113 George Boole and the origins of propositional calculus113 The logical calculus114 Negation115 Explanation of symbols115 Conjunction and the empty se t116 The empty set

117 Disjunction118 Implication119 Who has drunk the brandy?120 The multiplication tables of propositional calculus: truth-tables125 Another solution to the problem of the brandy drinkers126 Who is the liar?129 How to argue by diagram135 A practical application: logiccircuits143 Games with probability143 The reality of chance and uncertainty144 Cards, dice, games of chance and bets: historical origins of the calculus of probability144 Chance phenomena145 A clarification146 Sample space148 The measure of probability148 Horse races

149 The concept of function149 The algebra of events and probability games150 The complementary event and its probability measure150 The probability of the union of two events151 The probability of the intersection of two events152 The probability of a choice153 Drawing a card from a pack154 Joint throw of coin and die154 Dependent events156 Independent events159 What is the probability that George and Bob speak the truth?160 Probability and empirical science160 Probability and statistics160 Sample and population161 Guess the vintage161 Conclusion

163 Appendix: games with logic and probability163 Note164 Games with logic167 Games with probability

177 List of main symbols used

179 Bibliography

181 Index


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INTRODUCTION

Anyone who regards games simply as games and takes work too seriously has grasped little ofeither. So wrote the German poet Heinrich Heine a century ago. In today's world the division ofwork and play persists. Old prejudice still holds that the playing of games is an activity for children,not useless perhaps, but certainly not the responsible and serious work of adults.

Heine intuitivelyforesaw what modern psychology has since asserted. Games are not onlynecessary for the development of a child's self-awareness; they are also needed by adults,especially when their work is repetitive and uncreative. The word games, as it is used in this book,

is general and covers a variety of quite cornplex activities. However, it is those games based onmathematical or logical principles that are among the most absorbing and creative. Indeed, thegreat mathematicians and scholars of the past often applied their skills to the solution of logical andmathematical games.

This book is a collection of logical and mathematical games both ancient and modern. It is notsimply a recital of mathematical pastimes and curiosities set down at random; rational criteria havebeen used to link the riddles, mathematical problems, puzzles, paradoxes and antinomies.Sometimes the link is an historical reference, at other times it is a conceptual link. Althoughwhenever possible we have repeated some elementary rules, the reader needs no special orcomplex knowledge. The collection concentrates on some of the simplest and most widely known

logical and mathematical games; past events and the people who produced or studied such gamesare mentioned to make the story more accessible.

Games are presented in a certain sequence. Those mathematical concepts that may be beyond theaverage reader are stated as clearly as possible. We have omitted complicated games such aschess and draughts and avoided the use of symbolic languages. Technical terms are used onlywhen they are absolutely necessary.

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The last two chapters are the most difficultas they include certain abstract philosophical questioninvolvingnew mathematical concepts. However, they are rationally linked to what comes before thand they indicate that at critical points in the development of mathematics, problems and logicaldifficulties were resolved through games that captured the imagination of scholars. Furthermore,even the most abstract mathematical question, logicalparadox or antinomy, becomes clearer if itformulated as a game.

It should be mentioned in passing that games are valuable aids in teaching mathematics; amathematical device, a riddle or a puzzle, can engage a child's interest more effectively than apractical application, especially when that application is outside the child's experience.

The penultimate chapter offers those readers who may be unversed in logic and mathematics somof the basic concepts and methods on which modern formal logic has been built. The book Isdesigned to inform and educate and is enriched by illustrations of games and riddles that introduthe reader to propositional calculus, the first item of logic with an important place in modern culturThe last chapter, which seeks to complement the one preceding it,begins with ordinary gamesof chance to demonstrate the importance of the concept of mathematical probability to anunderstanding of various objective and subjective facts.

Because the last two chapters tend to be abstract and difficult, we have added a list of concreteproblems, examples, and games, along with their solutions, to permit the reader to test his graspthe more theoretical. We happen to believe that the solutions to such games and problems, evenelementary, would be hard to reach without an adequate theoretical foundation. The book concludwith an extensive bibliography to guide the reader in the search for new games and a deeper insiinto historical and philosophical questions.

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GAMES WITH NUMBERS

The symbolic language of mathematics Is a kindof brain-relieving machine on which we easily and often perform symbolic

operations that would otherwise tire us out (Ernst Mach D Reider)

An historical note

From our earliest schooling we have been taughtto operate with whole numbers, fractions, nega-tive numbers and the like; perhaps only a few ofus, however, have asked ourselves what numbersare or represent. Numbers developed with manand have marked his life from the beginning of

civilization. Indeed, the world is based on num-bers. Numbers began as symbols invented byman for a variety of uses, perhaps most immedi-ately for counting the elements in sets of things."2" could mean two cows or any other two things:two donkeys, two rocks. "2 + 3," assuming "3"meant three donkeys, could represent the sum oftwo cows and three donkeys, or itcould representa different kind of set-two things of one kind andthree of another. Numbers are mental construc-tions that can indicate material objects without

noting their particular features. They are instru-ments that enable us to make rapid calculationsand present quantitative expressions in a simplesynthetic way.

In the course of history different peoples haveused different symbols to represent numbers. Theancient Romans, for example, indicated "two' as

II, "three' as ll; while V, the sign for "five," sym-bolized the five fingers of one hand, and the signXsymbolized two hands, one across the other, fortwice five. We have since adopted another set ofsymbols of Indo-Arabic origin. Why? At first itmight even seem that these later symbols aremore complicated. Is t not easier to grasp 1,1,111,and V, han 1,2, 3, and 5? Yet the Arabic notation

has displaced the Roman one, primarily due tothe practices of medieval Italian merchants, andparticularly to the influence of the Pisan mathe-matician Fibonacci, born in 1179. In fact, themathematician's name was Leonardo da Pisa,however he acquired the nickname Fibonacci be-cause he was the "son of Bonacci," a well-knownmerchant and official in 12th-century Pisa. Theelder Bonacci traded with Arab countries in NorthAfrica and the East, and was accompanied by hisson on his frequent trips; hence Fibonacci atten-

ded Muslim schools and adopted their algebraicmethods together with the Indo-Arabic system ofnumerals. He later recorded his education inarithmetic, algebra and geometry in his bookLiber Abaci (1202), and demonstrated the sim-plicityand practicality of the Indo-Arabic systemas opposed to the Roman system.

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Games with numbers

Greek numerical notationRoman numerical notation

I1111III

VVIVII

VillVilIXXL

CD

=2=3-4

5-76

7=8

9=10=5 0=1 00=500

' = 1II'=2

3I"3As=4e' = 5(' =6

, =7a' =80' =9I' = 10

x' = 20A' - 30,u' =40v' =504' =60cC=70t' =80

Y = 90- 100

a' = 20 0

M = 1000LX = 60

DC = 600XL = 40XC = 90CD = 400

CM - 900MM = 2000

II 2000C = 100,0001 =1,000,000

1983 = MCMLXXXIII

and sampi, -, for 900. The syis decimal, with letters for unittens and hundreds

In the West, however, many men of science,trade and letters opposed the 'new fashion," andit was a while before it took root. In Florence, forinstance, the Statutes of the Art of Exchange pro-hibited bankers from using Arabic numerals. Onthe whole, people were hostile to the Arabic sys-

tem as it made reading commercial records moredifficult, but in time the new fashion establisheditself. The reasons itdid are linked to the nature ofmathematics itself, namely simplicity and econ-omy. A symbolic system of ten signs (0, 1, 2, 3, 4,5, 6, 7, 8, 9) serves to represent any number,however large or small, because in he represent-ation of numbers, the meaning of the numeralschanges according to their position. Thus, in thenumber 373, the two numerals 3, hough the samesymbol, mean different things: the first indicates

hundreds, the last indicates units, while the 7 ndi-cates tens. There is no other symbolic system sosimple or effective.

The Romans, and before them the Egyptians,Hebrews and Greeks, used a clumsy numericalsystem based only on a principle of addition. TheRoman number XXVIII, for example, means

ten + ten + five + one + one + one. The exprsion of numbers by a few symbols that changemeaning according to position was apparenused by the Chaldaeans and BabyloniansMesopotamia. Later it was developed by thedus who transmitted it to the Arabs and they

turn, passed it on to mathematicians of medieEurope.The introduction of Indo-Arabic numerals w

positional notation greatly influenced subsequdevelopments in mathematics. It simplified mematical concepts and freed them from thecumbrances stemming from representing mematical operations inmaterial terms. The Greand Romans, for example, used complex gmetric systems for multiplication; hence the ccept of raising a number to a power (as prod

of so many equal factors) could not be undstood or made simple, especially when dealwith numbers raised to powers higher thanthird. To illustrate (Fig. 1), if he number threeresents a line three units long, and 3 x 3 = 32resents an area, and 3 x 3 x 3 = 33 a voluwhat meaning might be attached to 3

10

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vat x'i,,yga'etcetera Left: The Greek number system,

consisting of the 24 etter alphabetand the three signs: stigma, a', forthe number 6, koppa, A, for 90 ,

300400500

6007008009001000800060,00090,000

=100,000

=1=49131

=80 5=1420=13,101


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Games with numbers

Opposite right: The Roman numbersystem aso decima, used fewersymbols: I, V, X,L, C, D, M, for1, 5,10, 50,100, 500,1,000respective y Multiples of units, tens,hundreds, thousands are repeatedup to four times (thus 3= 1II,200= CC) A number to the right ofa bigger one is understood asadded thus LX =60 DCC= 700 Anumber to the left of a bigger oneis understood as subtracted, thusXL 40 A horizontal ine over anumber mu tiplies it by 1,000, forexample C = 100,000 A numberenc osed by H is multiplied by100,000 thus X = 1,000,000 Th ssystem spread throughout theRoman world and persisted unti itwas replaced in the 13th centuryby the Indo-Arabic system

fig. 1

23 =3x3

3 33=3x3x3

3

1

3=33

3 x 3 x 3 x 3 or 35 = 3 x 3 x 3 x 3 x 3 ? UsingIndo-Arabic numerals, we find that they are sim-ply numbers.

A first curiosity

To be sure, man first used numbers to solve hispractical problems more quickly, but we like tothink he also used them to entertain himself. Onthis assumption, we shall begin our book with arather popular game that requires only the mostelementary numerical calculations. Take the set ofdigits 1, 2, 3, 4, 5, 6, 7, 8, 9. The game is o insertsymbols for mathematical operations between thenumerals so the result will equal 100. We are notallowed to change the order of the digits.

Here is one possible solution:

1 +2+3+4+5+6+77+(8x9)= 100.In the last part of the expression we have usedmultiplication, but the game is more interesting ifthe operations are confined to addition and sub-traction. Here is a solution:

12+3 -4+5+67+8+9= 100

11

3

3

The reverse game can also be played, with thedigits decreasing in order: 9, 8, 7, 6, 5, 4, 3, 2, 1.Now reach a sum of 100 using the fewest " +- " and

signs. A possible solution is:

98-76+54+3+21 = 100

If you are familiar with the properties of num-bers, you can solve the following as well. Findthree positive integers whose sum equals theirproduct. One solution is:

1 x2x3=1 +2+3=6

Note that 1, 2, 3 are the factors of 6, which is theirsum. We continue the game by finding the num-ber after 6 equal to the sum of its factors. Thenumber is 28, as the factors of 28 are 1, 2, 4, 7,14and:

1 +2+4+7+14=28

Such numbers form a series (after 28 comes496) called "perfect numbers." Itwas the rnathe-matician Euclid, famous for his Elements of geom-etry and a resident of Alexandria during his mostactive years (306-283 B.C.), who first created a


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Games with numbers

Archimedes' spiral (below) andnatural spirals (left, a nautilus shell

in sections) can be expressedFibonacci numbers

formula for the structure of perfect numbers,namely:

N = 2 (2 - 1)In his formula, the second factor, (2' - 1), must

be a prime number, that is, divisibleonly by itselfand unity. Thus n must be such that 2n 1 is

prime. It is easy to see that the latter is not primeifn is not prime. The reader should try to use thisformula to find the next perfect number after 496.After that, the calculations become rather lengthy.

Here is a table for the first nine perfectnumbers:

n 2 1 2- 1 Perfect numbers

1 2 2 3 6

2 3 4 8

3 5 16 31 496

4 7 64 127 8128

5 13 4096 8191 33550336

6 17 65536 131071 8589869056

7 19 262144 524287 137438691328

8 31 1073741824 2147483647 2305843008139952128

9 61 - 26584559915698317446546926159538421 76

We observe that all perfect numbers obtainby Euclid's formula are even and always end ior 8.

Fibonacci numbers

Among the many arithmetical and algebrquestions studied by Fibonacci, that ofquences deserves special attention, as it was basis for his interesting problem of the rabbSuppose we put a pair of adult breeding rabbin a cage to produce offspring, and that eamonth they produce another pair, which, in tubreed after two months. (This is hypotheticalcourse, as rabbits do not reach maturity beffour months.) Ifall the rabbits survive, how mawillthere be at the end of one year? The solut

is indicated in Fig. 2. We start in January withinitial pair A. InFebruary there willbe two pairsand their offspring B. In March A produceswhich makes three. However, in April,A produD, while B, now mature, produces E. In May,more complicated still: A produces F, B produG, and C produces H. Continuing in his fashi

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Games with numbers

the number of pairs produced in successivemonths is: 1, 2, 3, 5, 8, 13....

The law linking the numbers is easily detected.From 3 on, each number is the sum of the two

2 + 3 5+81 2 3 5 8 13

1+2 3+ 5

preceding it. Hence we can easily find thenumbers for the later months: July, 8 + 13 = 21;August, 13 + 21 = 34; September, 21 + 34 = 55;October, 34 + 55 = 89; November, 55 + 89 =144; December, 89 + 144 = 233. At the end of theyear there will be 233 pairs. Once the formationlaw is found, the sequence can be continued

indefinitely.Fibonacci did not explore the question of num-ber sequences more deeply, and it was not untilthe 19th century that mathematicians began tostudy their formal properties. In particular,Franqois Edouard Anatole Lucas investigated theFibonacci series, where starting with any two in-tegers the next term is the sum of the two before.The table shows the first twenty terms of the seriesstarting 1, 1 and 1, 3.

Fibonacci series have always captured theimagination of

mathematicians and enthusiastswho have tried endlessly to unearth their hiddenproperties and theorems. Recently, such serieshave been useful inmodern methods of electronicprogramming, particularly in data selection, therecovery of information, and the generation of ran-dom numbers.

A curious calculating device: the abacus

Man has always tried to do sums with greaterspeed. The Babylonians cut permanent signs onclay tablets to hasten calculations. Subsequently,the abacus was invented-where and when is notknown, perhaps in ancient Egypt. The abacuswas the first calculating machine and it was aningenious instrument. Numbers were represented

23

58

14321345589

144233377610987

1 597

2 5844 1816 765

347

1 118294776

12 319 9322

521843

1 3642 20 73571

57789 34 9

15127

as objects (pebbles, fruit stones, and piershells for example) and placed on small stfixed to a support. The word abax, abakos mea "dust-covered tablet" on which geometricalures can be traced or calculations perform

and it probably came into ancient Greek fromHebrew abaq, meaning "dust.' Thus the woriginated in the Near East.

Although the mathematicians of ancGreece were familiarwith the discoveries of Miterranean peoples, and enhanced themoriginal notions of their own, their mathematiadvances had no discernible impact on the strture or workings of Greek society. Indeed, sadvances were seen as little more than ilectual exercises. We know, too, that new

entific and technical knowledge were seldused to achieve greater productivity or freedfrom physical labour; again they were treatedply as expressions of man's creative ability.prejudice impeded the progress of mathemain Greece and explains why many of the marithmetical and algebraic discoveries came t

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Games with numbers

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from the Indian and Arab algebraists working be-tween A.D. 400 and 1200. Their discoveries werebrought to Italy and thence to the West by thetraders in the maritime republics.

Mathematical knowledge spread widely after

the Protestant Reformation and the invention ofpaper and the printing press. Indeed, it was Mar-tin Luther who insisted that the first arithmetic text-books be printed. Indian algebraists, and later theArabs, had demonstrated the advantages of thenew positional number system: calculations couldbe simplified much to the relief of those who usedthem in trade and commerce. Again, it wasthrough commerce that the abacus found its wayto the West, and today this simple calculator is stillused in Russia, China and Japan to total bills in

shops and restaurants, and frequently to teacharithmetic.Abaci have varied between peoples and peri-

ods. The abacus with beads on small sticks (pp.18-19) is only one type, probably of Chinese ori-gin. The Arabs developed others of a differentconstruction and one still in use consists of a kind

of grid. It is best explained by carrying out a mul-tiplication, say 3,283 by 215. Draw a rectangle ofas many small squares as the two factors havefigures, in this case 4 x 3, with their diagonalsvertical and horizontal. Divide the unit squares

vertically and extend the traces to a base line(Fig. 3). We have put the two factors on thesides-the four-figure number on the longer sideand the three-figure number on the shorter side.The result is a grid which is now filled with theproducts of the figures at the edge. For example,the one farthest right contains 5 x 3 = 15, the onefarthest left 2 x 3 = 6; the units are in the rightdivision and the tens in the left division. When thegrid has been filled, we add on the base line,carrying tens as needed. The result is 705,845,

which can be checked by the ordinary method.Fig. 4 shows a similar abacus on which 1,176

(continued on page 20)

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Games with numbers

What numbers are.The photographs (left) show threeMunich beer mats, on which waitersmark each beer ordered by thecustomer The bill is then tallied bysimply mu tiplying the price of oneglass times the number of marks onthe mat Such methods are ancientand date back to the mythicalorigins of mathematics

The notion of a markcorresponding to a unit eventual yproduced the natural, or countingnumbers In order to simplifycounting operations a special set ofsymbols was gradualy devisednamely what are known as ciphers,figures or numerals Naturanumbers were used in barter(exchanging one artice for another)and in childrens games whereobjects were classif ed, put insequence or set out according torules of proximity, continuity andboundary es

From the natural numbers and theoperation of adding, man hasgradually constructed the entiresystem of numbers (of pp. 21, 104)as wel as the other calculatingoperations (note that, formally, thepositive integers are ratios ofnatural numbers to the naturalnumber 1, a definition that was notput forth until the 19th century)

Opposite: A register used byilliterate Sicilian shepherds Bynotching sticks the shepherdsindicated the number of an ma s-sheep or goats-each owned, the

animals' births, the dairy productsproduced and so on This calendar,valid from September 1st for oneyear, was sett ed every August 31stAs the date approached, a literate

man was appointed to inscr beeach stick with certain sgnsindicating its owner's name antype of animals he possessed

I

heads of beasts eacmember owned,

births

males

r1w dairy products, curd,cheese, etcWhile the number of adult

animals was almost constant fyear, dairy products and newvaried from month to month (mwere sold, females were kept)the register was updated everymonth

Although the people who usthis instrument could nether rnor write, they cou o count intand they correated signs andobjects in a highly complex faInstruments such as this formbasis of mathematical and logthinking.

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Games with numbers

Addition and subtraction on theabacus.This abacus (opposite) is currentlyused n Ch na (the who e instrumentappears on the book jacket) Eachvertca column of beads starting onthe right w th the units indicates thedig ts of numbers in pos tionalnotation Each bead under the barstands for 1 it 100 and so onEach bead on the bar stands for 5 ofthe items in ts co umn Thus thenumber shown here is 173

To add. start from the rightSuppose we need to add 148 and451 Form the number 148 (A), addone unit, five tens and four hundredsproducing 599 (B) To subtract westart on the left and do the reverseTake 293- 176 Form 293 (C) takeaway one hundred, and seven tens-by removing the 50 and two 10s(D) to subtract six units borrow aten in the un t co umn by taking awayone ten in the column of tens andadding two f ves in the unit co umn(E) Now we can take away six unitsand the resu t is 117 (F)

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Games with numbers

and 6,895 can easily be multiplied. First, write theproduct of 1 by 6,895 in the first row (tens in thetop half and units in the bottom half). Put the prod-uct 1 x 6,895 in the second row, 7 x 6,895 in thethird row, and 6 x 6,895 in the fourth row. Sumdiagonally from the right and carry to the left asneeded. A schema much like this served Pascaland Leibnitz four centuries later when they con-ceived the idea of the first calculating machines.

The origins of algebra

The term "algebra" derives from the Arabic a -jebr, which the mathematician Al-Khowarizmiadopted to explain his ideas for solving what wecall equations. Later the term acquired a widermeaning and today it includes a broad range ofmathematics.

Mohammed ibn Musa Al-Khowarizmi, an Ara-bian astronomer and mathematician (diedca.A.D. 850), was active in the 'House of Wis-dom" in Baghdad, a cultural center establishedabout A.D. 825 by the Caliph Al-Mamun. Al-Khowarizmi wrote various books on arithmetic,geometry and astronomy and was later cele-brated in the West. His arithmetic used the Indiansystem of notation. Although his original Arabicbook on the system, probably based on an Indiantext, is lost, a Latin translation survives as Algo-rithmi: De numero indorum (about Indian num-bers). The author explains the Indian numericalsystem so clearly that when the system eventuallyspread through Europe, it was assumed theArabs were its inventors. The Latin title gives usthe modern term "algorithm"-a distortion of the

name Al-Khowarizmiwhich became Algorithmused today to denote any rule of procedureoperation in calculations.

Al-Khowarizmi's most important book, Alwa'1-muqabalah, literally "science of reducand comparing," gave us the word ''algebrThere are two versions of the text, one Arabicthe other the Latin Liber algebrae et almucabwhich contains a treatment of linear and quratic equations.

These works were of major importance inhistory of mathematics. Indeed, al-ebr originmeant a few mathematical steps and transmations to simplify and hasten the resolutionproblems.

Let us now turn to what we learned in schand begin with an equation of the first degr5x + 1 = 3(2x -1). An equation is generallyequality with one or several unknowns. It tralates into numbers a problem whose solution csists of finding those values of x that make equality true. In our example, we must findvalue of x that makes the expressions on eitside of the "equal" sign equal.

Al-Khowarizmi's mathematical workscontainthe solving procedures we learned mechanicain school, for example, reducing terms and traferring a term to the other side with a changesign. Hence, in our case, adding 3 and subtraing 5x on both sides, and then changing sidgives us x = 4, which solves the equation. Putt4 for x in the first equation, 5x + 1 = 3(2x - 1),find 21 = 21. Clearly, to solve an equation itransform it into other, and simpler equations uwe reach the solution.

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Games with numbers

SYSTEM OF NUMBERS

integer

fractional(eg. 1,1, ...

Y

rational

irrational(eg 2, T, ...)

2) add 3;

Algebra and its laws have often spawned tricksand games that seem to smack of magic. In fact,they are readily explained by algebraic laws.Imagine that we have asked someone to play thisgame:

1) Think of a number;

3) multiply by 2;

4) subtract 4;

5) divide by 2;

6) subtract the original number.

21

positive

(eg.+1, +2, +3, ...

negative(eg -1, -2, -3, ...)

real

complex

imaginary(eg I= i, 74=2i, ...

Games with algebra


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Games with numbers

Whatever the original number, the result mustalways be 1. Surprising? Not if you give the simplealgebraic explanation. The principles are elemen-tary but deep. Take the present game; the resultdoes not depend on the original number, whichneed not be known. The expression "any num-ber" can mean two different things in algebra,either a variable number which can have variousvalues, or a given number which is undetermined,namely a constant whose value has not beenfixed. For clarity, variables are shown by the lastletters of the alphabet (x, y, z . . .), and constantsby the first letters (a, b, c . . .). Thus if we write3 + x, with x integer and variable, we have forx - -1,3+(-1)= 2forx =Owehave3+0 =3;forx = 1, 3+1 =4.

A variable in an equality, say x in 6 = 5 + x,

becomes an unknown (a value not at first known),indicating the value required to verify the equality.Returning to our game:

1) Take a number, x;

2) add 3, x + 3;

3) multiply by 2, 2(x + 3);

4) subtract 4, 2(x + 3) - 4;

5) divide by 2, (2(x + 3) - 4)/2;

6) subtract the original number x, (2(x + 34)2 -x. In algebra, this last expression repsents the sequence of verbal moves. Whatevethis expression equals 1.

An expression such as (2(x + 3) - 4)/2 - xis called an identity. The difference betweenand an equation is readily explained. In an idtity the two sides are always equal, whatevervalue of x, while in an equation this is notReturning to our first equation, 5x + 1 = 3(2x -there is only one value of x for which the two siare equal, namely the single solution of that liequation, which, as we saw, is x = 4. For any ovalue of x, the sides are unequal. For examplwe put x = 0, we have 1 :& 3 (the sign me"different from").

By applying such elementary principles ogebra other games can be invented. For examthe following always results in 5:

1) Take a number, x (say, 6);

2) add its successor, x + (x + 1) (h6 + 7 = 13);

3) add 9, 2x + 10 (here, 13 + 9 = 22);4) divide by 2, (2x + 10)/2 (here, 2212 = 11);5) subtract the original number, x + 5 -x(here, 11 - 6 = 5).

(A): 2(x + 3) 4 =2 + 2x is an identity; (B): 5x + 1 = 3(2x - 1) is an equaif we set x = 0 in (A) we obtain 2= 2; in (B) we obtain 1 *if we set x = 1 in (A) we obtain 4= 4; in (B) we obtain 6 *

if we set x = 2 in (A) we obtain 6 = 6; in (B) we obtain 11 *if we set x = 3 in (A) we obtain 8 = 8; in (B) we obtain 16 *

if we set x = 4 in (A) we obtain 10 = 10; in (B) we obtain 21 =if we set x = 5 in (A) we obtain 12 = 12; in (B) we obtain 26 *

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Games with numbers

The algebraic expression reduces to 5, whateverx. The game is a bluff, the trick lying in the intri-cate instructions which are designed to compli-cate matters. Actuallyeach step is simple, but weare not always alert enough to see the mainpoints. In the present instance, the trick is to takeaway the arbitrary original number, that is the sub-traction x - x. By starting the whole process fromthe other end, any number of such games can beinvented. Let us construct one that always resultsin 13. Forany real number there are infinite identi-ties. Take 13 = 7 + 6, for example. Since x - x =0, we can add this to the right-hand side withoutupsetting the identity, 13 = 7 + 6 + x - x. Thiscan now be rewritten as 13 = (2(7 + 6 +x))/2 - xbecause multiplyingand then dividing an expres-sion by the same number leaves it unchanged.Next we can make things more complex by multi-plying out the bracket: 13 = (14 + 12 + 2x)/2 -x,which can be recast as 13 = (2(x + 7) +12)/2 - x. The game then is this:1) Take a number, x (suppose we take 10);2) add 7, x +7 (here, 10+7 =17);3) multiply by 2, 2(x + 7) (here, 2 x 17 = 34);4) add 12, 2(x + 7) + 12 (here, 34 + 12 =46);5) divide by 2, (2(x + 7) + 12)/2 (here, 46/2 = 23);6) take away the original number, (2(x + 7) +12)/2 - x (here, 23 - 10 = 13).

The rules of algebra are such that it can appearwe are able to read peoples' minds. Try thisexercise:1) Think of a number, x (suppose we take 6);2) double the number, 2x (here, 2 x 6 = 12);

3) add 4, 2x + 4 (here, 12 +4 = 16);4) divide by 2, (2x + 4)/2 (here, 16/2 = 8);5) add 13, (2x +-4)/2+ 3 (here, 8 + 13 = 21).We now ask the player for his answer, namely 21 ,

and quickly tell him he started with 6. Since thefinal algebraic expression reduces to x + 15, weknow that, in this instance, x + 15 = 21, so x = 6.Algebra, not mind reading.

There are countless variations of this garne, asthe natural numbers that can be subtracted fromx, which is itself a natural number, are endless.The number subtracted should not be too small,say at least 20. The first step would be x + 20, anexpression which will now be transformed. Forexample:

x +20=x +6+14 =3(x +6)/3+14(3x + 18)/3 + 14.The game then consists of these steps:

1) Think of a number, x (say, 8);2) multiply by 3, 3x (here, 8 x 3 = 24);3) add 18, 3x + 18 (here, 24 + 18 = 42);4) divide by 3, (3x + 18)/3 (here, 42/3 =14);5) add 14, (3x + 18)/3 + 14 (here, 14 + 14 -=28);6) subtract 20, (3x + 18)/3+14 -20=x (here,28 - 20 = 8); which produces the number origi-nally in mind.

Odds and evens

From our early efforts in arithmetic we learned todistinguish between odd and even numbers; thelatter are divisible by 2, the former are not.

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Games with numbers

Let us examine the algebraic notation of aneven number and its properties. Take any integerx; 2x then is even. Thus 14 is even, for we canwrite it as 2 x 7. If 2x is even, then 2x + 1 is odd.For example, 15 = (2 x 7) + 1.

This hatches some amusing games. Let aplayer take an even number of coins in one handand an odd number in the other. Ask him to dou-ble the number of coins in his left hand and triplethe number in his right, and to reveal the total ofthe two numbers. You can then tell him whichhand holds the odd number of coins and whichhand has the even number. Ifthe sum is odd, theodd number of coins is in the right hand; if thesum is even, the odd number of coins is in the lefthand. For example: if we have three coins in theleft hand and six in the right then 2 x 3 + 3 x 6 =24. The sum is even, and the odd number of coinsis in the left hand. What is the trick? We needalgebra to grasp it. As before, we can fol-low the operations step-by-step. There are twopossibilities:

1) The odd number of coins is in the left hand.2) The odd number is in the right hand.Call the number of coins in he left hand L and thenumber in the right hand R. Then:

1) Odd number of coins in the left hand,L= 2x + 1, R = 2y, where x,y are two unknownintegers whose actual value does not matter. Thesum to be considered is 2L +3R =4x + 2 + 6y = 2(2x + 1 + 3y) which is divisible by2, and hence even.2) Even number of coins in the left hand, L = 2x,R =2y + 1, and2L + - 3 R 4x +6y +3=2(2x +3y + 1) + 1, which is odd. This completes theproof.

The successor of a number

Those algebraic expressions that almost looksimple at first, can actually suggest a varietentertaining mathematical games successnumbers, for example. These are numberscome directly after one another: x + 1 followx + 2 follows x + 1, and so on.

Take five successive numbers and add ttogether:

x + (x + 1)+ (x + 2) + (x + 3) +(x +4) =5(x +2)

This produces the next game.

1) Tell someone to think of a number, x (suppit is 252);

2) now ask the player to add to it the nextnumbers, 5(x + 2) (here, 252 + 253 + 25255 + 256 = 1,270);3) ask for the result and from that you can recothe original number. All youhave to do is dividfive and subtract 2, for 5(x + 2)/5 - 2 = x (h1,270/5 - 2 = 252).

A shortcut in calculations

The world of numbers is vast and filledwith pobilities. With a bit of inquisitiveness one canate games simply by devising new steps, orways to work out complicated and lengthy su

For one of these games you need two playAsk each to write down a four-figure number opiece of paper. Suppose the numbers are 1and 1,887. One player (no matter which) is tasked to work out the product in the usual wMeanwhile, you subtract 1,887 from 10,000,1 from 1,233, which gives 10,000 - 1,88

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Games with numbers

8,113, 1,223- 1= 1,222. The second player isnow asked to multiplythese two numbers. Finallythe two players are told to add their results. Beforerevealing them, however, you can announce thatthe sum is 12,221,887 (indeed, 2,307,801 +9,914,086 12,221,887).

To clarify this let us examine the various stepsalgebraically. Let the two four-figure numbers bex and y.1 The first player works out xy,2) the second player works out (10,000(y -1 ) = Io,OOOy- 10,000 -xy + x;

x)

3) adding the two results yields 10,000y-10,000 -xy xy+x = 10,000(y - 1) +x.This final expression explains the trick: multi-

plying by 10,000 adds four zeros to the digits-forexample, 13 x 10,000 =130,000. Thus y - 1gives the first four figures and x the remainingfour. In our case x = 1,887 and y 1 = 1,222,producing 12,221,887.

How much money is in your pocket?

Substituting a letter for a number-x, or any otherletter may seem almost elementary, but it wasactually a major step in the development of math-ematics, as it helped to illuminate the formal fea-tures of numbers and raised analysis to a moreabstract level. When we see "652," we auto-matically thinkof a number. If,however, we see analgebraic expression, such as 1Ox + 9, it is lessclear that it too is a number.

We know that in algebra x can take any numer-ical value. If it equals 4, then the number justmentioned will be 49; ifx equals 1, he number is19, and so on. This gives rise to yet more games

which may seem perplexing at first. In his exam-ple we see that the values of x appear in the tensof the answer, as is obvious if we take away theunits. Consider another example.

We tell someone we can guess the amount ofsmall change in his pocket if he will do thefollowing:

1 Start with the total sum, s (say, 35 cents);2) multiply by 2, 2s (here, 2 x 35 = 70);3) add 3, 2s + 3 (here, 70 + 3 = 73);

4) multiply by 5, 5(2s + 3) = IOs + 15 (here,5 x 73 = 365);5) subtract 6, 1 s + 9 (here, 365 - 6 = 359).We ask for the result, take away the units and are

left with the sum of 35.Other expressions too can generate this kind oftrick, indeed a host of tricks. For example, use anynumber x and proceed as follows:

1) Take a number, x;2) add 2, x + 2;3) double, 2x + 4;4) subtract 2, 2x + 4 - 2 = 2x + 2;5) divide by two, (2x + 2)/2 = x + 1;

6) subtract the original number, x + 1 --x = 1.The fifth step gives the vital clue: To get x wemerely subtract 1.

How to guess a birth date

In he preceding algebraic expressions there wasonly one unknown and the trick was built aroundit. In he same fashion, we can devise tricks using

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Games with numbers

expressions with two unknowns and find twonumbers,

Consider an exercise that allows us to deter-mine a person's birthday. First assign the num-bers i to 12 to the months, starting with January.Let m be the month and d the number of the daywe are seeking. Now put the person throughthese steps:1) Multiply by 5 the number of the month, 5m(suppose the birthday is 13 June, then5 x 6 = 30);2) add 7, 5m + 7 (here, 30 + 7 = 37);

3) multiply by 4, 20m + 28 (here, 4 x 37 = 148);4) add 13, 20m + 41 (here, 148 + 13 = 161);5) multiply by 5, 100m + 205 (here, 5 x 161805);6) add the number of the day, 1 Om +205 + d(here, 805 + 13 = 818);7) subtract 205, 100m +d (here, 818-205613).

Now ask for the number. The hundreds give themonth, namely 6 for June, while the rest, 13, givesthe day. Try a different one. Suppose we are toguess the date on which the Bastille fell (14 July1789), marking the outbreak of the FrenchRevolution.1) Multiplythe month by 5, 5m (here, 5 x 7 = 35);2) subtract 3, 5m - 3 (here, 35 - 3 = 32);

3) double, 1Om - 6 (here, 64);4) multiply by 10, 1 Om - 60 (here, 640);

in the tens and units, while 100m is found inhundreds, preventing the two from overlappinNow we simply read off m and d. Of course,must remain below 100, which limits the gameage, shoe size and so forth.

Guessing age and size of shoes

Here is an analogous game with a few confusiodeliberately added. Suppose we are to guesssize of a person's shoes as well as his age.proceed thus:

1) Multiplythe number of years (a) by 20, 20aa is 20, we have 20 x 20 = 400);

2) add the number of the present day (20a + d (here, supposing it is the 9th, 400 + 9409);

3) multiply by 5, 1 00a + 5d (here, 5 x 4092,045);

4) add the shoe size (s), 1Oa + 5d + s (ifs is then 2,045 + 11 = 2,056).Now we subtract five times the number ofcurrent day, which is known, leaving 100a +The hundreds give the age and the rest givesshoe size (here, 2,056 - 45 = 2,011). The persis 20 and wears size 11 shoe.

Where is the error?5) add the day,640 + 14 = 654).

1OOm - 60 + d (here,

Given this number, we now add 60, leaving

1OOm + d (here, 714). The values of d are found

Behind every mathematical game there lieswile. Many such games rely simply on peopbeing unable to follow the various algebra

steps. However, we can invent very subtle tri

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Games with numbers

based on the procedure itself. Suppose we wishto prove that 1 = 2. Take any two numbers x andy and suppose:

1) X =Y;

2) multiply by y, xy = x2;

3) subtract y2 , xy -y 2 = x2 y 2 ;4) factorize, y (x - y) = (x + y) (x - y);5) divide by (x -y), y =x +y;6) by 1), x =2x;7) divide by x, 1 = 2.

Each step seems correct; yet there is an error,an illogical step. When there is a contradiction inmathematics, the mistake can be found in the

procedure or in the premises. If a game is irri-tating to play, it could be that either it is faulty (thepremises are wrong), or the player is not stickingto the rules (the procedures). In his instance, thelogical error lies in 5) when we divide by (x -y),which because of 1) is zero. Clearly it makes nosense to divide by zero. We can now see that thecontradiction was produced by introducing an er-ror into the procedure.

In the following numerical expressions thereare two mistakes for the reader to detect:

1) 2+1 (-1) 4;2) 6+ 1/3 2;

3) (3 + 1/5) (3 + 1/8) = 10;4) 18 -(-8)=26;5) -32 x (27 - 27) 32.

This is the solution. The errors are in 2), where theresult is 18 (dividing by 1/3 is multiplyingby 3); andin 5), where the result is zero (multiplyingby zero,

represented here by 27 - 27, is zero).

Positional notation of numbers

This method (cf. p. 10) was used by the ancientIndians and was spread throughout medievalEurope by the Arabs. At the time it representedenormous progress in mathematics. Today we areso habituated to using Arabic numerals that weseldom realize the system's advantages. To do sowe need only recall the Roman system which waslong, cumbersome and a ready source of errors.Arabic figures are less intuitively obvious, butfrom the start they have exhibited a peculiar fea-ture on which mathematical thinking rests: ever-increasing simplicityand generality. Take a num-ber in Arabic figures, say 6,245. Here 6 indicatesthousands, 62 hundreds, 624 tens and 6,245units. We can write it also as 6(1,000) + 2(10 0) +4(10) + 5(1), or 624(10) + 5(1).

Let us now consider a four-figure numberalgebraically, writing thedigits as x3, X2, X1, XO. Wecan then write the number as x3 (1000) +X2 (1 00) + x1 (10) + xo (1). The first term indicatesthe figure with a positional value of thousands, thesecond hundreds, the third tens, and the fourthunits. We can split this into x 3 (999 +- 1) +x2 (99 + 1) + x1 (9 + 1) + x., or, rearranging,9(11 1x 3 + 11x 2 + X,) + X3 + X2 + X1 + X0.

Given that x3 is the number of thousands x2 thenumber of hundreds, x, of tens and xO of units,consider the following set of instructions:

1) Take a four-figure number (say, 3,652);2) write down the figure of thousands, X3.1(here, 3);3) write down the figure of hundreds, x -10+x2-1 (here, 36);

4) write down the figure of tens, x 3 - 1 0 0 +

x2-10 +xi*

1 (here, 365);

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Games with numbers

Too often we th nk of *he ru es oalgebra as simply abstract atdforget that matherenat cs began fvery pract ca and concrete reasTake the product of two inomrath s s easier to understand f *t

nked w th the probeem of dir dlandLeft A geometric i lustrat on of tSimp e a gebra c pmob em Ot sqU(a b) We know troan algebra

th s asa - b) - a a - 2atthe d agrarn we t rd It s the smsquare at the top eft

(a-b)2= a2+ b2- 2ab

and unlike ordinary language, they are specificand unambiguous. Translating from mathematicalinto everyday language is therefore a particularlyuseful exercise, especially at school age. Takethe expression x/3 + 5 = x12 + 6, for example.This equation and its solution can help us to for-mulate a problem first in everyday language, andthen synthetically in the language of mathe-matics. In ordinary language the equation istranslated: A third of a number increased by fiveequals half that number increased by six, if thatnumber is minus six, mathematically, x/3 + 5 = x/

2 + 6, yields x = -6. Similarly, the expressi(x - 2)/4 = (5 - x)/6 can be read as: A quartertwo less than a number equals

one-sixth ofdifference between five and that number.Let the reader try to translate the expressi

AB = AC. IfAB and AC are two segments, wethat the segment AB is congruent with (equalthe segment AC. If ABC is a triangle, AB -tells us that the triangle with vertices at A,B,Cisosceles. IfBC is a segment, we can say thais the middle point (Figs. 5, 6, 7).

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Games with numbers

a+b

(a+b)2 =a'+ b2+2abIn this diagram the connectionbetween an algebraic express onand a geometric one is moreimmed ate When we multipy (a +b)

with itself we get the squares ofeach term plus twice their product(a +b) 2 a 2 +b2+2ab

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Games with numbers

fig. 6

A B

A-Cfig. 5

fig. 7

C

B

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GAMES WITH GEOMETRICALFIGURES

Godgeometrizes conbr ally (Plato)

Geometry and optical illusions fg 89

In Fig. 8-9, which is longer, AP or CD? In fact ABneither. They are equal. Now take Fig. 10. CDlooks longer than AB, but again they are equal, asa ruler will show. These are optical illusions. CThrough our sense organs we perceive vital infor- ;; 0mation about our surroundings. That data thentravels to the brain where it is processed and sentto us as sense experience. There are visual, audi- fig 10 Atory, gustatory, olfactory and tactile sensations, Hdepending on the sense involved. However, oursenses can deceive us and give us an incompletepicture of reality. Moreover, our senses can beconditioned by previous experiences or habit andthus create illusory sensations. Look at Fig. 11. CYou probably notice a triangle. None is drawn, butthe more one looks the more one seems to per-ceive a triangle, even if reason tells us the draw-ing consists of circular sectors.

LI

continued on page 38)I " "Il"" 1- I1 ~ I - -...... -1 1

D

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Games with geometrical figures

Below Th s pant by the Dutch artistMaur ts Come ius Escher I18981972). Ascent and Descent wa sinspired by the amb gu ty in certa ngeometrica figures The monks in theouter row seem to ascend endlesslywh le those n the inner row seem todescend end ess y ((c Bee drechtAmsterdam 1982)

Right The steps on which Escher soptical us on s based In threed tensions we cannot represent orbu id a sta rcase that ends where Itbeg ns In a two-d mensiona pictureEscher overcame these m nationsbya ter ng certain f curative signa s andv usual ata

Opposite Bridget R eys Catar1967 London (Brit sh Counciperm ss on of Rowan Gal ery )Op Art (Optical Art) painting plopt ca I usion Such phenomenwarn us that our perception ofmage can be d ferent from ts

which s perhaps what attractsattent on

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Opposite Regular similar geometricfigures converge to the center of thedome of Sheik Loftollah's mosque atIsfahan (Iran) This is a typicalexample of Arabic art, in whichgeometrical themes often occur Inthe Arabic world, the abstractsciences, particularlymathematicsand geometry, were based onreligion In Islam Allah represents lifeitself Man, or plants or animals areonly single parts of the universe andno one of them can represent thetotality Indeed Arabic artists seldomportrayed people or animals,choosing instead to create abstractgeometric patterns in which animalswere seen only in stylized formsAbove: Until about the second half ofthe 19th century the geometry in usewas basically Euc idean, namely theintuitive geometry we learn at schoolEuclidean geometry was regarded asan immutable and predeterminedway to grasp phenomena andexperience, witness the philosopherImmanuel Kant However, in thesecond half of the 19th century somemathematicians and scientists(Gauss, Lobachevsky, Bolyai)discovered that other geometriescould be constructed, leading tonon-Euclidean spaces, simply bydenying one or more of Euclid'sbasic principles In 1872 themathematician Felix Klein at Erlangenproposed a geometrical researchprogram to radically change

figurative geometry into a system oftransformations This means thereare many geometries. and togetherthey form a system in which eachcan be constructed from the simplestto the most general Euclideangeometry is a metric geometry and

belongsto

the groupof

displace-ments, itallows only isometrictransformations (displacementsrotations, symmetries) that vary theposition but not the size of anglesand lines If lines are allowed tochange in size, we have the group ofsimilitudes; if angles too can change,we have affine geometry, if parallelschange, we have protectivegeometry Finally, a figure may becontinuously distorted as long asconnected parts are not severed andpoints are not superimposed, this isthe group of homeomorphisms ortopology Each of these geometries isweaker than its predecessor butbroader as what remains are themore genera properties of figuresThe way the exact sciencesdeveloped at the time (Einstein'srelativity, quantum theory) confirmsthat the new ideas on spaceconstructed for non-Euclideangeometries were more in accord withreality than the absolute space ofclassical physics.Right: The perspective of an Arabportico, a clear example ofprojectivity

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fig. 12

[

fig 13

iiMany factors intervene in the processing of

data supplied by the senses; particularly impor-tant are our perceptual habits. The mechanismthat makes us see a triangle in Fig. 11 is the sameone that makes us see two letters in Fig. 12 but notin Fig. 13 .

Consider a curious experience we have all hadat the movies. In a Western, for instance, as wewatch the pursuit of a coach moving faster andfaster, suddenly the coach's wheel spokes seemto rotate backwards. Obviously our sensationsare confused; our eyes tell us the coach movesforward while the wheels suggest that it is goingbackward. When the chase ends, the wheels just

as suddenly resume their proper forward rotation.Our senses have deceived us and we have ex-perienced an optical illusion. The explanation isquite simple and is based on the physical prin-ciples of cinematography. The sensation of move-ment we perceive in he film is caused by the factthat the slightlydifferent individualframes that are

projected on the screen at the rate of 20-second, "persist" for an instant on the eye's rein an ordered sequence, hence the sensatiomotion.

Why then do we see the spokes move bward after a certain speed is reached? Weonly give a general indication, as a detailedcount involves the psychology of perceptionprinciples of optics too complex to outline brRoughly, this phenomenon occurs becausetime a spoke takes to move from one positiothe next steadily decreases, until it is less thanappearance of an individual frame on the scrAs a result, the pictures on our retina merge,ing the impression of a reversal of rotation.can be seen more or less in Fig. 14. The fdiagram shows the reversal point. Note thadistance between the two spokes increases.

In discussing this common optical illusiohave mentioned some physical principlesmechanisms of perception. Optical illusion

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fig. 16

indeed particularly important in many fields: art,psychology, mathematics, even philosophy. Phi-losophers ask to what extent our senses give us

information that correctly reflects the worldaround us and what subjective elements distortour perception of reality. In any event, optical illu-

fig. 15

sins have decisively influenced the psychologyof perception; they are a popular instrument forstudying how the brain organizes and interprets

what the senses convey to it.Look at the pins with their heads up in Fig. 15 .Now raise the book to eye level, hold ithorizontallyand close one eye. The pins seem to be standingup. Among the illusions studied and analyzed byGestalt psychology (the psychology of the per-ception of shape, adopted by the Germans MaxWertheimer, Kurt Koffka, Wolfgang Kbhler, KurtLevin among others) are those concerning figuresthat can be perceived in two equally valid ways.

Fig. 16 shows the new flag adopted by the Ca-

nadian House of Commons in 1965. The middlepanel has a maple leaf on it, but ifwe concentrateon the white background surrounding the leaf, weseem to perceive two angry faces. Such figuresare called "reversible," because their mental rep-resentation can suddenly switch without anychange in the visual information to the eye.

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fig 17

A well-known figure of this type is 'Necker'scube," or transparent cube. In 1832 Louis AlbertNecker, a Swiss geologist, observed the "per-spective inversion"'and reported that some draw-ings of transparent rhomboidal crystals presenttwo different pictures in which front and back are

interchanged.*In the cube of Fig. 17, one side seems to be infront, but when you look at the cube steadily thesense of depth reverses and the side at the backsuddenly appears in front. To convince yourself,look at the corner A and watch how it umps fromback to front.

How do we explain this? Reversible figures pro-duce a set of data that can be given two equallyvalid interpretations; the brain accepts one firstand then the other. Another classic example is the

goblet of Edgar Rubin (a Danish psychologist). InFig. 18, initiallywe see two faces, then a goblet.The inversion of object and background is only

one reason for optical illusions; other elements ina figure can also produce ambiguity. Take the

'A rhomboid s a paralle ograrr n wh ch the angles are ob ique and adjacentsides are unequa

40


fig 19

{in I Q


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fig 20 9 I

U _UA I

II I

Ui

Uc==

flg ? I

t- picture in Fig. 19. It looks like a duck or a rabbit,fig. 24 - i p n p n r i r i i n n on wA/hpthpr we fnousi first nn the Itsft

side or on the right side. Indeed, the duck-rabbitwas devised by psychologist Joseph Jastrow in1900 to illustrate ambiguity.

Games with matches

The simplest figure games can be played justabout anywhere. Allwe need is a box of matches.Fig. 20 shows a coin inside a chalice formed byfour matches. By moving only two matches, howcan we reconstruct the figure to put the coin onthe outside? Fig. 21 shows the solution; the dottedlines indicate the initial position. The next figureillustrates a similar trick. Fig. 22 shows fivesquares formed by a certain number of matches.The problem is to remove one square by chang-ing the position of only two matches (no open-sided or incomplete squares are allowed). Onesolution appears in Fig. 23.

Next Ionk at the triannkl in Fin. 24. Bv removingfig. 26 . - - -- - - - -

four matches we always finish with two equilateral

41

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I

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e

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Ii

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I

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2

7

6

15

fig.29

1 15

15

15

15

1500:

4

3

8

15

triangles (Fig. 25). Now try to produce two trian-gles by taking away only two matches (the figuresmust have no open sides). One solution is shownin Fig. 26.

Lo shu, an ancient Chinese figure

Take a square of nine equal boxes and, usingnumbers one through nine, write a number ineach box in such a way that each row and columnadds up to the same sum. This is quite an absorb-ing and difficult task and will probably be solvedonly by trial and error. One attempt, shown in Fig.27, fails because the totals of the second andthird columns and the diagonals are different fromthe totals of the rows and the first column. How-ever, by interchanging 5 and 7 we reach the ar-rangement of Fig. 28, which is a solution. A figuresuch as this with rows, columns, and diagonalsadding up to the same sum, is known as a "magicsquare.' The Chinese were the first to discoverthe properties of magic squares which they calledLo shu. Legend tells us that the figure was re-vealed to man on the shell of a mysterious tortoise

which crawled out of the river Lo many centubefore Christ. Historically, Lo shu goes backfurther than the 4th century B.C. The Chinesetributed mystical significance to the mathematproperties of the magic square and madesymbol uniting the first principles that shaped

things, man, and the universe, and that remapart of them eternally. Thus, even numbers cato symbolize the female-passive or yin andnumbers the male-active or yang. At the centethe two diagonals is he number five, representhe Earth. Around itare four of the major elememetals, symbolized by four and nine; fire, byand seven; water, by one and six; and woodthree and eight. As Fig. 29 indicates, eachment contains measures of both yin and yfemale and male-the opposites in reality.

Magic squares: their history andmathematical features

A magic square exhibits the integers from 1 towithout repetition, in such a way that each rowto right) and column (top to bottom) and the

42

fig. 27 fig, 28

17

4

3

8

15

9

7

1

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Left Leonardo da Vnci s Canonedelle proporzioni de/ corpo umanoRue for the proportions of the humanfigure, according to V truv us Ga ler ede IAccademia Ven ceBelow Leonardo s sketch for a se -mcving vehicle powered by two largespr ngs and steered by a handlefixed to a sma rear wheel

Opposite A orecht Durer sMelencolir erigravng 15t14) Dthe Renaissance w th the renewinterest n the arts and so ences

art sts to show partial ar fasc oaw th mathematics and geometryLeonardo and Durer This picturreveal s a mag c square of orderto symbol ze melancho y thencons dered an energy zing cond t

The ast ne of the Square rd cashe year of compose t on 1514

which the number 15 can be produced from thefirst nine natural numbers (Fig. 30). Note that eachnumber appears only once. Therefore we mustarrange the triplets of equal sum (in Fig. 30, thefirst two, for example, share the nine) in such a

way that two of them have one number in com-mon. This is possible because the square haseight lines (three rows, three columns, two diago-nals) that must add up to 15, and these corre-spond to the eight possibilities of Fig. 30. To fill thecompartments, remember that each number canoccur only once. Take the central compartment: Itmust appear in four triplets (one row, one column,two diagonals). The only such number is 5. Nextconsider the corners: Each must contain a num-ber that appears in three triplets (one row, one

column, one diagonal). Fig. 30 indicates there areonly four such numbers: 8, 6, 4, 2.We could have reasoned the other way around

and counted how many times a number appearedin he set of triplets, and then deduced its positionin the square. For example, 9 occurs only twice,and therefore cannot be at the center or in a cor-

ner. Given the square's symmetry, there cantwo cases-mirror images of each other.

More intricate magic squares

The Renaissance was a period of culturalartistic revival throughout 15th and 16th cenEurope and touched all branches of learning.development of mathematics and geometryremarkable and, indeed, their influence extento the figurative and architectonic arts for wthey became a model and reference pointthese trends came together in the personLeonardo da Vinci (1452-1519), scientist, wman of letters, engineer, mathematician and

ist. For da Vinci, mathematics and geometry wclosely linked to mans' artistic and culturaldeavors. In a no longer extant treatise on painDe picture, he wrote: "Do not read me ifyouno mathematician." Actually Plato (428-348 Bmade the connection long before da Vincieven saw fit to place this warning over the

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2x15=30 2x15=30 2x15=30 2x15=30

trance to his school: "Let no one enter who mas-ters no geometry."

The link between mathematics, geometry andart also underlies the work of the German painterAlbrecht Durer, a contemporary of da Vinci. InDurer's noted engraving Melencolia, there is a

magic square, often considered the first exampleof one seen in the West. It is constructed so therows, columns and diagonals add up to 34(Fig. 31). Moreover, the four central compart-ments add up to 34. The second and third com-partments in the bottom row indicate the date ofcomposition: 1514. Aside from the intimacy be-tween the arts and sciences during the Renais-sance, perhaps another reason for DOrer's inclu-sion of a magic square in his engraving, is thatfourth-order squares were thought to possess

special therapeutic virtues. Indeed, astrologers ofthe period advised wearing them as amulets todispel melancholy.

Let us move on to other magic squares. Takethe third-order square of Fig. 29 and multiply itsentries by a constant to produce new entries.These will again add up to a constant. Indeed,

fig 33

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multiplying by 2 gives a sum of 2 x 15 =(Fig. 32). Similarly, Figs. 33 and 34 are constrted by multiplying by 3 and 4 the entries ofsquare in Fig. 29. If we exclude rotationsreflections, there exists a unique magic squarethe third order. Figs. 35 and 36 are only a sinthird-order square with a reflection about the c

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tral row, while Fig. 37 has been rotated around adiagonal. With higher orders, the number ofarrangements increases. A fourth-order squareallows 880 different placings of its 16 numbers,excluding reflections and rotations. This wasfirst discovered by the mathematician BernardFrenicle de Bessy in 1693. Figs. 38, 39 and 40show some of these solutions, with the sum 34. Itis not yet clear what mathematical law governs thedisposition of numbers in magic squares. Thequestion remains open, and the known solutionshave only been discovered by trial and error.

How many fifth-order m agicsquares are there?Until recently the estimate was about 13,000,000.In 1973, however, Richard Schroeppel, a pro-grammer with Information International, deter-mined the exact number with the aid of a moderncomputer (His findings were later published inScientific American, vol. 234, no. 1, Jan 1976).Without counting rotations and reflections, thereare 275,305,224 different solutions.

Diabolic squares

These are even more intriguing than magicsquares, because of their additional properties.Again consider Durer's square as it is rearrangedin Fig. 39 and repeated in Fig. 41. The sum of thefour central squares is 13 + 8 + 3 + 10 = 34, as isthe sum of the four corners and the vertical as wellas the horizontal off-diagonal squares. Suchmagic squares are called "pandiagonal." Thesame constant results if we add the set of fournumbers marked in Figs. 42 and 43.

Similarlywe can form fifth-ordermagic squareswith particular properties. In Fig. 44 the sum isalways 65. Considering corner numbers plus thecentral one, as well as diagonal numbers (includ-ing the central one), we always produce thesame sum: 9+13+17+1 +25=65, 7+5+21 + 19+ 13=65.

A square of order 5 in which any pair of num-bers opposite the center adds up to n 2 + 1 (n

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being the order) is called ''associative." Here,n2 + 1 = 26. Thus, horizontally, 20 is opposed to6, and 20 + 6 26; diagonally, 17 is opposed to

9 and 17 +9 = 26, 25 is opposed to 1, and25 + 1 = 26. The Lo shu of Fig. 29 also has thisproperty and is therefore associative. Indeed,with n = 3, n2 + 1 = 10, and in the square7+3=10, 4+6=10, 8+2=10, 9+1 =10. Afourth-order square may be either associative orpandiagonal, but never both. The smallest squarethat can be both is of the fifth order. If, as usual,we exclude rotations and reflections, there areonly 16 fifth-order squares with both propertiesaccording to Schroeppel's calculations.

In Medieval times, the Moslems imbued pan-diagonal squares of order 5 with 1 at the centerwith mystic significance, for number 1 is the sa-cred symbol of Allah, the Supreme Being. Theproblem of representing God and the concept ofGod occurs in all religions and theologies. Thesymbol that best evokes the unity of being is the

number 1. God is one. However, the Moslemception of God is such that no sign or pictureadequately represent Him, not even the most

stract and immaterial such as the numbeHence, in some magic squares the ineffableture of the Supreme Being is suggested by ling the central square empty.

Magic stars

Similar features are observed inother geometrfigures such as magic stars. Take twelve counnumbered from 1 to 12 (Fig. 45) and const

a star of David from two equilateral trian(Fig. 46). Now place the counters on verticesintersections so the numbers along each of thsides add up to the same sum. As before thisbe achieved by trial and error. In Fig. 47, theis always 26. However, if we add up the six vcesweget3+2+9+11 +4+1 =30.

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Below A Chinese magic circle ofgreat h stor cal interest executed bySeki Kowa In the 17th century

fig 48

Let us refine the star by displacements thatmake this last sum also equal to 26. Such a prob-

lem must be tackled systematically and demandsa plausible strategy. With the figure consisting oftwo equilateral triangles, in order to reach a sumof 26 at the vertices, we must put the internalnumbers aside for the moment (they form a hexa-gon shared by the two triangles). A rational pro-cedure might be to produce the vertex sum of 13for each triangle, so that 2 x 13 = 26. In Fig. 47,the inverted triangle gives a vertex sum of11 + 1 + 3 = 15. We therefore interchange 11 and10, 3 and 2. The vertex sum becomes

10 + 1 + 2 = 13. For the upright trianglewe can nolonger use 1 or 2 (indeed 9 + 4 + 2 = 15), or 8 or7; they would merely complicate matters. There-fore we try 6; we interchange 6 and 9, 3 and 2,and leave 4 untouched. Asmall rearrangement onthe sides then produces the solution shown inFig. 48.

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More about squares

There are still more magic squares. Take Fig. 49.Its square seems to have no formation rule, itsnumbers being haphazardly distributed. How-ever, the square has a property that furnishessome interesting tricks. Ask a player to performthe following:1) Take any number and eliminate all others in hesame row and column. For this we need countersor other markers to cover the numbers to be re-moved. Suppose the number chosen is 20 in thethird row and second column. Eliminate 6, 7, 5, 4and 33, 27, 29, 23, leaving the square of Fig. 50.

2) Repeat the maneuver on Fig. 50, and suppose14 is chosen. Eliminate 13, 12, 11 and 20, 16, 10 ,leaving Fig. 51.3) Repeat as before and suppose 15 is chosen.Eliminate 14, 13 and 19, 9. leaving Fig. 52.4) Repeat and suppose 17 is chosen. Eliminate18 and 7, leaving Fig. 53. Only 8 remains. Addingthis and the four chosen numbers (20 + 14 +15 + 17 + 8) we get 74. Repeating the whole pro-cedure with any other numbers, the result will al-ways be 74. What is the trick?

Consider how the square is constructed. Anynumber is the sum of two, one each from a groupof generators that together add up to 74 (Fig. 54):2 + 16 + 3 + 1 + 0 + 17 + 11 + 4 + 13 + 7 = 74.The two groups are shown in black along the firstrow and column; any number of the square is thesum of the generators against its row and column.

The trick then is o eliminate all numbers exceptone (and only one) in each row and column, andthat is achieved by the procedure stated above.The final sum then, is simply the sum of the twogroups of generators; a rather simple device inwhich the order of the square does not matter, nordoes the sum to be calculated. Any type of num-bers can be used: negative, positive, fractions orintegers.

An extraordinary surface

There are mathematical and geometric gamesthat can be resolved only by a proof or through aconcrete example. Take a square with 16-inchsides for instance, and subdivide it into four as inFig. 55. We can then transform it into the rectangleof Fig. 56. The four parts fit perfectly, yet the twofigures are unequal in area, for 16 x 16 = 256 in2and 10 x 26 = 260 in2. It appears that we mag-ically produced 4 in2 out of nothing. Here, too,there is a trick, as we can see by actually con-structing the figures. Take a large sheet of graphpaper, large-meshed if possible, and substitutefor each inch a certain number of squares(Fig. 57). Suppose the square has 8-inch sides,so there are 8 x 8 = 64 small squares. Cut out thepieces as required (Fig. 58). The rectangle willbe5 x 13 = 65, leaving us with one too manysquares. Ifwe lay out the pieces as in Fig. 59, it isat once clear that the sides of the two trianglesand trapezia do not form a diagonal of the rectan-gle: it is only the diagram that produces this illu-

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fig. 61 D

fig. 61b D

Euler's work on graphs appeared in 1736.Since then it has been usefully applied not only inmathematics but in other fields as well, In he 1 thcentury, graphs were used in circuitry and in the-ories of molecular diagrams. Today, aside frombeing a method of analysis in pure mathematics,the theory of graphs is used for the solution ofnumerous practical problems, for example intransportation and programming.

Euler was one of the most productive and orig-inal mathematicians in the history of science. Theson of a Calvinist pastor, he was barely twentywhen, in 1727, he was invited to join the Academyof Science in St. Petersburg (today's Leningrad).He had an encyclopedic mind and though a stu-dent of physics, astronomy and medicine, Eulerhad a particular fondness for mathematical prob-lems. His output was prodigious. It is said that hewrote constantly-while waiting for dinner to be

served, even while holding one of his manyspring Indeed his desk was always ladenwork awaiting publication. In 1 46 Euler lectat the Berlin Academy, but finding the culclimate and the appreciation of his work less

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f[g 63

th s instance the theory of graphsproves that the prop em of thebridges can be so ved

favourable, he returned to Russia and the court ofCatherine the Great. Even though Euler eventuallywent blind, he pursued his mathematical re -

searches intensely until shortly before his death in1783. Some time ago, Swiss mathematicians hon-oured Euler by beginning to collect and publishall his writings; some fiftyvolumes have appearedto date, and the total may well reach two hundred.

The bridges of Paris present a problem similarto the bridges of Kdnigsberg. Consider the lIe dela Cite in the Seine (Fig. 61). Here A has 8 bridge-heads; B,7; C, 10; D, 7; producing only two oddtotals. Therefore, there must be a solution, 'butwith certain restrictions as we shall see. The solu-

tion is easily found by starting from an area withan odd total and tracing a path crossing thegreatest number of pairs of bridges leading fromone area to another. A further solution is shown inFig. 61 b.

Elementary theory of graphs

When Euler grappled with the problem of Kbnigs-berg's bridges, he did not consider going there to

solve it. Instead, in the manner of modern sci-ence, he tried to formulate the problem in a gen-eral manner by tracing a schema (Fig. 62) inwhich banks and islands are shown as points,and the various bridges between them as lines.The problem then is: Starting from any of thepoints, trace the figure and return to the samepoint without retracing any line and without liftingpen from paper. It is impossible. To solve theproblem formally, consider some auxiliary con-cepts first. What does a graph amount to? Given

two or more points in a plane, we join them witharcs or curves or segments to obtain a figure wecall a graph. The points are called vertices ornodes, and the lines between them (of whatevershape) are called sides or edges. The number of

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fig 62

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B

Opposite r ght A 17th century pr ntof Paris show ng the Se ne and the I ede a Cite A prob em s mi ar to thatof Ken gsberg can be posed but n

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vertices and edges of the regular dodecahedronhas a Hamilton circuit (Fig. 68). Aclassic exampleis the following: Let the vertices A, B, C, Dstandfor four cities (Fig. 69). What are the possiblepaths that go through each vertex once and onceonly? Starting from A,we have the followingHam-ilton routes: ABCDA, ABDCA,ACBDA, ACDBA,ADBCA, ADCBA.Note that 1 and 6, 2 and 4, 3and 5, are pairs differing only in direction.

Save the goat and the cabbage

This is an old saying, but not everyone knows thatits origins are an ancient puzzle of some twelvecenturies ago. A man wants to transport a wolf, agoat, and a cabbage across a river in a boat thatbarely has room for him and the cabbage, andcertainly for not more than one of the animals.Moreover, he cannot leave the wolf alone with thegoat, or the goat with the cabbage. How can heget everything across the river without the wolf

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eating the goat, or the goat devouring thecabbage?

Graphs are essential for solving those puzzlesin which we must move from one place to anotherunder certain conditions. Let us represent the var-ious crossings, denoting the man by t, the wolf by1, he goat by p, and the cabbage by c. The firsttrip might be to take the goat across, since thewolf will not eat the cabbage. Starting with thegroup tlpc, we now have /c left. Next, t returnsalone which creates the group tic. Now t trans-ports either the wolf or the cabbage. In eithercase, he returns with the goat, so the group is nowtpc or t/p respectively. Now he takes the cabbageif he had already taken the wolf, or the wolf if hehad already taken the cabbage. When the manreturns alone to join p, the group becomes tp.They finally cross the river and that concludes theoperation.

A synthetic graph for these moves is shown inFig. 70. This simple example enables us to visual-ize a graph as a game. The vertices represent thevarious positions (the changes in the original

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group), and the lines between them are the movesallowed.

To clarify the advantages of graphs still further,try to solve the present problem intuitively bydrawings (Fig. 71). This turns out to be more com-plicated, so for the next game we will only usegraphs.

The jealous husbands

This is similar to the last problem and just as oldbut a bit more complex. Three honeymoon cou-ples reach a river and find a small boat that willhold only two people. The dilemma is made worseby the fact that the husbands are rather jealous.How can the entire party cross the river withoutleaving any bride alone with a man who is not herhusband? As before, let us simplify the problemand construct a graph. Let the couples be A, B,

C, and men be distinguished from women bysuffixes, u and d respectively. Thus a, andrepresent husband and wife of couple A, so using a synthetic notation A = au, ad), B =bd), C = (Cu, Cd).

The first vertex will be given by A, B, C, othree couples together. The problem is knothan the previous one because there are mcombinative alternatives, as can be seen in dein Fig. 72. First, two wives cross the river, yieldthree possible groups-ad, bdor bd, Cd, or cd,From the first vertex there are three sides to shthis. At successive vertices we show the chanof the original group until everyone has crossThus, A,bu, C means that on the starting banknow have everybody except the wife of coupleThe vertex marked a~, u, cu means that onstarting bank we have all the men, while allwomen are on the opposite bank. The fact thathe other sides converge here indicates the

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Left 1) Starting posit on witheveryone on the left bank, 2) he mantakes the goat to the right bank, 3)he returns alone, leaving the goat, 4)he takes either the wolf orthecabbage to the right bank, 5) if theformer, he returns with the goat andleaves the wolf, 6) if he atter, he

returns with the goat and leaves thecabbage, 7) he leaves the goat onthe left and ferries either the cabbageor the wo f, (depending on whether 5)or 6) was the case), to the right bank;8) he returns to the left bank to fetchthe goat; 9) final position witheveryone on the right bank

cessity of this stage. Indeed, one of the twowomen who had crossed first returns (producingA, bu, C or A, B, cJ) and helps the remainingwoman to embark, leaving the three men (au, bu ,cJ) alone. One woman then disembarks while theother returns to the three men. We now have acomplete couple (either A, or B, or C) along withthe two other men. The next move is: The two menembark, leaving the couple. One man disem-barks on the other bank while the second manreturns with his own wife, producing two couples(A,B, or B, C,or C, A) at the starting point. Thetwo men embark leaving their wives. From theother bank, the third woman, who had been therewith her husband, returns alone, and in wo furthercrossings brings the other two women across,thus reuniting the entire party.

Interestingly, even using identical rules, whenthe same problem involves four instead of threecouples it is unsolvable. Remember that only themen are jealous, which means that on neither

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fig. 75 W 1 5 fig. 76

knight's move, we can construct the graph ofFig. 74, where W. and W2 are the two whiteknights, and B1 and B2 the two black ones. Thecentral square is not numbered because it s nac-cessible to the knights. The graph of Fig. 74 isclear enough. For example, to shift W2 to square6, the piece must follow the route 3-4-8-2-6, whileB. reaches 3 by 6-5-1-7-3. Further, to shift W. to8, the route is 1-7-3-4-8, while B2 traverses8-2-6-5-1.

In the graph of Fig. 74 the sides intersect atvarious points that should not be considered ver-tices. If a graph can be drawn without such inter-sections it is called planar. The following is agraphic solution of the knight interchange by pla-nar graphs. The only restriction is that we mustisolate the moves of symmetrically oppositeknights (Figs. 75-76). The graphs can be read ineither direction, clockwise or counterclockwise.

A wide range of applications

The theory of graphs, born from giving mathe-matical forms to puzzles and first used in geome-

try and mathematics, was applied in numerareas of science as it was in practical life.cause of its formal properties, it was swiftlydeoped as a way to simplify and visually presotherwise complicated problems. Since thecentury, it has proven immensely fruitfulin repsenting problems of electric circuitry.

Fig. 77 shows the graph for combining thswitches with two lamps. The arrows indicatethe sides are oriented; a graph is directional wits sides can be traversed only by followingrows. If the sides are not directional we canalong them in either direction and the figurehave no arrows. Graphs serve a numberpurposes. Among other things, they are widused for road routing, floor plans and econoprogramming.

Because graphs are simple and immedithey are often used to visualize such compsituations as relations between people or groof people. The stages of a football championsinwhich a number of teams play each other mibe an example. Another might be a simple rouproblem: the road between London and Dove

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fig. 77i;: :aximum biological birthrate

01

Graphs simplify and clarify situationsand have become essential tools inmany fieldsLeft:A graphic description of thefactors affecting the world's birthrateFertility depends on two factors the

efficiency of birth control and thedesired birthrate These in urndepend on other factors: per capitaservices and industrial product, andthe average life-span.

this case, two vertices would represent the twocities, and the sides would indicate any two roadsbetween them. Ifwe take part of a superhighway,its separate lanes are directional and marked withone-way arrows. Of course many ordinary roadsare also two-way, and in hat sense

Math and Logic Game

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