Figures for Fun: Stories and Conundrums

Citation preview

Yakov Perelman

Figures for Fun

Stories & Conundrums

FOREIGN LANGUAGES PUBLISHING HOUSE MOSCOW

YAKOV PERELMAN

'

•

•

S T 0 R I E S A N D C 0 N U N D R U,M Sv; . ,...

'

,

l (

.

FOREIGN LANGUAGES PUBLISHING H 0 U S E MOSCOW·I957

CONTE~TS

Page

..,

._.REFACE

CHAPTER I Dratn Teasers for Luncl1 11 13

1 A Squurel 1n the Glade 2 School Groups 3 'Vho Counted ~~ore? 4 GTandt atbet &Dd Gt ands on 5 Ratlway Tickets 6 A D1t1glble s Fhgbt 7 Shadow 8

14 14.

14 1(

15 16 16

l\.1 a\.ebes

9 The "'\Vonderful" Sturn p 10 ihe December Puzzle 11 An Arithmetical Tr1ck Answers 1 to :t1 12 The Allsstng D1g1t 13 Wbo Has It?

~Ia them a hcs

Dominoes 14 A Cham of 28 P1eces 15 The Two Ends of a Cba 1n 16 A Dom1no Trick 17 A Frame 18 Seven Squares 19 :Mag1c Squares 20 Progress1on tn Doro1noes

17 17

lB 26 26

1n

G1:mc~

29' 29' ~

29

29 30 30

31

31 36

The F1fteen Puzzle 21 The Fus t Pr obi ~m 22 The Second Problem 23 The Tbtrd Pr ob 1em Answers 14 t 0 23

3G 3G 36 CHAPTER

Ill

\ nothcr Dozen l'uzzlcrs

24

25 26 27

41

Slrtng Socks and Glo' c~ Longevlt) of Ila1r

41

\Vages

42

42 42 42 42

Skung

28

29 T\\ o \\ orl..crs 30 TJ- p1ng 31 T" o Cog \ Vh~els 32 Hov- Old Is He' 33 Another AgE" R1ddl~ 34 Shopp1ng Answers 24 l 0 34

42 42

43 43 43

CH.APTER IV'

Counhng '35 .36

Do You h.no\\ llow to Count? \Vhl Count Trees 1n a Forest'

u2

CHAPTER V

na rflers '' 1 th 37 JH 39 40 41 42 43

44

Hundred Rubles for Fl\e 1Jne ""Thousana Twenty Four Th1rt}

Numbers 54 ...dl 00 55

Restoratton of Dtgtts \Vhat Are the D1g1ts'

55

DlVlSlOD

fiD

Dv1d1ng by

11

55

D5

45

46 47 48

Ansv.

Tr1cly \lult1pl rat1on A :l'cumber Triang1e Another Number Tru1.ngle A Aiag1c Star ers 37 to 48

EG 66-

oi.J. 56 o7

CHAPTER VI Number G1ants

49 A Profitable Deal 50 Rumours 51 The Btcycle S'\\lndle 52 Rewatd 53 The Legend About Chess 54 Rap1d Reprodactton 55 A Free D1nner 56 A Tr1ck w1th Coins 57 A Bet 58 Number G1ants Around and Ins1de Us

63 68

71 74 80 84

89

94 99 102

CHAPTER VII

'V•thoul lnstruments of ~Iensurement 59 60

Calculat•ng D1stance by A Lt\ e Scale

106 107

Step~

CHAPTER VIII

Geon1etr1c Brr1n TeMers 61 62

109 109

A Cart

Through a 1tlagn1fymg Glass 63 A Car'Qenter s LeYel 64 How !tlpny Edges? Cr~cent

65

A

66 67

A ].tntch T1'1ck Another ~latch 'rr1ek The 'Vay of the Fly F1nd a Plug 1'he Second Plug The Thtrd Plug

68 69 70 71

109 110 111 lll 111

tll 112 112

112 5

72 73 74 75 76 77 78 79 80

A Co1n Tt1ck The II etgbt o£ a To,ver S1mtlar F1gures The Shadow of a \Vtre A Br1ck A Gtan~ and a Pygmy Two \Vater melons Two ~felons A Cherry

112 112 113 113 113 113 113 114 114

.Bj

T.be

EJJfel Tow£'.r

114

82

Two Pans

114

83

In Wmter

111

84 Sugar \.nswers 61

114 114:

to 84

CIIAPTER IX The Gcon1.etry of Ratu and Snow

85 86 87

12J 12 128

Pluvtometer llow ~luch Ratnfall? How ~luch Snow? CHAPTER ~In themat1cs

A.

and the Deluge 13Q

88 The Deluge 89 \Vas the Deluge Possible-;. 90 \Vas There Such an Ark?

133 134

CIIAPTER

XI

Thlrty Dtftercnl I roblcms gt 92 93

99 95 96 97 98 99 100

136

A Chain Sp1ders and Beetles A Cape a Hat and Galoshes CbJ.$keR ~ml Duck BggsAn A1rp Tr1p l\Ioney Guts Two Draughts Two D1g1ts One Ftve 9 s

136 136

J37 13 137 137

137 137

137 6

101 102

'fen Dtg1ts Four \Vays

four 1 s 104 !'tiysterJousJ._.DI\ IS Jon 105 Another Dl' 1S1on 103

106 'Vhat \Vlll You Have? 107 Somethtng of the Same Sort 108 A Plane • 100 A Aldlton Thmg~ 110 The Number of 'Vays

111

The Clock Face

137 137

137 138

138 138 138 138 139 139

139 13~

1 12 E1gbt-Po1n ted Star 113 A Number 'Vheel 114 A Tripod 115 Angles

140

116

On the Equator S1x Rows

140

How to D1V1de That?

140

117 118

119 The Cross and the Crescent 120 The Benedtktov Problem Answers 91 to 120

131'1 140

140

141

141 142

PREFACE

To read and enJoy th1s book 1t \Vlll su{ficc lo pos~ess a modest knowledge of mathemattcs, 1 e knowledge of arithmetical rules and ele mentar) geometry ''ery few problems requ1re the ab1ltt) of forming and solv1ng equations-and the snnplest at that The table of contents, as you may see, IS quite dtversified the sub Jects range from a motley collection of conundrums and malhemat Hjal stunts to useful practtcal problems on count 1ng and measurtng Tl1o .author bas dono ever;} t1Itng to make his book as fresh as possible, a votdrng repet1 t Ion of all that has already appeared 1n his other \vorls (Tnch.s and Arnusements Interesttng Problems, ett.) TJ1o reader w1ll find a hundred or so brain-teasers that have not heen 1ncluded 1n earller bools Chapter VI-- "Number Giants "-1s adapted from one of the author's carl1er pamphlets, '"1th four new stories added t

•

CHAPTER I

BRAIN-TEASERS FOR LUNCH It \Vas raining.. . . . \Yo had just sal do\vn for I unch at ou1 holiday .. home \Vhen ono o£ the guests asked us whether we would 1ike to hear what had happened to him in tho morning. Everyone assented, and he began. 1. A SQUIRREL IN TilE GLADE.-"! hnd quite a bit of fun plnying hide-and-seek with a squirrel,,, he said. "You kno,,t that lillie round glade with a lone birch in the centro? It was on this tree thot a squirrel \vas hiding from me . As I emerg£~d from n thicket, I sn\v it.s snout and two bright. I itt le oyes peeping from behind the trunk .. I \van l• ed to soe tlte little animalt so I started circling round along the edge of the glade,. mindful of keeping t1Ie dis tan co in order not to scare it . I did four rounds, hut the 1ittlo cheat kept. hacking a\Yay from met eyeing me suspiciously from behind the tree . Try as I did, I just could not seo its back . u uBut you bave just snid yourself that you circled round the treefour times," one of tlto listeners interjected . "'Round tho tree, yes, hut notf round the squirreL. u "But tho squirrel was on the l roo \vasn' l it? '" uso it was . n "\Veil, thnt means you circled r~~nd the ~quirre~.l too . ., · "Call that circling round lho squirrel \vhcn I didntt, sco its bnck? ,. u\Vhat has its back to do 'vith tile 'vho]c thing? The ,squirrel \Vas ,on lho tree in tlio centre of tho glnde nnd you circled round thr tree. In ot l1cr \vords, you eire) cd round the squ irrcl. '' · · J

A

~

i"ll

"Oh no,. I dtdn't .. Let us assun1e that I'm ctrcl1ng round )OU and you 1\.eop turntng, s1Io'" 1ng me Just ) our ft;1cc Call that cJrcl 1ng round you?'' 'Of course, ''hat else can ) ou call tt?' "Yo~ mean l'nl ctrcl1ng ro\1nd) o\1 thougl\ I'm never llelllnd ~ fhc \\ay I under sl ttnd It, 1t •s mo\ 1ng Ul suth a mn nncr so as I o li;\ee l h o oh JcCL I'm n10' tng around fron1 all srdes Am I r1ght, profe~sor~ lie turned to an old man at our table "Your 'vholo argument lS esscnt1all \ one nbout n \\ordt tho pro fessor replied \Vhat you should do first IS ngrcc on the definition of •circling • How do ) ou undersldnd tho \\orcls ~c1rclo round nn obJect"? Tl1cro arc l\vo "a)S of understanding that F1rsll) J tt's mo\Ing round an obJect that IS 1n ll1c centre of a c1rclc Sccondl~ 1L's moving round an object 1n suclt a '' ay os to sec all Its stdes If J' ou zns ISI 011 the first mean1ng, tlten you 'vnlkcd round the squirrel four l1mcs If It's the second that you hold to, then )OU d1d nol \\nll. round Il at all There's 1eall) no ground for an argwnent l1erc, that 1s, 1f you t\vo speak tl1e ~arne language and understand \\ords 10 tho same ~y' . 'All r1ghl, I grant lltere aro t" o meanings llut \Vhiclt JS the correct one'.,, That's not tbe way to put the quesLton ) ou can agree about anllhtng The question 1s, 'vh1ch of the two meanings IS the more general!' accepted? In m~ optnton,. 1t's the first and here's why The sun, as you kno\\, does a complete cirCuit 1n 26 dnj s t• Does the sun revolve', ~~or couTse, 1t does, l1ke the earth Just. 1mag1nc, for Instance, tl1at 1t \vould takenot26 days, but 365 1/4 dajs, 1 e t a ''l1ole 1 eart to do so If th1s \vero the case. the earth would see onl~ one Side of the sun, that 1s, onlj Its 'face ' And yet, can anyone claim that the oartl1 does not revolve round the sun? u uYest no\V It's clear that I Circled :round the squirrel after all ,.,. t

11

t.

"I've a suggesttonl comrades l.. one of the company sltouted "It•s raining no\v, no one IS going out, so Jet•s play r1ddles The squirrel riddle \Vas a good heg1nning Let each tllin1.. of some brainteaser ,., ''I gtve up tf they have an) thing to do \"\Ith algebra or goometrJ," a ) oung \voman said 'l\fe too,, another Jotned In. "No, \VC must all play, but \Vc'll promise to refrain from an) alge bra1cal or geometrical formulas exccpt 1 perhaps, the most elementary ones Any obJeCtions? 'None f), the others chr,.russed 'Let's go ,. uone more thtng Let professor be our JUdge ,. 2. SCHOOL GROUPS ~ ''Ve have five e\.tra curricular group\\ you count th 10gs1 then ) ou do not l.now how to 1

counl Th1s method

Inconvcntent, bothersome and at ttmes pla1nlj tmpract.tcable It. 1sn 't half bad tf ) ou ha' e to count na1ls or clothes, they can eas1ly be sorted out But JUSt unagtnc you are a forester and are asked ho\v mnn) p1ncs, firs birches and aspens there are on each bectare lVell, oere Lt IS Impt%'"5Ible to SOrt them aut Ot"' group them b) fom1l~ \VItal nre j ou go1ng to do count the p1nes, birches. firs and aspens soparatel y? If ) ou do that. you '\\ tll have to walk around the forest four t1mes There IS an caster \V3) of do1ng 1t-zn JUSt one go I shall show ) au bov. thts IS dono '\ 1th natls and scre"--s 4-19~~

IS

49

F1g 21 Strokes must be JOtted down tn

fives Fig 22 HO\\ to count wttb squn.rcs

To count na1ls a11d scre\VS In a box Without sorttng them out, need, first of all, a pencil and a sheet of paper l1ned out as folio''~ Nalls

)OQ

Screws

Then you start counting You tale a tbtng ouL of the hox and If 1t IS a na1l you put a stroke tn the appropriate column You do the same tn tha case of a scre,v, and thus continuo until there IS notbtng le[t 1n the ho~ In the end you "tll have as many strot.es 1n the ''na1l" column as there \\ere nails tn the bo~ and dttto for the 'scre\\" column After that, all you have to do Is add them up The add1t1on of these strokes may be simplified and speeded up If J ou JOt them down lll fi\ es tn the rann of II t tie squares (fig 21) Squares o.f this sort are best grouped In pa1 ..s, 1 c • after the first ten strokes JOt do"'ll the elc,entb 1n a ne\v column \\"11cn there nrc two squares In tbo second column, start the third, etc You \\Ill thcn1. l1ave your strolcs ns sho,vn in fit! 22 11

50

It Is very ensy to count them, for you \\Ill see r•ght a" ay that there nrc three lots of 10 stroJ...es each one square of 5 and ono Incomplete figure of 3 strokes, 1 c' 30+5+3=38 You can uso other figures too For 1nst ance n full square Is o[ten used to represent 10 (fig 23) In count1ng trees of different families you follo\v the same rule, only In thts case you Will ba' e say, four columns 1nstead of t '' o It IS also more convcn· 1ent to h ,_vc hor1 zon tal and not vert 1c ul col urnns Tal.e ftg 24 !Jelo'' as a specunen Ftg 2~ shO'\\n what th1s form "Ill look 1I1.c when filled 1n. After that 1t 15 very easy to find tbc total of each eolumn Fn·~

53 79

B1rch~

46

P1nes

·----· F1 g 23 E acb square represents 10

37

'\spr-ns

Thts 1s also the method emplo) ed by med1cal workers 1n counting red and white corpuscles 1n a drop of blood

Btrcnes Aspens

'Vomen can save a lot o[ t1me and labou-r by adopt1ng tbts method In tlemtzing their washing No" you kno\v ho" best lo count dtffcrcnt plants growing on a p1ot 1 au draw a form \\TJt1ng dolvn each different plan~ j , "'"' £

-51

column, leavtng a few columns 1n reserve for any other plants you may eomc across, and then start counting A spec1men form IS gtven 1n fig 2u Then you proceed tn exactly the same way as 'vhen you counted trees In the forest ~

P1nlls

0

Firs

tz1 0 0 0

~

0

0

~

0

0 121

121 t21 l2l ta

0 0

t2l t

t21 eJ 121 0

Btrches

0

121

~

~

I(J

Aspens

0

0

tzl 121

I2J

t2l

0

n IZJ

I2J

0 13

t2l 0

0

r

Ftg 25 How the form looks when filled 1n

36 WRY COUNT TREES IN A FOREST? Indeed. \vhy' C1ty dwellers, as a rule, thtnk tb1s IS Impract1cable In Lev Tolstot's Anna

J)onde/lonJ

Buttercups Plontotns -

laster Bells Shepherds-

purses

Ftg 26 II O\\ to count. pian ts

Karenlna, Lcv1n. who 1s qutte a farmer, talks wtth Oblonsl"-y, who 1s about to sell a forest "Have you counted the trees?,. he asks the latter &t\Vhat' Count my trees?., Oblonsky 1s surprised Count the sand on the seasbore, count the rays o[ the planets-though a lofty gen1us m1gltl • 52

"\Veil," Levin Interrupts h1m. "'I tell you the lofty genius of Ryahinin succeeded~ A merchant never purchases Without counting~" . People count trees 10 a forest to determine how many cubic metres: of ttmber there are To do that. they do not count all the trees, Just part of them, say on 0 25 or 0 5 hectare, taltng care to choose a place with average dens1ty of gro\vth and average SIZe of trees.. For thts, one must, oi coursot have an experienced eye It 1s not enough s1mply to lno\v how many trees of each famtly there are It Is also necessary to kno"v how thick they are Thus, the form W1ll probably have more than the four columns v..,.e g1vc 1n our s1mpl1fied version You may well 1mag1no ho\\ ronny times we v."ou]d have to wall around the forest to count the trees 1n the ordinary way and not 1n the way we have explained here As you may see, count1ng 1s easy and s1mplc when you have to count things of the same 1-,.~nd. ''Then they are not, \\C have to use the method we have JUSt sho\\'U you-and many have no Idea that such a method exists.

CHAPTER V

BAFFLERS WITH NUMBERS 37. HUNDRED RUBLES FOR riVE.-A stage magician once made the follo\ving attracttve proposal to his audience "I shall pav 100 rubles to an) one \Vho g1ves me 5 rubles 1n 20 coins50 lopek. 20 kopek and 5-kopek co1ns One hundred for fivef Any takers~"'

The audttorium was s1lent Some people armed themselves With paper and pencil and \Vere ev1dentiy calcu]atJng the1r chances No one, 1t seemed, was Willtng to take the magiCian at h1s word '"I see you find 1t too much to pay 5 rubles for 100,"' the magictan went on uAll r1ght I'm ready to take 3 rubles In 20 co1ns and paj you 100 rubles for them Queue upr ,, But no one wanted to queue up The spectators 'vere slow 1n taltng up thts chance of mak1ng "easy" money "What?! You find even 3 rubles too much \Veil, I•Jl reduce 1t b) another ruble---2 rubles tn 20 co1ns How s that," And st1ll there \Vere no takers The magiCian continued "Perhaps J ou haven t any small change? Jtts all r1ght 1 11 trust you Just \VrJte down how many co1ns of each denomination you •JJ g1vo me J On m) part, I prom1se to pay 100 rubles to each reader who sends 1

me such a l1st of co1ns 38. ONE THOUSAND.-Can you ''Tlte 1,000 by using e1ght Identical d1gtts (1n add1t1on to dtgtts you may usc stgns of operation)' 54

39. T\\'Eft,TY FOUR -It 1s 'ery easy to \Vrite 24 b) ustng three 8's 8 +8 +8 Can ) ou do that by using three olhor tdcnt 1cal digtts? There 1s more lhnn one solution to thts problem ~0 TfiiRTY.-Tho number 30 may easily be \\ritten by lhrco Sll s 5 X 5 +5 I l ts h ardcr to do tt by us1ng three other 1den tical digits Trv . 1t You rna\ find SO\ era! sol ut 1ons 41. RESTORATION DIGITS -In the followtng multiplication more than half of the d1g1ts aro expressed by x's

or

x1x 3x2 x3x 3x2x x2x5 1 x8 x30 Can ) ou restore the miss1ng digtts' 42 \VllAT ARE THE DIGITS >-Horc 'The tush_ lS to find the IDlSSIDg digltS

IS

another stm1lar problem

x xs 1X X 2x x5 13xO XXX

4x77x 43 DIVISION -Restore the m1ssing dtglls ID the following problem

x2x5x 325 1

X X X

1X X

xOx x x9x x x5x

x5x 4.4. DIVIDING BY 11 -Wr1te a number of n1nc non repet1t1ve dtgltS that IS dlVlStble by 11 \'Vrtte the btggest oi such numbers and then the c::mallest 55

4.5

TRlCK \

1\IULTIPLICATION

Lool carefuli) at tho follow

tng c""tamplc 48

X 159=7

632

The Interesting thing 1s that all the nine dtg1ts arc different Can you g1ve several other srmllar examples? If they do extst at all how many of them are there'

F1g 27 \Vr1te the d1g1ts 1n the c1rcles

F1g 28 A mn.g1c star

46 A NUl\IBER TRIANGLE -\Vr1te the nme non repet1ttve d1g1ts 1n the circles o£ this triangle (fig 27) In such a way as to have a total of 20 on each stde 47 ANOTHER NU:MBER TRIANGLE -\Vr1te tho n1ne 11on re pet1t1ve d1g1ls 1n the Circles of the same triangle (fig 27) hut this ttm.e the sum on each s1de must be 17 48 A l\IAGIC STAR -The s1x pointed star (fig 28) 1s a magte" one the total 1n every row 1s the same

4+6+ 7+9=26 4+8+12+2=26 9 +5 +10+2 =26

11+ 6+ 8+1=26 11 + 7+ 5+3=2b 1 +12+10+3=26

The sum of the numbers at the po1nts however IS dtfferent

4+11+9+3+2+1 =30 56

Can you perfect the star by placing the numbers In the Circles 1n such a way as to make the1r sum 1n every row and nt the points read 26'

Answers 37 to 48 37 All the three problem s are Insolub le The mag1c1an and I could well afford to prom1se any rr1ze money for the1r solutio n To pro\ e that, Jet us turn to algebr a and analyz e all three of them Payment of 5 nlbles Let. us suppos e that 1t Js possibl e and that for th1s 1t wtll be necess ary to have x numbe r or 50 1.opo1. co1nst y number of 20 kopek co1ns and z. number of 5 kopek coins \Vc nov, have the follow1ng equatio n

50x+2 0y+5z =500 (or 5 rubles) Simpli fying this by 5 we obtain

10x+4 y+z=1 00 1\loreover, accordtng to the problem, the total number of coins 20, therefo re, we have anothe r equatio n

lS

x+y+ z=20 Subtra cting thts equatio n from the first we have

9x+3y =80 D1v1d1ng this by 3, we get 3-z: +y=2 6 , /3 But 3x, 1 e , the numbe r of 50 lopek co1ns multip lied h) 3, ts, of course , an 1nteger So IS y, the numbe r of 20 lopel co1ns 'llu~ sum of t bese two numbers cnnnot he a fract•ona1 number Therefore. 1t Js non-

sense to presum e tlu1t the problem can he solved It IS JnsoluLlc In the samo manne r the reader mny con' 1ncc himself thnt the 'rc duccd paymcnL,. prob1ems nrc ltlcv..t~c 1nsolub le In the first case (3 rubles) "'o get the follov. tng equat1on

3r+y= 13 57

1 /3

And 1n the second (2 rubles)

3x+y=6

2

/:

Both, as you sec, are fractional numbers So, the maglczan rJsked absolutely natb1ng by olfer1ng a brg money pr1ze for tho solut1on o[ these problems He \vould never have to pay for 1t It \vould have beo11 another case 1f one had to gl\ et sa) , 4 rubles-and not 5, 3 or 2_..1n 20 co1ns It would have been easy to solve the problem thcnt and 1n seven dtiTerent ways at that

*

38

888 +88 +8+8+8=1.000

39. Here aro two solut1ons

22+2=24, 33 -3=24 40. Thero are three solutions

6 X6

33 +3=30,

6=30

33--3=30

41. The mtssing d1gtts are restored gradually when we use the following method For convenience's sake let us number each ltne

x1x 3x2

I II III

t

•

x3x

IV

3x2x x2x5 1 x8 x30

v

VI

It IS easy to guess that the last d1gtt In I1ne III Is 0, that IS clear from the fact that 0 IS at the end of line VI ll'e nezl aeterml~FJB !:be me.aDJJJg oJ She J~st- X JD J.~Da 1 Jl- }S a DJgJ! that gtvcs a number end1ng wtth 0 1£ multtplted by 2 and wrth 5 tf multiplied by 3 (the number 1n ltne V ends \Vlth 5) There IS only one d1g1t to do that 5 • Here ts one of posstble solutrons s1x 50 kopek cotns. two 20-kopek co1ns and twelve 5 kopek cotns

58

It IS not dt[ficult to guess what htdes behind x tn l1ne II 8 for It IS only th1s dtgtt multtplled by 15 that gtves a nrunber ending w1th 20 (l1ne IV) rlnall) It becomes clear that the first X Ill llne I IS 4 Ior onlj 4 multtplted by 8 g1ves a number that begins \VIth 3 (ltnc IV) After that there wtll be no d1t ficulty In rcstorJng the remaining unknown d1gtts 1t \VIII suffice to multtpl~ the two factors which we have fully determined In the end we have the following e"tample of mu1tlp1Icatlon

415 382

810 3320

124J 158530 42 The same method appl1es to the solution of this problPm 'Ve get

325 147

2275 1300

325 47775 -43

Here

IS

the problem '' 1lh all tho dtgt ts restored

52650 1325

162

325 2015

1950 650

GSO 44 To solve th1s problem wo musl lno\\ the rule govcrntng the dtvlstbtltt)' or 3 number by 11 A number IS dtvtslhle hv 11 I£ the dliTCr 59

cnce between the sun1s of tho odd d1g1ts and the even d1gttst counting from the rtght. IS d1 VlSiblc b) 11 or cqt.ol to 0 For example,. let us try 23,658,904 Tbe sum of lllo even d1gats

J\nd tlu) sum of the odd d1g1ts 1s

2 +6 + 8 + 0 = 1(J The difference (subtracting the ~ma1lcr number from the b1gger, of course) 21 1G=5 1s not d1 vtstbie by 11 Thuc; the number 1s not d1Vls1ble b) 11 etther Let us try another number say t 7 ,.344 535

3+4+3=10 7+4+5+5=21 21-10=f1

Stnco 11 1s dtvisible by 11, lhe whole number Is d1 VISible too No\v 1t 1s easy to guess 1n what order \\C must place our 111ne d1g1ts to get a number that IS divisihle by 11 Here IS an example 352~049 '786

Let us ver1f'

1

t

3+2+4+7+6=22 5+0+9+8=22 The dlfierence (22 22) ts 0 'l he number '"e have taken dtVlSlble by 11. Th'e hlgg~t ~{ su~h. ~u.m\w:ts 1s

Tho smallest 102~34 7,586

60

tS)

there[are,

45. A patient reader can find n1ne e~amples of this sort Here the) are

12 x483 42 X 138 18 x297 27 X 1.98 39 X 186

=5_.79& =5r796 =5,34b =5,346 =7 ,254

48 x15f3) = 13 persons 68

In the tr tur n, eac h of the ntn o persons wh o hav e Ien rne d tbc ne~~ las t pas s 1t on to thr ee frte nds By 8 45 a m the ne\\~ 1s kn0'\\'11 to

13 +(3 X 9) =4 0 residents

F1g 36 The whole to'\\~ wJl l kno w the ne\\yS by 10 30

J. 1g 35 I:ac h tell s tho ne, ,s to thre e oth ers

I£ lllc rum our cont1nues to spr ead 1n the sam e ma nne r, 1 e ~ tf c' cry min 15 t nex the 1n 1th v. ers oth eo thr to on It ses pas 1t rs hea o wh one ute s, the res ult wil l be as follo\vs Dy 9 a m the new s 'vll l be known to 40 + (3 x27);::=f21 pcr~ons 121 + (3 x81} =36 4 persons n n , , By 9 15 a m 364 + (3 x243} -r= 1,093 persons , , , , D ~ 9 30 n m

In oth er wo rds , '\\ 1th1n one and a hal f hou rs tl1c nc~s V..Ill be l.nov:n to \\111 1 n pop .. n to" a for cb mu too m see not s doe nt Th s son per 00 1,1 nlmost uln tto n of 50,.000 Iu fac t, som e ma j th1 nl It "Il l tal c quJ lc n lon g t1mc bef ore the ''ho le to' ' n lno "s 1t Le t us sec ho~ fnst 1t "'tl l can t 1n uc to spr ead B) 9 45 a m the nc\ \s v. J11 b~ J..nov. n to 1,0fl3-r (3X 72n )=3 280 lll0,\ '"11 to 3 280 + (3 X2 187 )=0 ,81 1 B.} 10 n m thea nc" s w11I be .

69

pc~ns pcr~on5

In the next 15 minutes it 'viii be the property of more than hnl[ o( the town's population:

9,841 +(3 X 6,561) =29,524 persons And thts means that before 1t 1s 10 30 am tbc news that on)\ one man knew at 8 a m. \VIII bo known to the cnt1ro to\V"n 'II

II

Let us see now how that IS calculated The whole tl11ng bo1ls do,,n I 0 tho nddlt ion or the follo\VIng numbers 1 +3+(3 x3) +(3 x3 x3) +(3 x 3 x3 x3), ctr Perltnps there IS an easier \Va)' of computing tl1rs number, l1kc the one we used before (1 +2+4+8,

etc )' There 1s, If \VC talc Into account the following peculiarity o[ t be nt1mbers wo arc ndd1ng.

1=1 3=1 x2T1 9={1 +3) x2+1 27=(1 +3+9) x2+1 81 =(1 +3+9+27) x2+1.

rlg. 37' llow runtour sprc·llls

clc.

In other 'vords, cacl1 number ts cqunl to double tho totnl o( tbc prc~cd1ng plus 1. lienee, to find tl1o sum ot all our numbars, from 1 to nny number, it is enough to ndd to this Jnqt number balf o£ 1tsclf (minus 1). For

tnstnnco 1 the sum totnl of 1 +3 +9+27 +B 1..J...2q3+ 729 equals 729+balf of 728. i.e . , 729+3fil.t=l,093 , Ill

In our cnsct cnch resident. pn..c;s~s the ncl\~ to on1y lbrcc otbcrs . Dut if tho resident~ of Lhc l0"\\'11 ''ere more tnll.nti\ c nnd sl1nr~d tt not v."itlJ iO

thre e, but wit h five or eve n ten , the rum our "?"auld spr ead muc h fast er . In the case of five, the pic ture wou ld bo as follows: At 8 a.m . the nc"~s is know n to . • • • .. • By 8.15 a. m. • . • • .. • 1+ 5 By 8.30 n . m . • .. .. .. • 6+( 5 x5) By 8 . 45 a.m . • .. • 31+ (25 x5) • By 9 a.m . • ' 156 + (125 x5) • • • By 9.15 a.m . .. 781 +(62 5 x5) • By 9.30 am . • 3,90 6(3, 125 x5) .. • • ~

•

• •

• • •

I

. 1 pers on 6 pers ons 31 ,, i56 ,..

781

3,906 19,531

.,, ,.,

"'

In sho rtt. it wou ld be kno wn to eve ry one of the 50,000 resi den ts before 9 . 45 n. m. It wou ld spr ead a lot fast er if eac h man sha red the nc"'~ \Vit h ten othe rs . Her e we wou ld get thes e ver y fast -gro win g num bers : At 8 a.m . the ne,vs would be know n to · · · · .. · 1 pers on 1 +10 11 pers ons By 8.15 a.m . . . . 11 + 100 By 8.30 a.m . . . . '~ 111 111 + 1,000 , 1,111 By 8.45 a.m. . . . 1,11 1+1 0,00 0 ,, 11,111 By 0 a.m . . .

The nex t num ber is evidently· 111,111, and tha t sho\vS tha t the wbo1e tow n l\'ou ld hal"e hea rd the news sbo rtly afte r 9 a . m. The ne~"S, in this case , '' ould hav e tak en a litt le over an hou r to spre ad thro ugh out the tow n. 51. 1'1IE BICYCLE S\V IND LE .-In pre -rev olu tion ary Rus sia there were fi.Irus whi ch reso rted to an inge niou s wa)~ of disp osin g ot nveragcqua 1ity goo ds.. Tbe who le thin g wou ld beg in \Vit h an ad som ethi ng like the foll owi ng in pop ular new spa pers and mag azin es: 1

A BICYCLE FOR 10 RUBLES! •

You can gef a bicy cle for onf y 10 rubl es Take adv anf age of this rare cha nce 10 RUBLES INSTEAD OF 50

CONDITIONS SUPPLIED FREE ON APP LIC ATI ON

71

There were many, of course, \vho fell for the bait and \\rote for the cond1t1ons In return they would rece1ve a detailed catalogue \Vhat the person got for his 10 rubles was not a bicycle but four coupons whtch he "as told to sell to h1s frtends at 10 rubles each The 40 rubles he thus collected be remttted to the company wh1ch then sent hun the bicycle And so, tho man reall) pa1d only 10 rubles The other 40 came from the poclets of his friends True, apart from pa) 1ng these 10 rubles, the purchaser had to go through qu1te a bit of trouble nnd1ng people 'vho 'vould bu) the other four coupons but then that d1d not cost htm anything lVhat \Vere these coupons? \\7JJ.at advantages dtd the purchaser get for h1s 10 rubles? He bought himself the right of exchangtng this cou pon for five simJlar coupons 1n other v. ords he paid for the opportunttj of colle.ct1ng 50 rubles to purchase a btcycle 'Ahtch 1n realtty, cost him only 10 rubles, the SlJm he pa1d for the coupon The new pos sessors of the coupons, In thetr turn, rece1' ed .five coupons each for further distribution, etc At the first glance there was nothing fraudulent In the whole affair The adverttser kept h1s promise the bicycle really cost 1ts purchaser only 10 rubles Nor was the nrm los1ng any mone) -1t got tl1e full price for Its goods And yet, the thtng \Vas an obv1ous swindle For this avalanche," as 1t 'vas called In Russia, caused losses to a great many people ''ho were unable to sell the coupons the) had purchased. It was tbe~t. peoplo who pa1d the firm the difference Sooner or later, there came a moment when coupon holders found 1t Impossible to dtspose of the coupons That this was bound to happen you can see 1f you arm yourself '' 1th a penc1l and a sheet of paper and calculate ho\v fast the number of coupon holders 1ncrea~ed The first group of purchasers, receivtng their coupons direct [rom the firm, usually had no d1lficulty 1n finding atber buyers Each member of tillS group drew four ne" partiCipants 1nto tho deal The latter had to d1spose of thetr coupons to 20 others (4 x5) and to do that they had to convince them of the advantages of the purchase Let us suppose that they "ere successful and that another 20 new part ICI pants v.. ere recruited 72

The avalanche gathered momentum, and the 20 ne\v holders of coupons had to distribute them among 20 x5=100 others So far each of the origtnnl holders had dra,vn 1+4+20+100=125 others 1nto the game, and of these 25 received h1cycles and the other 100 were grven the hope of getting one-a 11ope for '\\htch they patd 10 rubles eacl1 The avalanche now smashed out of n narrow circle of fr1cnds and spread throughout the to"ll v..here, ho" ever, 1t became Increnstngly hard to find new customers The last 100 purchasers bad to sell their coupons to 500 new VICtims \Vho, 1n their turn, had to rccrutt another 2,500 The tov..11 was be1ng flooded with coupons and It was bccom1ng a d1lficult thing Indeed to find people \V1ll1ng to buy them You will see that the number of the people dra" n Into the 'bargain'" Increases along rum our spreading lines (see ahovcj Here Is the pyr-

amid of numbers

\Ve

get 1

4 20

100 500

21500 12,500 G2t500 If the town 1s b1g and tl10 number of hie) cle-r1d1ng people IS L2,500t then tbe avalanche siiouid peter out In the Etlt round By that timeevery person w1ll have been dra\\"'11 Into the scheme But onl) onefirth " 1ll get b 1 cycles, the rest. w1l I be 1n poc::;sec;s 1on of coupons 'vb 1ch they have no earthly chance to dtsposo of In a t 0 ,,"11 w1th a h1gger population, even 1n a modern cap1tal "1th mllllons of people, tbe end comes only a few rounds later, because the pj ramtd o£ numbers gro\\S with Incredible speed Here are tbo figures from the ntnlh round up 312,500 1,562,500 7,812t500

39)002,500 73

In the 12th -round, as you see, the scheme \Vlll have 1nve1gled the population of a whole country t and 4/5 w1ll have been swindled by the perpetrators of the fraud Let us see \vhat they gatn They compel four ftfths of the population to pay for the goods bought by the rema1n1ng fifth,. 1 e , the former become the benefactors o£ the latter Moreover they get a whole army of volunteer salesmen-and zealous salesmen at that tt Russian wr1t er JUstly called the affatr "the avalanche of mutual fraud " And all that can be said of the th1ng IS that people,. who do not know how to calculate to guard themselves against frauds, are usually the ones who suffer 52. RE\VARD. Here IS what legend says~ happened tn anc1ent Rome.*

I The Roman general Terenttus returned home from a VICtOrious campaign w1th tropbtes and asked for an audience \Vlth the emperor The latter rece1 ved htm very ktndly J thanked htm for what he had done for the emp1re and promised h1m a place In the Senate that "ould befit hts dign1ty But that was not the reward Terent1us wanted "I have won many a victory to enhance thy m1gbt and glorify thY name,'' he sa1d ''I have not been afraid of death, nnd had I more l1ves than one I would have w1liingly sacrificed them for thee But I ron tired of fighting I am no longer young and tho blood 1n my ve1ns lS no longer hot It ts time I ret1red to the home of my ancestors and enJoyed life ,, "What wouldst thou ltl.C then, Tcront1us' u the emporor nslt..ed "1 pray thy IJldulgence, 0 Caesar! I have boen a \Varrior almost all my l1fe, I have sta1ncd my sword \Vlth blood, but I have had no ttmc to build up a fortune I am a poor man • - ' "Continue, brave Tercnt1n~." the en1peror Interrupted b1m. • ThlS 15 a hbcrai translation from a Lnt1n manuscript in the keeping of a pr1 va tc library 1n Britain

74

'If thou "'ouldst reward th) servant," the encourage d general went. on, ''lot thy generosity be1 p me to l1ve my lnst da3 s 1n peace and plenty I do not. seel. honours or a h1gh position tn the almtghty Senate I should ltke to retire from power and soctcty to rest Jn peace 0 Caesar, g1vo me enough money to l1ve the rest of my da)s 1n comfort '' The emperor the legend says, was not a generous man He was a m1sert 1n fact, and 1L hurt htm to part with money He thought for a moment before answer1ng the general "\Vhat IS the sum thou wouldst. consider adequate )' he finally asked "A mill ton d enar11, 0 Cae~ar ' The emperor aga1n fell stlent The general watted, h1s head lo" Thou art a great 'Valtant Terent1us t' the emperor satd nt last general and thy glorious deeds tndecd deservo a worthy rc"' ard I qbalJ g1ve thee rtches Thou wtlt hear my deciSion at noon tomorro" Tcrentius bowed and loft II

The next day Terenttus returned to tho palace "Ha1l,. 0 brave Terent1us '., the emperor satd The general bowed reverently "I have come, 0 Caesar to hear thy deciSIOn Thou hast gracious]\ promised to reward me ' "Yes,' the emperor answered, 'I would not \\llnt a noble wnrr1or ltke thee to recetve a niggardly reward Harl to me In m) treasun there are 5 mtll1on brass co1ns v.. orth a mlll1on denar11 NO\\ l1stcn caretully Thou w1lt go to my treasur~. talc one co1n nnd br1ng 1t here On the next day thou n 1lt go to the treasury aga1n and talc nnotltor coin \vorth t~ 1cc the fi~t nnd pi nee 1t bestdo the first On tile third day thou "'1lL get a cotn ~ orth four ttmcs the first., on the fourth dny c1gh t tJm cs, on tbc filth s1xtecn ttmcs, and so on I shall order to ha\e COlllS or tbc rcqutrcd value minted for tbco C\Cry dn~ And so long ns thou hnst tlte strength, thou mn)st take tho cotns out of my treasury But thou must do 1t thyself, w1thout nn~ holp And "hen thou canst no longer I 1ft the co1n, stop Our agreement "1ll hnvo ended then, but nil thccotns thou v.1lt hnvctnt..c n out wtll be tb~ rc\\nrd ,. 75

Terent1us listened greedtly to the emperor In his 1mag1natton be sa\v the huge number of co1ns he would take out of the treasury 'I am thankful for thy generosity, 0 Caesar," he answered bapptly '"Thy reward 1s '' onderful Indeed' ,,

l

1

}I

.. Fl'j

38 The £rst COin

•

F1g 39 The seventh F1g 40 The n1nth COlU

COin

III

.c\nd so Tcrcnt1us began his dati) ptigrunng~s to tbc treasury ncar the emperor's audtcncc hnll, and It "as not dJlficult to hr1ng the first rotns there On tlu~ first day Tercnt1us tool. a ~mnll co1n I hat ·''as 21 mm tn \nd lo try nll of them ll \\auld take almost 10,000 \ears-as you .. '' 1ll sec Pcrhnp~ ) ou mn\ not 1Jc1tc\ e thnt thero arc so many ways lor ten persons to stt nround a table' To mnh.e 1t ns s1mplc as posstble, lot, us .stnrt n Jt h t !Jrc~ obJects that uc shall cal} A. Bj and C 1 find ts ho\\ many \Vays thcro arc of rcnrrang \ \ hnt \\C \\nnt to 1ng these ObJeCtS rtrst, let US pul (, nstde nnd do It With JUSt two -obJects \Vc \VLil see that thcr{\ ar~ oniJ t '\o \Vn) s of rearrangin g jhcm

} lft 52 Only

l\VO WD)S

or rlarranglng two

th1ngs

No\v let us add C to each of thcso pau~ 'Vo cnn do ent ways 1 \Vc can put C belund the patr,

1t 111

three dtffcr-

2 Before the pn1r, and 3 Bellt,een tho two obJects Thoro nrc ev1dcntly no other \Vays of placing 1t And SJnce ''e have J.wo pn1rs, AB and BA \VC have 2 x3=G ways of rcarrang1ng the obJects Tho '\ ays these obJects are rcartange d arc sho\vn 1n fig 53 Let us procced- wtth four obJects .tl, B C, and D For the time bctng we shall put D as1de and do all tha rearrangem ents with three . 'l.ta'lt\. B ~ ODJCC\ s \v 0 a'\ rca ny h.ntJ\\ \nu\. ~l>'n"C~t: 711~ '(.. *«C..J~ "0~ ~-o. . ~-g many -,.vays are tl1cro to add the fourth obJeCt (D) to enfh of the 6 arrange ments of the other three obJects~ Let us see Wo t..an 1 Put D behlnd the three obJects 2 Before them, 3 Between t1lc first and second ob;ec.ts, and 4 Bet1~;een Lho second and third obJects 91

Therefore, we have

0 x4=24 arrangements, ~

and stnce 6 ==2 x 3 and 2= 1 X 2. then the number of all the arrangements may be wrttten as follows ~

1. x2x3x4~24

----

-

•A

8

A

c

6

c

B

8

c

A

A

(}

c

A

8

A

...

Fig. 53~ Three thu:--gs. can be n.r-range d in s 1.X wn ys ~ ,_.J

Now 1£ we apply the same method Wllh five objects, we get the following·

1 x2 x3x4x5=120 And for six: •

1 x2 x3 x4 x5x6=720t etc .

Let us now return to the l~n ~ oung men . The number oi poss1b1o arrangements in this case-Jf we talc the trouble to cnlculotc 1t-"ill be as follows .

tx2x3x4x5x6x7x8x9x10 92

'fho rcsul~ ,vtJl bo the number ne mcntloned above

3.628,800 CaJculatJon \Vould ltavc been far more complicated ti half of tho ) oung people '' cro g1rls, and they \vould \vant to s1t With each young man 1n turn Although the number of arrangements 1n thss ease would ho much smn11cr, 1t \\'Ou]d bo harder to compute It Let one j oung mnn s1t do'Wll. \\hcrevcr be \Vants at the table The other four 1 lcnving empty chntrs between them for the g1rls, can stt do\vn 1n 1 x2 X3 x4=24 d1ffcrcnt ways Stncc there aro 10 cltairs, the first young mnn can s1t down tn 10 different ways1 therefore, there arc 10 X 24 =240 different ~'nys In \\htch tho young men can occupy thc1r scats nround tho table Ilo\\ 1n1.ny wn)S nro thoro tn whtch tho five girls can occupy the empty seats between the young men'> Obviously 1 x2 x3 x4 x5=120 "nys Combining each of the 240 postttons or each young man \Vlth each or Lhc 120 posttions of cnch gtrl. \Vt.. come to the number of possible arrnngemcnts" whtcb IS . 240 X 120 =28,800 Tins, of course, 1s very much less than the 3 628,800 arrangements Cor the young men and would take slightly Jess than 79 years And that • means that the young peoplo \VOUld get 8 free dtnner from the heir of the \Vnltcr, 1f not from tbc 'vatter himself, b) the time they were about 100 years oid-prov1ded they ltvcd that long Now thnt wo havo learned ho\v to calculate the number of ar.range

or

mcnts, we can determine the number of combinations blocks 1n a uF1ftecn Puzzle * box In other words, we can compute the number of problems th1s game ean set a player It IS easy to see that the task 1s to detormtne the total number tn \Vhlcb t.he blocks can he rearranged To do thatt we know, we must effect tiie following mufttpfu~atton

1 x2 x3 x4 x5 x6 x7 x8 x9 x 10 x 11 x 12 x 13 x 14 x15 Tho answer

JS

1,307,674,365,000 • Tho square In the lower right hand corner must. al"\\ays rema1n vacant

93

Half of th1s hugo number of problems are Insoluble There are, therefore,. more than 600,000,000,000 problems for \Vllich there are no solut1ons The fact that people never e\cn suspected that e"t:plains the craze for tl1e 'F1£teen Puzzle " Let us also nato that 1£ 1t \Vere posstblc to sh1fl one blocl every sec ondJ. 1t would take more than 40~000 years to trJ all the possible com btnations, and that 1f one sat at It uninterruptedly As we come to the close of our talk about arrangements~ let us solve a problem r1ght out of school l1fc Let us suppose thera are 25 pupils 1n a class Ho\v many 'vays are

there to sent them? Those who have well understood tl1c problems we explained above '' •ll not find any d1fficul ty In sol v1ng th1s one All we ha\ e to do IS to multtply the 25 numbers, lhus 1X 2X 3 X 4 X 5X G X 23 X 24 X 2~ Matbemat1cs shows many ways of simplifying varJous operatJons, hut there 1s none for the one mentioned above The only 'vay to do 1L correctly ts to multiply all these U)Jmbers And tl1e only tb1ng that will save t1me IS an appropriate arrangement of mult1pl1ers The result. IS stupendous-It runs Into 26 d1g1ts-so stupendous that 1t IS beyond • our power of Imagination Here 1t 1s 15,511 ,210,043.330t985,984,ooo Of all the numbers we have encountered so far th1s one, of course, IS the b1ggest and therefore takes the palm as the number gtant Compared with It, the number of drops In all oceans and seas 1s qu1te modest 56. A TRICI{ 'VITH COINS In my boyhood my brother, I reca1I. shoVved me an 1nterest1ng game \Vlth co1ns F1rst he put three saucers 1n a line and then placed five co1ns of different denomination {one ruble co1n, 50 1.opek co1n, 20 kopek co1n, 15 kopel co1n and 10 lopek coin* 1n the first saucer, one atop the other In the order gr' en The task was to transpose theso co1ns to the thtrd saucer~ observtng tht. following three rules

tooo

• The game can be played With any five co1ns of dulerent s1ze

94

1) It IS permit ted to transpo se only one cotn at a tune 2) It 15 not permit ted to pJ1cc a b1gber coin on a smatJer one and 3) It 1s pe:nm ttcd to use tho middle saucer temporanly observ mg tho first t\vO rul~s but In the end the co1ns must be 1n the thlrd saucer nnd tn tho origtna l order

(_" .L ·---I .

---......

F1g 54 1-.Iy brother sho\ved me an 1nte res ttng game

The rules my brothe r sa1d are qutte s1mplc as you see Now get to 1t I took the 10 lopel co1n and put tt Into the th1rd saucer then I placed the 15 lopek Into the middle saucer And then I got stuck \Vbera was I to put the 20 kopek eo1n' It was h•ggcr than both' \Veil' my brothe r came to my ass1stnnce Put the 10 l.opel. co1n on top of the 15 lopels Then you w1ll have tl1e thrrd saucer for the 20 J...opol_ COlD I d1d that But Jt d1d not. mean the end of my dJf.ficultJes 'Vhere to put the 50 kope1.. co1n? I soon saw the way out I put the 10 lopek coin Into tlte first saucer the 15 lapel co1n Into the thtrd and then transpo sed the 10 lopek cotn there too Now I could place the EO lopel cotn tn the second saucer Then after numerous transpo sitions I 95

succeeded 1n moving the ruble cotn from the first saucer and eventually had all tho p1le 1n the thtrd "Well, how mnn) moves d1d you make altogether?' • my brother askedt praising me for tho way I had solved the problem "Don't know I dtdn't count • 'All right, lets count It would be 1nterest1ng to len ow how to got 1t done wtth the least poss1ble number of moves Lets sup pose we had only two cotns-15 and 10 kopek and not five How many moves would you rcqutro then?t1 Three-the 10 kopek coin would go tnto the mtddle saucer, the 15 kopek co1n tnto the third and then the 10 kopeks over 1t Correct Lets add another co1n-the 20 kopel and see ho\v many moves we need to transpose the p1le F1rst we move the two smaller co1ns to the m1ddle saucer To do that as we know It, we need three moves Then we move the 20 kopek co1n to the thtrd saucer That,'s another move Then we move the two coins from the second saucer t to the th1rd and that s another three moves Therefore, we have to do 3 +1 +3=7 moves n "Let me calculate the number o( moves \Ve would requ1re lor four co1ns, ~ I Interrupted htm "F1rst, I move the three smaller coins to the m1ddle saucer That's seven moves Then I transpose the 50 kopek coin to the thtrd saucer That's another move And finally the three smaller coins to the th1rd saucer and that's another seven moves Altogether It will be 7 +1 +7=15 moves ' Excellent And what about five coins?,. Easy 15+1 +15=31,' I answered promptly 'Well, I see you•ve caught on But I II show you a still easter way of dotng 1t Take the numbers wa have obtained 3, 7, 15 and 31 All of them represent 2 mnlttplted by Itself once or severa] ttmes mtnus 1 1

Look . ,. And my brother wrote down the following table

3::::2x2-1

7=2x2x2 1 15:::::2 X 2 X 2 X 2-1 31=2 x2 x2x2 x2-1 96

"I see 1t no\v \Vo multiply 2 by 1tself as many ttmcs a~ there are co1ns to be transposed and then subtract 1 Now I know how to calculate tl1e number of mo\ es for any p1le of cotns For 1nstance 1 tf we have seven co1ns, the operation w1ll lool.. as follo\VS

2 x2 x2 x2 x2 x2 x2-1 =128-1=127 ,, "\Vell, now you l.now th1s anc1ent game," my brother sa1d "There's only one other rule that you should b~ar tn m1nd If the number of co1ns ts odd, then you put the first Into the third saucer, 1f 1t s even, you start Wtlh the second saucer " ~~Is the gnme really ancient? I thought It \vas your own' n I exclaimed "No, all I dtd \\as to modernize 1t \Vlth coins The game's '\erj) very old and comes from Jndta, I th1nk There's a very 1nterest1ng legend connected WtLh It In Benaros there's a temple and 1t 1s satd that \\hen Brabma created the world he put up three diamond sticks Lhere and around one of them be placed 64 gold r1ngs, WJth tho h1ggcst at the bottom and the smallest. on top The priests had to work day and night Without a stop, transposJng the r1ngs from one st1cl to another,. us1ng the third as an a1d-the rules were the same as 1n the case of co1ns they were allo\vcd to transpose only one r1ng at a time and forbidden to place a h1gger ring on top of a smJller one \Vhen all the rtngs are transposed, the legend says, the world wtll come to an end n .. uThen the world should have perished long ago, 1f one 1s to bel1eve th1s legend " "You thtnk the transpositiOn of these 64 r1ngs doc'3n 't ta.le much t1me, do you? , 'OI course, 1t doesn't Lets say 1t tales a second for each Jnove. That means tn an hour one can male a.GOO transpositions " • \Vcll?, 'lThnt II he about 100,000 a day and cbout 1.000t000 tn ten days, and I m suro ) au can transpose all of 1,000 rtngs w1th 1,000,000 moves ,. "You're wrong there To transpose these 64 rtngs you'll requtrc neither moro nor less than 500.000 mJll1on years',. J 7-1944

97

$

to 2: 'Va1t, I 11 tell

uBut why? The total number of transpostttons Will be equal

multtpl1ed by Itself 04 ttmes mtnus 1, that 15, to you tho result tn a second " uF1nc And whtlc you're do1ng all th•s multiplication JOb I II have enough ttme to attend to some bustness '

1ft

(,

F1g. 55 The prtcsts worked dal and ntgbt

My brother left and I busted myself 'vtth calculation First I found 18

the value of 2 and then multiplied the result -65t536-hy Itself and then the result agaxn by tlself and subLracted 1 'Vhat I got after that, was

18,446.744,073' 709,551 '615*

1\iy brother was r1ght, after all Incidentally, you mtght he Interested 1n learning how old our Earth 1s 'Veil, scientists have worked that out-though only approximately .. The Sun has existed .. The Earth •

Life on Earth Human be,ngs

•

•

10 000 000,000 000 years 2 000 000 000 )I 300 000 000 t 300 000 "

• We know thts figure 1t was the number of gratns Sessa asked as a reward for lnventmg chess 98

57. A BET.-¥le\\ere ha,tnglunchntourhollda) home when tho talk turned to dctcrminntion of the prababtllly of a cotncidence One of the fello"s, n young matbcmatJcinn, took out a co1n and sa1d • Look, I II toss th1s coin on the table wtthout lool.•ng What's the probability of a head-up turn?, nYou,d better cxplaln what probabtltty' IS,,, the rest chorussed ..

"Not cvcrvon o h.n o"'" s what 1 t 1s " .... uQh, that's Slmp)e '}here arc only tWO pOSSible ways

0

F1g 56 Head or tail

e

lTI

Which a

e

F1g 57 A dte

may fall (fig 56) Cither head or tall Qf these on]y one WJ]l be a fa\ourable occurrence Thus we come to the following relat1on: COlD

The number of favourable occurrences The number nf po~c,Jble occurrences

'The turn

l

fraction ...o- represents

the

1

== 2

probabtlJty

of

a

head up

n

"It's simple w1th a corn someone Interrupted ~Do It wrth something more compltcatcd-a dte, for Instance ' "All r1ght,,. the mathematician agreed .cLet's take a dte It's cubtcal 10 shape, with numbers on each of 1Ls faces (Iig 57) Now, what s the probability, say, of the number 6 turn1ng up? How many posstb1e occurrences arc thereJ There arc Sl"\: faces and,. therefore, any of the numbers from 1 to 6 can turn up } or us, the favourable occurrence "'til he \vhcn 1t JS 6 The probalnl1ty In thts case \VIII he ~ ~ "Is 1t really possible to compute the probability of any event?" one of the gtrls asled 'Tal\_e this, for Instance I've a hunch that the first person to pass our Window wtll he a man \'7bat,s the probahtl1ty that my hunch 1s correct?" 7• 99

1

'The probab1l1ty 1s 2' 1! we agroo to regard even a year old baby hoy o.s a man There. s an equal n urn b cr of men and "omen on our earth '' •And wl1nt S the ptobnbtll Ly that tho first tu o persons w1JI be men?" another nskcd ccHcre computatton wtll be more complicated Lets try all tho posstble comblnat1ons First, 1t s posstblc that they \VIII be men Second. lhe first may bo a man and tho second a woman Th1rd. 1t may be the other way round first tho "oman and then tho man And fourtlJly, both of them may bo \VOmen So the number of possible combtnaltons IS 4 And of these combinations only one IS favourable1

the .first ThUB the probabdtty Js! That•s tho soJutton of your problem ,. That's clear, hut then \VC could have a problem of tltree men 'Vhat s the probabiilty tn tbts case that tbc fi~t three to pass our w1ndow wtll bomcn c \Veil 've can calculate tbnt too Let's start WJtb computing the number ot posstble combtnattons For two passers by the num iber of combtnattons, as \VC have seen, lS 4 By addtng a th1rd passer by we double the number of posstble combtna.tlons because each of those 4 groups o[ l\\O passers by can he JOined Clll1cr by a man or a woman Therefore, the numbor of posstble comblnattons 10 1

thts caso w1ll bo 4 x2=8 The obvious probabtltty wtll be B' Since only one combination Will bo the one we "ant Its easy to remember tho method of computing the probabilities tn the case a[

two passers by the probabtltty 1

1

I

2 x2=s,

JS

I 1 I l 2 X2=4, for three lt 1s 2 X

for four the probabtlity '' 1ll be equal to the product of

4

halves, etc The probability as you may see, grows less each tune " "Then what w1ll 1t be for 10 passers by;J ~1'acc metta ~rrwat ts the (Jrauao~lltJ~ t}J.'&t lhe flrst ten passers OJ' are men' For that we have to find the product of 10 halves That Will 11

1

That means tf you bet a ruble that tt will happen, I can

be 10L4 wager 1 ,OCfJ rubles that 1t w1ll not.' • The bet IS tempting' • one of those present exclaimed 'I m more than w1lltng to put up a ruble to w1n a thousand ' 100

•

~