famous puzzles of great mathematicians

7/27/2019 Famous Puzzles of Great Mathematicians

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-f Great Mathematicians

Miodrag S. Petkovic



FAMOUS PUZZLES

ofGreat Mathematicians

Miodrag S. Petkovic

@AMSAMERICAN MATHEMATICAL S O C IE T Y

Pro\"ide:nce:, Rhode: Island



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FnnHJu:oI pllzzlHIi of gwat mnthmnaticit1nli I Miodmg S. Pntkovio!:.

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luciudL's bibliographical rC'ferencL's and index.

ISBN 978-C)..8218-·18U-2 (alk. paper)

1. l\1ntholllatical f ( ~ r o a t i o n s . 2. MnthoITlAtics···Popuhu works. 1. Titlo.

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To my family



CONTENTS

Preface

Art and Photo Credits

1. RECREATIONAL MATHEMATICS

2. ARITHMETICS

DiuphanllUi' ug(! (Dio/Jlw,ntut/)

Number of arrows (A'lahiitrim)

How many rabbit.s? (Fiborwcci. p. JJ)

Fibunacci s S ( ~ q u c I l C ( ~Chessboard parndox

Square 1l1lmbcl'8 problem (Fibonacc'i)

Moncy in It. pile (Fibollacci)

Magic configurntioll (Hui)

Triangle with integral sidt!s (Bachct.)

\¥cighlS problem (Fibo'IIacC'i. Tartaglia. Bachet)

MllcMahon ,s gencralllppronch

Division oC 17 hors(!S (TartU!llia. p. 2.'1)

\Vinc and w l l t ( ~ r (Tartaglia)

CUiJL"'t in hands (Reco1'tle)

Sides or two cubes ( i l ~ t e . p. 25)

Animnls on a field (Ncwton)

Polya's solution

Dorrie's variant

Olher variantsv

xiii

xvii

9

10

11

12

13

14

15

16

17

17

20

23

2·125

25

26

27

28

28

29



vi CONTENTS

Gathering an army (Alcuin of York)

Answers to Problems

3. NUMBER THEORY

Cattle problem (A rchimedes, p. 38)

Dividing the square (Dioplaantu.s)

Wine problem (Diophantus)

Amicable numbers (ilm Qorra)

Qorra'8 formula

Euler's rule

How many soldiers? (Bhaskara)

Horses and bulls - a Diophantine equation (Euler)

The sailors, the coconuts, and the monkey (P. Dime)

Dirac's solution

General solution

Unknown a d d r ~ (Ramanujan)

Stamp combinations (Frobeni'Us, Sylvester, p. 60)

Generalized problem

Answers to Problems

4. GEOMETRY

Arbelos problem (Archimedes)

Archimedean circles

Perpendicular distance

Two touching circles

~ H n i m a l distance (Heron)

A fly and a drop of honey

Peninsula problem

The same distance of traversed paths (Bmhmagupta)

Broken bamboo (Brahmagupta)

Height of a suspended string (klah«vira)

The diameter of the material sphere (ibn Qonu)

Dissection of three squares (Abu'l- Wafa)

Dudeney's six-cut dissection

29

30

37

41

44

45

45

47

47

48

50

52

56

56

57

61

62

64

67

68

69

70

73

73

75

76

77

78

78

80

81

82



CONTENTS

Dissection of four trianglt-s (Abu '1- Wafa)

Billiard problem (AU,.azen)

Distance of the optimal viewpoint (Regiomontallus)Rugby football I)roblem

Saturn problem

The minimal sum of distances in a triangle

( S t e i f l c r ~ Fcnnat t Torricelli, Cavaliet-i)

Volumes of cylinders and spheres (Kepler, p. 93)

Dido's problem (Ste'ine1", p. 95)

Di\rision of space by p l a n ~ (Steiner)

Division of plane

Number of hounded regions

Road s)'stem ill a S<llUlfC (Steiner-)

Kissing circles (Soddy, Deticllrtes . Kowa)

Kazayuki's problemF i v ( ~ kissing s p h e r ~n-dimensional kissing spheres

T h ( ~ shortesl b i S L ~ t i t l g arc of area (Polya)

Answers to Problems

5. TILING AND PACKING

r..·josaics (Kel'ler)

Escher's mosaics

NOllperiudic tiling (Pellrose. Conway)

Penrose's kite-dart t.iling

Conway's p i n w l l ( ~ ~ l tiling

l\·laximulDa r t ~ a

by pentaminoes (Klntth)Pentamillo tilings

Squaring t.he S<luarc

Kissing spheres (D. Gregory, Newton)

The densest. s J > h c r t ~ packing (Keple,., Gauss)

Kepler's conjecture

Circle packing

Hales' solul.ioll

vii

83

83

8688

89

91

94

96

98

99

101

101

104

108109

110

110

112

119

121

124

126

126

128

129131

132

134

137

137

138

141



viii CONTENTS

Cube-packing puzzles (Coll'way) 142

Answers to Problems 144

6. PHYSICS 151

The gold crown of King Hi("ro (Archimedes) 152

The I<mgth uf traveled t.rip ( O , · e . ~ m c ) 153

~ I e e l . i n g of shillS (Lucas. p. 155) 156

A girl and a bil'd ( \ l01l NC·lI.marl1l. I'. 157) 158

Von Neumann's ( ~ n s y S l ~ r i e s 158

l\Iure difficult. p r ( ) h l ( ~ m s 160

The lion and the man ( L U t . l ( ~ u : o o d . Besico'uitc/a, R. Rado) 161

Answers to Problems 166

T. COMBINATORICS 171

Combination with flavors (AJahiiviro) 172

Married couples c r ( ) s . ~ the river (Bachet) 173

Four married c o n p l l ~ cross tbe river 175

.Josephus I>roblem (Bachet. an.d otller's) 176

Dudeney's version 177

Japanl'SC version 177

Graham-Knuth-Patasnik's v ( ~ r s i u n 178

Rings puzzle (Cardano, p. 18lJ) 181G r a ~ r - c o d e numbers 182

The problem of the misaddre.sSt.>(lletters (N. (/I) Dcnl.oulli. E'ulcr) 184

Eulerian squares (Euler') 186

Lat.in squares 187

Gracco- Latin squares 187

Euler's officers problem 188

Euler's conjecture 188

Parker's square of order 10 189

Kirkman's schoolgirls problem

(Kirkman.. Steiner. Sylvester. Cayley) 189

Steiner tri pIe system 190

General problem 192

Sylvester's problmn 192



C O ~ T E N T S ix

Counting prublem ( C ( l . ' l l ( ~ ! I ) 19:1

Rn<:c.os with t.it,S 19-1

Thl' Towel' uf Hanui (Lunl,'j) 196

D(,IUU'('S l('>llIpl(> 19X

Iliterdianging thf' d t e c k ( ~ r s (I) (Lur.w,) 199

Int,l'rclinngillg lhe dwck('I1; (II) (LIlCCl,'i) 20()

Shullt.ing pl'ohl(,11l (/ . ucfI.4j) 200

Pruhlt'1II uf marriNI cuuples (problellll' dC!i mellages) (LIICU,li) 201

TIl(' t.ree plcllltillg prohlc.'1lI (Syi-lJfslcr) 202D u d e n e ~ " s militnry puzzle 203

Dudency's fuul' pluJlting problem 204

DHfl'relll dist.anct-s (E,'dos, p, 20,,) 2U6

A l1swers to Problems 206

8. PROBABILITY 209

The prohlem of til(' point.s ( f ( ~ r ' m a t , PaM:al, I'. 21 I) 212

Paseal-Fl'I'nult's general prohlem 213

C a l l 1 b l i l l ~ gallic \villa dkl' (HuYfJf'Il,'1) 215

Gamhl('r"s ruill problem (PfMr.lll, FC"mat, Hl&u,qcru;) 21 i

C('lu'I'al pruhll>11l 218

Thl' P('ler!ihllrg paradox (N. (III) BenuJlllli. D, B ( ~ n l ( m l l i . Cnmw1') 220Derlloulli's moral expect.at.ion

C .. , t·mlller s varian.

The probability prohlel1l with t.he misaddressed lett.ers

(N. (II) 8e1'1wulli. EfllfT)

~ I a t c h h o x p r ( ) h l ( ~ 1 1 1 (BfU1fU:h)

Comhinatorial Sllm

A l1swcrs to Problems

9. GRAPHS

The prohlt!l1l of I(c)lligsherg's hridges (Eliler)

Diugnun- t.racing p1lzzle

Eul('r's thror(,111

Euler's pat.h

22l

222

222

224

225

225

229

2:10

23l

231

231

famous puzzles of great mathematicians

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