Famous Puzzles of Great Mathematicians

345 Pages·2009·18.92 MB·English

by  Miodrag Petković

FAMOUS PUZZLES
of Great Mathematicians
Miodrag S. Petkovic
® AMS
AMB.IU Cd.N M~TI:I£.MA1'JCAL SOCI£T1'
Providence, Rhode Island
2000 MC1tlr.ema:ti"" Subject Cla:11rifimtion, Primary OOA08, 97 A20, 01A05, 01A70, 05A05,
05045, 05C90, ll004, 11009, 51E:l0, 51Ml6, 52015, 52C22, 97040.
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Library of Congress Cataloging-in-Publication Data
Petlowit, Miodrag.
Famous puzzles of great mathematicians / Miodrag S. PetkoviC.
p. em.
[ucJudes bibliographical references and index.
ISBN 978-0-8218-4814-2 (alk. paper)
1. Mathematic:al recreations. 2. Ma.thEJmatics-Populnr works. I. T itle.
QA95.P4358 2009
5l0-dc22 20090l'l018
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CONTENTS
Preface
Xlll
Art and Photo Credits xyu
1. RECREATIONAL MATHEMATICS "I
2. ABTTHMETICS 9
Diophantus' age (Diophantus) 10
Number of anows (1\fahiivim) 11
How many rabbits? (Fibonacci, p. 11) 12
Fibonacci's sequence 13
Chessboard paradox 14
Square numbers problem (Fibonacci) 15
Money in a pile (Fibonacci) 16
Magic configuration ( Hui) 17
Triangle with integral sides (Bachet) 17
';>,'eights problem (Fibonacci, TM·tagl·ia., .Bachet) 20
MacMahon's general a.pproacb 23
Division of 17 horses ( Ta1·taqlia, p. 23) 24
Wine and water ( Ta·1·taqlia) 25
Coins in hands (Reconie) 25
Sides of two cubes ( Viete, p. 25) 26
Animals on a. field (Newton) 27
P6lya's solution 28
Dijrrie's variant ?8
Other va.rjants ?.9
v
vi CONTENT S
Gathering an anny (AlC1Ain o.f Y01·k) 29
Answers to P roblems 30
3. N UMBER THEORY 37
Cattle problem (A1-chimedes, p . .18) 41
Dividing tile square (Diophcmt1!S) 44
Wine problem (Diophantus) 45
Amicable numbers (ibn Qo1•ra) 45
Qorra's formula 47
Euler's rule 47
How many soldiers? (Bhaskam) t18
Horses aod bulls - a Diopbaotine equation ( E'(tler-) 50
The sailors, the coconuts, and the monkey ( P. Dime) 52
Dirac's solution
General solution ,56
Unknown address (Ra.ma:nu;jan) 57
Stamp combinations (F1-obenius, Sylveste1·, p. 60) 61
Generalized problem 62
Answers to P roblems 64
4. GEOMETRY 1)7
Arbelos problem (A1-chimedes) 68
69
A rch jJnf::dea.u cjrch~:o:
Perpendicular distance 70
Two touching circles 73
Minimal distance (Heron)
A fly and a, drop of honey 75
Peninsula problem 76
The same distance of traversed paths (Brahmaqupta) 77
Broken bamboo (B?-a,hmaguptct) 78
Heigbt of a suspended string (Ma.hiivim) 78
The diameter of the ma.terial sphere ('ibn Qor1"a) 80
Dissection of three squares (Abu'l- Wafa) 81
Dudeney's si:x-cut dissection 82
CONTENTS VII
Dissection of four triaJ1gles ( A.bu 'l- Waja.) 83
Billiard problem (Alhazen} 83
Distance of the optimal viewpoix1t (Req·iomontanus) 86
Rugby football problem 88
Saturn problem 89
The minimal sum of clistaJ1ces in a. triangle
( Steine1·, Fe1mat, To!'ricelli, Cavalie1·i) 91
Volumes of c:ylinders and spheres (Kepler, p. 93) 94
Dido's problem (Ste·iner·, p. 95) 96
Division of space by planes (Steine1') 98
Division of pla.ne 99
Number of bounded re2'ions 101
Road system in a. square (Steind) 101
Kissing circles (Soddy, Descartes, K owa) 104
I<azavuki 's problem 108
Five kissing spheres 109
n-dinlensional kissing spheres 110
The shortest bisecting arc of area (Polya) 110
Answers to Problems 112
5. TILING AND PACKING 119
Mosaics ( K eple1·) 121
Escher's mosaics 124
Nonperiodic tiling (Pem-ose, Conway) 126
Pemose's kite-dart tiling 126
Conway's piuwheel tiling 128
Maximum area by pentaminoes (Knuth) 129
Pentamino tilings 131
Squaring the square 132
Kissing spheres (D. Newton) 134
Gn3QO!~lf.
The densest sphere packing (I< eple1·, Gauss) 137
Kepler's conjecture 137
Circle packing 138
HaJes' solution 141
viij CONTENTS
Cube-packing puzzles (Conway) 142
Answers to P roblems 144
6. PHYSICS 151
The gold crown of King Iliero (A1·chimedes) 152
The length of traveled trip ( Oresme) 153
Meeting of ships ( L11cas, p. 155) 156
A girl and a bird ( Von Ne'UmrLnn, p. 157) 158
Von Neumann's easy series 158
i'vlore difficult problems 160
The lion and the man (Littlewood, Besicovitch, R. Rado) 161
Answers to P roblems 166
7. C O M BINAT O RJCS 171
Combination with flavors (Mahavim) 172
Married couples cross the river (Bachet)
Four o1arried couples cross the river 175
Josephus problem (Bachet and othe1·s) 176
Dudeney's version 177
Japanese version 177
Gra.ha.m-Knuth-Patasnik's version 178
Rings puzzle ( Ca1·dano, p. 180) 181
Gray-code numbers 182
The problem of the misaddressed letters (N. {II) Benw1•lli, Eule1·) 184
Eulerian squa.res (Eu.Le1·) 186
Latin sq nares 187
Graeco-La.tin squares 187
Euler's officers problem 188
Euler's conjecture 188
Parker's square of order 10 189
Kirkman's problem
schoolgirl~
(Ki1·kman, Steine1·, Sylveste1·, Cayley) 189
Steiner triple system 190
Genera.! problem 192
Sylvester's problem 192
CONTENTS ix
Counting problem (Cayley) 193
Races with ties 194
The Tower of Hanoi (Lucas) 196
Benares temple 198
Interchanging the checkers (I) (L~1cas) 199
Interchanging the checkers (II) (Ltt.ca.s) 200
Shunting problem (L·ucas) 200
Problem of ma.Hied couples (probleme des menages) (L·ucas) 201
The tree planting problem (Sylveste•t') 202
Dudeney's rnilitaJ·y puz.zle 203
Dudeney's four planting problem 204
Different distances (Er'diis, p. 205) 206
Answers to Problems 206
8 . PROBABII.ITY 209
The problem of the points (Fe1'mat, Pascal, p. 211) 212
P<<Scai-Fermat's genera.) problem 213
Gambling ga.me with dice ( Hu.ygens) 215
Gambler's ruin problem (Pascal, Fer-mat, Huygens) 217
Genera.) problem 218
The Petersbmg paradox (N. (III} Ber·noulli, D. Bernot1lli, Cmme1·) 220
Bernoulli's moral expectation 221
Cramer's va.riaJ1t 222
The probability problem with the misaddressed letters
(N. (II) Bemoulli, Eule1·) 222
Matchbox problem (Banach) 224
Combinatorial sum 225
Answers to Problems 225
9. GRAPHS 229
The problem of Konigsberg's bridges (E!lle't) 230
Diagram-tra.cing puzzle 231
Euler's theorem 231
Euler's pa.th 231