FAMOUS PUZZLES of Great Mathematicians Miodrag S. Petkovic ®AMS AMB .IU Cd.N M~TI:I£.MA1'J C AL SOCI£T1' Providence, Rhode Island 2000 MC1 tlr.ema:ti"" Subject Cla:11rifimtion, Primary OOA08, 97 A20, 01A05 , 01A70, 05A05, 05045, 05C90, ll004, 11009, 51E:l0, 51Ml6, 52015, 52C22, 97040. Por additiona l infon1ut.tion and upd ates o n t his book: visit www.ams.o rg/ b ookpages/mbk-63 Libra ry of Congress C ataloging-in-Publication Data Petlowit, Miodrag. Famous puzzles of great mathematicians / M iodrag S. PetkoviC. p . em. [ucJudes bibliographical references and index. ISBN 978-0-8218-4814-2 (alk. paper) 1. Mathemat ic:al rec reations. 2. Ma.thEJmatics- Populnr works. I. T itle. QA95.P4358 2009 5 l0-dc22 20090l'l018 Copy ing and re print ing. (udh,;dua.l readers of this pubHcatiou, and nonp1·ofit libra ries acting for t hem, are 1>ermiUed to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the <; \Ust..ornary tl<:knowl(;ldgment of tbe $tll..m;e }$given. Republi<:a.t.ion, l5'y.sterntttic copying, or muJt.iple reprodutt.ioo of any rrmterittl in ~hi1; publit<-ttiou is permitted only under license from the American Mathematical Society. R<E-quests for such permission should be addressed to the Acquisitions Depa.rtment, American !\{atbematical Society, 201 Charles Street, Providence, Rhode Island 02004-2294 USA. Requests can also be made by e-mail 'o r eprint - per!QieeionCama. or g. @ 2009 by the American Mathematical Society. All l'ights reserved. The Americtl.u Ma.tbemat.k<-tl Society r<:! t<-tius oll rights except Lbose granted to the United ShAot.es Government. P rinted in tbe U 11ited States of America. §The pa.per used in t his book is ac id~ free and falls ,... , it.b in the guidelines estabJish0<l to AllSure pe•·m a.nenca aod d ura bili ty. V isi t the AMS home page a.t ht t p://www.ams .or g/ 10 98 7 6 5 <I 3 2 I 14 13 12 :II :10 09 To my family 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 wit h 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 t o P roblems 30 3. N UMBER THEORY 37 Cattle problem (A1-chimedes, p . .18) 41 Dividing t ile square (Diophcmt1!S) 44 Wine problem (Diophantus) 45 Amicable numbers (ibn Qo1•ra) 45 Qorra's formula 47 Euler's rule 47 How many soldie rs? (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 Unknow n address (Ra.ma:nu;jan) 57 Stamp combinations (F1-obenius, Sylveste1·, p. 60) 61 Generalized problem 62 Answers t o P roblems 64 4 . GEOMETRY 1)7 Arbelos problem (A1-chimedes) 68 A rch jJ n f::dea.u cjr c h~:o: 69 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) 10 1 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 t o Problems 11 2 5. TILING AND PAC KING 119 Mosaics ( K eple1·) 12 1 Escher's mosaics 124 Nonperiodic tiling (Pem-ose, Conway) 126 Pemose's kite-dart tiling 126 Conway's piuwheel t il ing 128 Maximum area by pentaminoes (Knuth) 129 Pentamino tilings 131 Squaring the square 132 Kissing spheres (D. Gn3QO!~lf. Newton) 134 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 . PHYSI CS 151 The gold crown of King Iliero (A1·chimedes) 152 The length of traveled t ri p ( 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 schoo lgi rl~ problem (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 (Sylvest e•t') 202 Dudeney's rnilitaJ·y puz.zle 203 Dudeney's four planting problem 204 Different distances (Er'diis, p. 205) 206 Answers to Proble m s 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 22 1 Cramer's va.riaJ1t 222 The probability problem with the misaddressed letters (N. (II) Bemoulli, Eule1·) 222 Matchbox problem (Banach) 224 Combinatorial sum 225 A nswers to Prob le m s 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 x CONTENT S Crossing over 15 bridges 232 Tait's net 23a Hamilton's game on a dodecahedron (Hamilton, p. 2."J,'J) 234 Icosian puzzle 234 Hamiltonian cycles a nd the Tower of Hanoi 236 Hamiltonia-n cycles on the Pla.tonic solids 237 Dira.c's theorem 238 Ore's theorem 238 King Arthur's knights 239 A man, a wolf, a goat and a C<tbbage ( Alcu·in of Yin·k) 240 A stout family crosses the river ( Alc1,in of Yo1·k} 242 Explorers and carmibals 2 •1:~ Seven towns and one-way roads (Enlos) 244 Poinsot's diagram-tra.ciug puzzle (Poinsot) 245 Complete graph K 7 and dominoes 2t16 Milk puzzle (Poisson) 247 Gra.ph solution 248 Listing's diagram-tracing puzzle (L-iS'ting) 249 A nswers to P roble m s 250 10. C HESS 257 Knight's re-entrant route (de Moiv'l'-e, de M ontm01"t, Vandetmonde, Eule'l'") 258 De Moivre's re-entraJlt route 259 Eoler)s re-ent.ra.nt route 21)0 Vandermonde's approach 260 Schw•m k 's thP.orP.m 21)2 Semi-roa.gic re-entrant route 262 Non-attacking rooks ( E·uler) 265 The eight queens problem ( Ga·u.ss, p. 268) 269 n -q ueens problem 271 Number of solutions 272 The longest uncrossed knight's tom (Knuth} 2n Guarini's knight-switching problem 274 A va.riant of the knight-switching problem 276 CONTENTS XJ Answers to Problems 2·76 11. MISCELLANY 28~3 Problems from A lenin of Ymk 283 Problems from A lm'I-Wafa. 283 Amusing problems from Fibona.cci 284 Problems from Bachet 285 Huygeos' probability problems 285 Problems fro m Newton 286 Problems from Euler 286 APPENDIX A: Method of continued fractions for solving Pell's equation 289 APPENDIX B : Geometrical inversion 293 APPENDIX C: Some basic fa.cts from graph theory 294 APPENDIX D : Linear difference equations with constant co emJr cwn• t.