http://dx.doi.org/10.1090/mbk/063 FAMOUS PUZZLES of Great Mathematicians Miodrag S. Petkovi´c A MERICAN M ATHEMATICAL S OCIETY Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 00A08, 97A20, 01A05, 01A70, 05A05, 05C45, 05C90, 11D04, 11D09, 51E10, 51M16, 52C15, 52C22, 97D40. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-63 Library of Congress Cataloging-in-Publication Data Petkovi´c, Miodrag. Famous puzzles of great mathematicians / Miodrag S. Petkovi´c. p. cm. Includes bibliographical references and index. ISBN 978-0-8218-4814-2 (alk. paper) 1. Mathematical recreations. 2. Mathematics–Popular works. I. Title. QA95.P4358 2009 510–dc22 2009011018 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. 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Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 To my family CONTENTS Preface xiii Art and Photo Credits xvii 1. RECREATIONAL MATHEMATICS 1 2. ARITHMETICS 9 Diophantus’ age (Diophantus)10 Number of arrows (Mah¯avira)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 Magicconfiguration(Hui )17 Triangle with integral sides (Bachet)17 Weights problem (Fibonacci, Tartaglia, Bachet)20 MacMahon’s general approach 23 Division of 17 horses (Tartaglia, p. 23)24 Wine and water (Tartaglia)25 Coins in hands (Recorde)25 Sides of two cubes (Vi`ete, p. 25)26 Animals on a field (Newton)27 P´olya’s solution 28 D¨orrie’s variant 28 Other variants 29 v vi Contents Gathering an army (Alcuin of York)29 Answers to Problems 30 3. NUMBER THEORY 37 Cattle problem (Archimedes, p. 38)41 Dividing the square (Diophantus)44 Wine problem (Diophantus)45 Amicable numbers (ibn Qorra)45 Qorra’s formula 47 Euler’s rule 47 How many soldiers? (Bh¯askara)48 Horses and bulls – a Diophantine equation (Euler)50 The sailors, the coconuts, and the monkey (P. Dirac)52 Dirac’s solution 56 General solution 56 Unknown address (Ramanujan)57 Stamp combinations (Frobenius, Sylvester, p. 60)61 Generalized problem 62 Answers to Problems 64 4. GEOMETRY 67 Arbelos problem (Archimedes)68 Archimedean circles 69 Perpendicular distance 70 Two touching circles 73 Minimal distance (Heron)73 A fly and a drop of honey 75 Peninsula problem 76 The same distance of traversed paths (Brahmagupta)77 Broken bamboo (Brahmagupta)78 Height of a suspended string (Mah¯avira)78 The diameter of the material sphere (ibn Qorra)80 Dissection of three squares (Abu’l-Wafa)81 Dudeney’s six-cut dissection 82 Contents vii Dissection of four triangles (Abu’l-Wafa)83 Billiard problem (Alhazen)83 Distance of the optimal viewpoint (Regiomontanus)86 Rugby football problem 88 Saturn problem 89 The minimal sum of distances in a triangle (Steiner, Fermat, Torricelli, Cavalieri)91 Volumes of cylinders and spheres (Kepler, p. 93)94 Dido’s problem (Steiner, p. 95)96 Division of space by planes (Steiner)98 Division of plane 99 Number of bounded regions 101 Road system in a square (Steiner) 101 Kissing circles (Soddy, Descartes, Kowa) 104 Kazayuki’s problem 108 Five kissing spheres 109 -dimensional kissing spheres 110 The shortest bisecting arc of area (Polya) 110 Answers to Problems 112 5. TILING AND PACKING 119 Mosaics (Kepler) 121 Escher’s mosaics 124 Nonperiodictiling ( Penrose, Conway) 126 Penrose’s kite-dart tiling 126 Conway’s pinwheel tiling 128 Maximum area by pentaminoes (Knuth) 129 Pentamino tilings 131 Squaring the square 132 Kissing spheres (D. Gregory, Newton) 134 The densest sphere packing (Kepler, Gauss) 137 Kepler’s conjecture 137 Circle packing 138 Hales’ solution 141 viii Contents Cube-packing puzzles (Conway) 142 Answers to Problems 144 6. PHYSICS 151 The gold crown of King Hiero (Archimedes) 152 The length of traveled trip (Oresme) 153 Meeting of ships (Lucas, p. 155) 156 A girl and a bird (Von Neumann, p. 157) 158 Von Neumann’s easy series 158 More difficult problems 160 The lion and the man (Littlewood, Besicovitch, R. Rado) 161 Answers to Problems 166 7. COMBINATORICS 171 Combination with flavors (Mah¯avira) 172 Married couples cross the river (Bachet) 173 Four married couples cross the river 175 Josephus problem (Bachet and others) 176 Dudeney’s version 177 Japanese version 177 Graham-Knuth-Patasnik’s version 178 Rings puzzle (Cardano, p. 180) 181 Gray-code numbers 182 The problem of the misaddressed letters (N. (II) Bernoulli, Euler) 184 Eulerian squares (Euler) 186 Latin squares 187 Graeco-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 triple system 190 General 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) (Lucas) 199 Interchanging the checkers (II) (Lucas) 200 Shunting problem (Lucas) 200 Problem of married couples (probl`eme des m´enages) (Lucas) 201 The tree planting problem (Sylvester) 202 Dudeney’s military puzzle 203 Dudeney’s four planting problem 204 Different distances (Erd¨os, p. 205) 206 Answers to Problems 206 8. PROBABILITY 209 The problem of the points (Fermat, Pascal, p. 211) 212 Pascal-Fermat’s general problem 213 Gambling game with dice (Huygens) 215 Gambler’s ruin problem (Pascal, Fermat, Huygens) 217 General problem 218 The Petersburg paradox (N. (III) Bernoulli, D. Bernoulli, Cramer) 220 Bernoulli’s moral expectation 221 Cramer’s variant 222 The probability problem with the misaddressed letters (N. (II) Bernoulli, Euler) 222 Matchbox problem (Banach) 224 Combinatorial sum 225 Answers to Problems 225 9. GRAPHS 229 The problem of K¨onigsberg’s bridges (Euler) 230 Diagram-tracing puzzle 231 Euler’s theorem 231 Euler’s path 231 x Contents Crossing over 15 bridges 232 Tait’s net 233 Hamilton’s game on a dodecahedron (Hamilton, p. 233) 234 Icosian puzzle 234 Hamiltonian cycles and the Tower of Hanoi 236 Hamiltonian cycles on the Platonic solids 237 Dirac’s theorem 238 Ore’s theorem 238 King Arthur’s knights 239 A man, a wolf, a goat and a cabbage (Alcuin of York) 240 A stout family crosses the river (Alcuin of York) 242 Explorers and cannibals 243 Seven towns and one-way roads (Erd¨os) 244 Poinsot’s diagram-tracing puzzle (Poinsot) 245 Complete graph 7 and dominoes 246 Milk puzzle (Poisson) 247 Graph solution 248 Listing’s diagram-tracing puzzle (Listing) 249 Answers to Problems 250 10. CHESS 257 Knight’s re-entrant route (de Moivre, de Montmort, Vandermonde, Euler) 258 De Moivre’s re-entrant route 259 Euler’s re-entrant route 260 Vandermonde’s approach 260 Schwenk’s theorem 262 Semi-magicre-entrant route 262 Non-attacking rooks (Euler) 265 The eight queens problem (Gauss, p. 268) 269 -queens problem 271 Number of solutions 272 The longest uncrossed knight’s tour (Knuth) 273 Guarini’s knight-switching problem 274 A variant of the knight-switching problem 276 Contents xi Answers to Problems 276 11. MISCELLANY 283 Problems from Alcuin of York 283 Problems from Abu’l-Wafa 283 Amusing problems from Fibonacci 284 Problems from Bachet 285 Huygens’ probability problems 285 Problems from 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 basicfactsfrom graph theory 294 APPENDIX D: Linear difference equations with constant coefficients 296 Biographies – a chronological order 299 Bibliography 311 Name index 319 PREFACE Mathematics is too serious and, therefore, no opportunity should be missed to make it amusing. Blaise Pascal Mathematical puzzles and games have been in evidence ever since man first began posing mathematical problems. The history of mathematics is replete with examples of puzzles, games, and entertaining problems that have fostered the development of new disciplines and sparked further research. Important connections exist between problems originally meant to amuse and mathematical concepts critical to graph theory, geometry, optimization theory, combinatorics, and number theory, to name only a few. As a motivating force, then, the inclination to seek diversion and enter- tainment has resulted in the unintended revelation of mathematical truths while also tempering mathematical logic. In fact, Bertrand Russell (1872– 1970) once noted: “A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science.” Perhaps the popularity of mathematical puzzles and games endures be- cause they fulfill the need for diversion, the desire to achieve mastery over challenging subject matter or simply to test our intellectual capacities. Of equal importance, mathematical amusements also offer an ample playing field to both the amateur and the professional mathematician.