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Famous Puzzles of Great Mathematicians PDF

345 Pages·2009·18.92 MB·English
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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. Por additional infon1ut.tion and updates on this book: visit www.ams.org/ bookpages/mbk-63 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 Copying and reprinting. (udh,;dua.l readers of this pubHcatiou, and nonp1·ofit libraries acting for them, 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 publit<-ttiou ~hi1; 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 reprint -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. Printed in tbe U11ited States of America. §The pa.per used in this book is acid~ free and falls ,. .., it.bin the guidelines estabJish0<l to AllSure pe•·ma.nenca aod durability. Visit the AMS home page a.t ht tp://www.ams.org/ 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 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

Description:
This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. The selected problems do not require advanced mathematics, making this book accessible to a variety of readers. Mathemati
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