Mathematical_Recreations-Kraitchik-2e

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Index 827Composite magic squares, 170-174Condorcet, 136Congruences, 41, 62--63, 77Constellation, 175Convergent, 100Courser, 279Coxeter, Ho Mo So, 208Craps, 123-126Crossings, problems of, 214--222Cryptarithmetic, 79-83Cuboid, 107-108Cunningham, 70, 74Curious problems, 22-43Cyclic numbers, 74--76Cylindrical chess, 277DDabbaba, 278Daily promenade. 226Dawson, chess innovations, 278Decanting, reapportionment by, 29-31Deductive reasoning, 13-21Delannoy, 302, 304Devedec, 150-152Diagonal, magic squares, 143, 189Dice, problems on, 118--119, 123-127Difficult crossings, 214--222Digits, location of, 48--50negative, 51Diophantus, 25, 44, 66, 69Dissections, 193-197Distances, shortest, 18--21Distributions, 226-230Dodecagon, 206,207Dogs and wolf, 310Domain, 260Dominoes, 298--304Draughts (see Checkers)Duke Mathematical Journal, 198Earth, girdling, 39Error curve, 121-123Euler, 28--29, 70, 73, 74, 78, 209Euler squares, 179-182Fairy chess, 276-280Fathers and sons, 37Fermat. 28, 70, 74,92. 117, 119,211EFFermat numbers, 73-75Fers, 278Fibonacci, Leonardo, 26-27Fifteen puzzle, 302-308Figurate numbers, 66--69Flanders, Donald Ao, 7Fleisher, 182Fool'" mate, 267Four-story towers, 311-312Fourth dimension, figurate numbersin, 68Fractions, ancient problems, 22-43Freak problems, 44Frederick II, emperor, 26Frenicle, 191Frogs and toads, 313-314Frolow, 169Fry, T. Co, 139GGambler's ruin, 140Gambling, systems, 131Games, arithmetical, 83-93permutational, 302-323positional, 267-302problems on, 267-323unfinished, 117Gauss, 74, 110Genealogy, 16Geodesic, 18Geometric recreations, 193-213Geometry, arithmetic and, 95-108Gergonne, 318Go, game, 279Go-bang, 280Goormaghtigh, 318Graces, 24Gracco-Latin squares, 179-182Grru3Shopper, chess piece, 279Grru3Shopper, game, 310-311Great bishop, 279Greek Anthology, problems from, 23-26Gregorian calendar, 109-116Gregory XIII, 110Gros, 90-91Guiser, 86HHalf-bishop, 279Halma, 311Halton, 122, 123Hanoi, tower of, 91-93Heath, R. Vo, 57,72,171,177,209
328 IndexHeawood, 211Heineman, D., 7Heracles, 24Heronian triangles, 104-108Hexagonal numbers, 67Hilbert, 193Hindu problems, 32-33Hopscotch, 29(}-292Horses, betting on, 134-135Huber-Stockar, 266Husbands and wives, 37IIcosahedron, 208Inheritance, problems of, 29, 31JJapan, games, 279-280Jealous husbands, 214-221Jinx, 281John of Palermo, 26-27Johnson, Alvin, 7Joining points, 296Josephus, 94Josephus' problem, 93-94Julian calendar, 109-116KKing, algebra of moves, 273-274Kirkman, 230Klein, Felix, 209Knights, domination of board by,256moves, 273-277problem of the, 257-266Konigsberg bridges, 209-211Kowalewsky, G., 311Kraitchik, 8, 74LLagrange, 135Landry, 70, 74, 78Lasca, 287-290Lasker, 287Latin squares, 178, 246Latruncules, 279Lattices, 152-157Leaper, 279Legal chess, 278Lehmann, 182Lehmer, 70, 78Leibnitz, 298Le Lasseur, 70Leonardo of Pisa, 26-27"Liber Abaci," 26-27Ling, 281Loubere, de la, 148-149Loyd, Sam, 267-268Lucas, 78, 92quadrille, 302-303MMacMahon, 53Magic constant, 143Magic series, 143, 176, 183-186Magic squares, 142-192composite, 17(}-174composite order m, 162-167domino, 302elementary construction methods,148literal, 186-187, 192order 4, 182-192prime order p, 157-161third order, 146-148Map-coloring problem, 211Marceil, 321Marked squares, 296Marseille chess, 278Martin, 317Matador, 300Matches, 23(}-237Mathematics, nature of, 13M~r~, Chevalier de, 117, 119Mersenne, 7(}-72Mersenne numbers, 7(}-73Meuris,33Minos, 79--80Miscellaneous problems, 44-51Mobius, 277Mobius band, 212-213, 277Monge's shuffle, 321-323Monmort, 298Morehead, 74Mosaics, 199--209Mouse trap, 295Muggins,3ooMultigrade, 79Multimagic, defined, 144M ultimagic squares, 176-178Muses, 23, 24, 230NNaval battle, 283Neckties, paradox, 133-134
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MathematicalRecreationsMAURICE KRAI
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TOMR. DANNIE HEINEMANwho encouraged
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8 Mathematical RecreationsMr. W. W.
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10 ContentsPAGECHAPTER SEVEN.1. Def
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CHAPTER ONEMATHEMATICS WITHOUT NUMB
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M athematics Without Numbers 15he r
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Mathematic8 without Numbers 17word
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Mathematics witlwut Numbers 19way a
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Mathematics without Numbers 21~ I(h
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Ancient and Curious Problemsto the
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Ancient and Curious Problems 25Any
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Ancient and Curious ProblemsSolutio
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Ancient and Curious Problems 29Semp
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A ncient and Curious Problems 31Ans
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Ancient and Curious Problems 33monk
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SolutionNo.Ancient and Curious Prob
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Ancient and Curiou8 Problemab+da+ci
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Ancient and Curious Problems 89and
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Ancient and Curious Problems 41t =
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thenA ncient and Curious Problemsx-
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Numerical Pastimes 45desired number
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Numerical Pastimes 47Mr. Thebault g
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Numerical Pastimes 49threes I find
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Numerical Pastimes(3) 16ab + c = 16
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Numerical Pastimes 53If the standar
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Numerical Pastimes 55consists of k
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Numerical Pastimes 57sponding to th
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Numerical Pastimes 592. Think of a
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Numerical Pastimes 61the result. Or
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Numerical Pastimes 68If we subtract
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ABCDEFGHIJKLM IIN 0 P Q R STU V W x
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Numerical Pastimes 67progressions.
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Numerical Pastimes 691 (n + d - 2)!
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Numerical Pastimes 71Two or more pr
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Numerical Pastimes 73= 8. m2(m + 1)
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Numerical Pastimes 75Thus, in order
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Numerical Pastimes 778. AUTOMORPHIC
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Numerical Pastimes 79primality has
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Numerical Pastimes 81products with
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Numerical Pastimes 838. QUI)TRO UVE
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Numerical Pastimes 85There are many
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Numerical Pastimes 87if a player A
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Numerical Pastimes 893. CHINESE RIN
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Numerical Pastimes 91rings. If the
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Numerical Pastimes 93others getting
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CHAPTER FOURARITHMETICO-GEOMETRICAL
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SinceArithmetico-Geometrical Questi
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A rithmetico-Geometrical Questions
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Arithmetico-Geometrical Questions 1
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CHAPTER FIVETHE CALENDARTIME can be
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The Calendar111MONTHMJanua~ .......
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The Calendar 113scale). For the mon
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The Calendar 115"other point." In a
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CHAPTER SIXPROBABILITIESTHE serious
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Probabilities 119possibilities prod
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Probabilitiesl~lone suit will not b
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Probabilities 123into some one of w
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Probabilities 1~5Finally, the proba
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Probabilities 127It is clear that t
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Probabilities 129deck is usually no
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Probabilitiesunpartments, so that t
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Probabilities 133from the pieces. T
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x·whence ~ = S + g.Piwins, we must
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Probabilities 137with a throw of ta
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Probabilities 139do. Does there exi
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Probabilities 141staking .1 of what
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Magic Squares 143fore we can readil
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Magic Squares 145change of rows may
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Magic Squares 147b + h = c + g = d
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Magic Squares 149fill in the remain
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Magic Squares 15101-1 -1-1-1-1 "-1-
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Magic Squares 153in one-, two-, or
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Magic Squares 155We shall call the
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Magic Squares 157tinct in the modul
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Magic Squares 159shows how this met
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that is,Magic Squaresb+p-Iu+p-1w ==
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Magic Squares 163semimagic if t, u,
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Magic Squares 16541, and 14. Hence
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Magic Squares 167There are no regul
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Magic Squares 169One can also borde
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Magic Squares 171element of the giv
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Magic Squares173113 124 119 126 1 1
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Magic Squares 175possible to form a
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Magic Squares 177square of order 12
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Magic Squares 17912. EULER (GRAECO-
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Magic Squares 181themselves must th
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Magic Squares 183Figure 58. If (Fig
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Magic Squares 185must take. You can
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Magic Squares 187From this one can
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Magic Squares 189number of distinct
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Magic Squares 191We may remark in p
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CHAPTER EIGHTGEOMETRIC RECREATIONS1
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Geometric Recreations 1952. Dissect
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Geometric Recreations 197SoZ1dion:
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Geometric Recreations 1992. MOSAICS
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Geometric Recreations~OlMosaic ofMo
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Geometric Recreations111 1 1 3-+-+-
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Geometric Recreations ~05have edges
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Geometric Recreations 207solutions,
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Geometric Recreations ~09the 20, an
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Geometric Recreations ~11closed --
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Geometric Recreations ~13example, s
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Permutational Problems ~15child, ar
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Permutational Problems 217the minor
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Permutational Problems 219The two h
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Permutational Problems 221Thus it t
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Permutational ProblemsInitially the
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Permutational Problemsget onto the
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Permutational Problemseter give 2n
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Permutational Problems 229go back t
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Permutational Problems 231When two
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Permutational ProblemaRn = nC2 = n(
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Permutational Problems ~35n = 171,1
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Permutational Problems 2371,2, 3, 4
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The Problem of the Queens~391. THE
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The Problem oj the Queens 241tions.
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The Problem of the Queens ~43no sol
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The Problem of the Queens 245Such a
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The Problem of the Queens 247up int
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The Problem of the Queens ~49FIGURE
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The Problem of the Queens 251On the
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The Problem of the Queens 253n-1 n-
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The Problem of the Queens '255ular
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CHAPTER ELEVENTHE PROBLEM OF THE KN
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The Problem of the KnightU92. W ARN
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The Problem of the Knight ~61tour i
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The Problem of the Knight 263tal me
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The Problem of the Knight265There a
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CHAPTER TWELVEGAMES1. POSITIONAL GA
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Games 269The checker man moves one
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Games 271left," namely" no change."
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Games 273U- 2 V-1 + u- 2 v + U-1V-2
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Game8 275(16 moves for the 8 pawns
Page 276 and 277: Games 277rules of play, or the piec
Page 278 and 279: Games 279The centaur, which combine
Page 280 and 281: Games 2816. REVERSI. This is played
Page 282 and 283: Games 283tween 2 and 4 balls in eac
Page 284 and 285: Games 285This great power granted t
Page 286 and 287: Games ~87that White can always win,
Page 288 and 289: Games 289if Black), the guide alone
Page 290 and 291: Games 291Here are a few games. (W s
Page 292 and 293: Games ~98units. The strength of a s
Page 294 and 295: Games '295end of the game. When all
Page 296 and 297: Games 297first one to be unable to
Page 298 and 299: Games 299draw. In case of tie the o
Page 300 and 301: Games 301in the original game, exce
Page 302 and 303: Games 303initially the cubes are se
Page 304 and 305: Games 805aspects of the theory of p
Page 306 and 307: Games 307n (> 1) objects are even,
Page 308 and 309: Games 309the two sets of pawns inte
Page 310 and 311: Games 311Grasshopper is a variant o
Page 312 and 313: Games 313one tries to construct squ
Page 314 and 315: Games 315umn, one takes that opport
Page 316 and 317: GamesRl Bl ~ B2 R3 B3 ~ B4~ B3 Rl B
Page 318 and 319: Games 319cards by pairs in order an
Page 320 and 321: Games 321up after the fifth deal in
Page 322 and 323: Games 3~3of the cuts, we find that
Page 324 and 325: THE FALSE COIN. 825Hence we can ide
Page 328 and 329: Needle problem, 132Negative digits,
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