Mathematical_Recreations-Kraitchik-2e

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Needle problem, 132Negative digits, 51Neuter, 279New School for Social Research, 7Nim, 86-88Nomogram, 110Normal case, magic squares, 143Norton, W. W., 8Notation, scales of, 49-58Numa, calendar, 109Numbers, battle of, 83-86guessing, 56-66mathematics without, 13-21Numerical pastimes, 44-94oOctagon, 204One-sided surfaces, 212-213Order, magic squares, 143Orthogonal, magic squares, 143, 189Over ten, game, 126-127Ozanam,34PPaillot, 302Pallas, 24Pandects, problem of, 28Panmagic, defined, 143Panmagic squares, 170-176Parallelograms, Heronian, 107-108Parker, 8Partner changes, 226-230Pascal, 66, 117, 119Pascal's triangle, checkers and, 269Peg solitaire, 297Pell's equation, 100Pendus, les, 291Pentagonal numbers, 67Perelman, 57Perfect numbers, 70-73Permutational games, 302-323Permutational problems, 214-238Permutations,linear, of n things, 239-240Perpetual calendars, 109-116Pervouchine, 74Pettie, 291Philosophers, 15Pigeolet, 82Pillai, Siva.sankara.na.rayana, 290Plana, 70Plane figures, dissection, 193-198Poisson, 136Index 8~9Polish bank, 127Polycrates, 23Polyhedron, 18Polytechnic School Game, 295Pool, in games, 127Positional games, 267-302Poulet, 70, 215Powers, 78Prices, problems on, 33-37Prime numbers, 78-79Primitive triangles, 95-104Probabilities, 117-141Problems, solution by false methods,42-43Punga,282Pyramidal numbers, 68Pythagoras, 23Pythagorean relation, 95-108QQuadrilles, 303-304Queens, domination of board by, 255-256problem of, 238, 247-256Qu~telet, 138RRailroad trains, problems on, 38Ramesam, 70Raynor, 174Reapportionment, 29-31Reuscble, 70Reversi, 281Right triangles, 95-104Rings, Chinese, 58, 89-92Romulus, 109Rooks, domination of-board by, 256problem of, 240-247Rose-Innes, 66Roulette, 130-132Round dances, 228-230Round-robin, 231ROma,282Russian bank, 128SSaint-LagUe, 221Saint Petersburg paradox, 135-140Scales of notation, 49-58Scripta M athematica, 208Seelhof,74
830IndexSemimagic, defined, 143Sempronius, 28-29Ships, problems on, 38Shunting problems, 222-226Sivasankaranarayana Pillai, 290Smith,198Solid figurate numbers, 68Solitaire, peg, 297Sphere, mosaic on, 208-209Sphinz, 7, 78-80, 82Square numbers, 67-69Squares, dissection of, 194-198in line, 296marked,296Stone, 198Tarry, 176, 179, 220--221Ternary scale, 52, 57-58Tetrachrome, 312-313Thebault, 47, 48Themison, 24Theodore, 26Thieves and loot, 221Three piles, 317Time, problems on, 24-25, 39-40Titus, 28-29Topology, 209-213Torus, 145, 211Tossing up, 85Tournaments, 230--237Tower of Hanoi, 91-93Transformations, square, 144-146Trente et quarante, 130TTriangles, arithmetical, 103-108Heronian, 103-108primitive, 96problem, 132Pythagorean, 95Triangular numbers, 45, 67-69Tricolor, 293Trigonometric relations, 102-104Trimagic, 144, 176, 177Triple hopscotch, 291-292Trolley cars, problems on, 38Tutte, 198UUnfinished game, 117-119Uniform motion, problems, 17,38-39,41Unterrichtsbliitter fiir Mathematik undNaturwissen8chaften, 28VVatriquant, 79-80Violle, 169WWarnsdorf's rule, 259Webster, 77Weights and measures, problems on,41,52-55Western, 74Window reader cards, 62-65Wolf and goats, 309Wolf-cabbage-goat, 214
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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
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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 326 and 327: Index 827Composite magic squares, 1
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