Mathematical_Recreations-Kraitchik-2e

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Numerical Pastimes 67progressions. These form the plane figurate numbers, orfigurate numbers of the second dimension.i Figure Plane figurate numbers General term--I Triangle 1 3 6 10 15 ... tn(n + 1)2 Square 1 4 9 16 25 ... n23 Pentagon 1 5 12 22 35 ... tn(3n - 1)4 Hexagon 1 6 15 28 45 ... n(2n - 1)5 ... 1 7 18 34 55 ... tn(5n - 3)... ... ... . ..i ... 1 2 +i 3 +3i 4 +6i 5 + lOi ... tn[in - (i - 2)]FIGURE 8. Representation of Plane Figurate Numbers.The names triangular, square, pentagonal, hexagonal, andso on, are justified by the arrangements of the correspondingnUP.lbers of units shown in Figure 8.3. If we now form the successive sums of the terms of the
68 Mathematical Recreationsplane figurate numbers, the resulting numbers are the solid(pyramidal) figurate numbers, or figurate numbers of thethird dimension.i Solid figurate numbers General term1 1 4 10 20 35 in(n + 1)(n + 2)2 1 5 14 30 55 In(n + 1)(2n + 1)3 1 6 18 40 75 tn2(n + 1)4 1 7 22 50 95 in(n + 1)(4n - 1)5 1 8 26 60 115 in(n + 1)(5n - 2)i 1 3 +i 6 +4i 10 + 10i 15 + 20i in(n + 1)[in - (i - 3)J4. We can penetrate the fourth dimension by taking thesums of the terms of the solid progressions, which we call thefigurate numbers of the fourth dimension.i Figurate numbers of the fourth dimension General term1 1 5 15 35 70 n41n + l)(n + 2)(n + 3)2 1 6 20 50 105 n12(n + 1)2(n + 2)3 1 7 25 65 140 n4i(n + 1)(n + 2)(3n + 1)4 1 8 30 80 175 n2'6<n + l)(n + 2)5 1 9 35 95 210 ~(n + 1)(n + 2)(5n - 1)24i 14+ilO+ 5i 20 + l5i 35+35i··· ~(n +1)(n+2)[in- (i - 4)J4!6. This procedure can be generalized so as to form whatwe shall call figurate numbers of dimension d and difference i,
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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
Page 16 and 17: Mathematic8 without Numbers 17word
Page 18 and 19: Mathematics witlwut Numbers 19way a
Page 20 and 21: Mathematics without Numbers 21~ I(h
Page 22 and 23: Ancient and Curious Problemsto the
Page 24 and 25: Ancient and Curious Problems 25Any
Page 26 and 27: Ancient and Curious ProblemsSolutio
Page 28 and 29: Ancient and Curious Problems 29Semp
Page 30 and 31: A ncient and Curious Problems 31Ans
Page 32 and 33: Ancient and Curious Problems 33monk
Page 34 and 35: SolutionNo.Ancient and Curious Prob
Page 36 and 37: Ancient and Curiou8 Problemab+da+ci
Page 38 and 39: Ancient and Curious Problems 89and
Page 40 and 41: Ancient and Curious Problems 41t =
Page 42 and 43: thenA ncient and Curious Problemsx-
Page 44 and 45: Numerical Pastimes 45desired number
Page 46 and 47: Numerical Pastimes 47Mr. Thebault g
Page 48 and 49: Numerical Pastimes 49threes I find
Page 50 and 51: Numerical Pastimes(3) 16ab + c = 16
Page 52 and 53: Numerical Pastimes 53If the standar
Page 54 and 55: Numerical Pastimes 55consists of k
Page 56 and 57: Numerical Pastimes 57sponding to th
Page 58 and 59: Numerical Pastimes 592. Think of a
Page 60 and 61: Numerical Pastimes 61the result. Or
Page 62 and 63: Numerical Pastimes 68If we subtract
Page 64 and 65: ABCDEFGHIJKLM IIN 0 P Q R STU V W x
Page 68 and 69: Numerical Pastimes 691 (n + d - 2)!
Page 70 and 71: Numerical Pastimes 71Two or more pr
Page 72 and 73: Numerical Pastimes 73= 8. m2(m + 1)
Page 74 and 75: Numerical Pastimes 75Thus, in order
Page 76 and 77: Numerical Pastimes 778. AUTOMORPHIC
Page 78 and 79: Numerical Pastimes 79primality has
Page 80 and 81: Numerical Pastimes 81products with
Page 82 and 83: Numerical Pastimes 838. QUI)TRO UVE
Page 84 and 85: Numerical Pastimes 85There are many
Page 86 and 87: Numerical Pastimes 87if a player A
Page 88 and 89: Numerical Pastimes 893. CHINESE RIN
Page 90 and 91: Numerical Pastimes 91rings. If the
Page 92 and 93: Numerical Pastimes 93others getting
Page 94 and 95: CHAPTER FOURARITHMETICO-GEOMETRICAL
Page 96 and 97: SinceArithmetico-Geometrical Questi
Page 98 and 99: A rithmetico-Geometrical Questions
Page 100 and 101: Arithmetico-Geometrical Questions 1
Page 108 and 109: CHAPTER FIVETHE CALENDARTIME can be
Page 110 and 111: The Calendar111MONTHMJanua~ .......
Page 112 and 113: The Calendar 113scale). For the mon
Page 114 and 115: 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
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Games 279The centaur, which combine
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Games 2816. REVERSI. This is played
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Games 283tween 2 and 4 balls in eac
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Games 285This great power granted t
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Games ~87that White can always win,
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Games 289if Black), the guide alone
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Games 291Here are a few games. (W s
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Games ~98units. The strength of a s
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Games '295end of the game. When all
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Games 297first one to be unable to
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Games 299draw. In case of tie the o
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Games 301in the original game, exce
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Games 303initially the cubes are se
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Games 805aspects of the theory of p
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Games 307n (> 1) objects are even,
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Games 309the two sets of pawns inte
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Games 311Grasshopper is a variant o
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Games 313one tries to construct squ
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Games 315umn, one takes that opport
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GamesRl Bl ~ B2 R3 B3 ~ B4~ B3 Rl B
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Games 319cards by pairs in order an
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Games 321up after the fifth deal in
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Games 3~3of the cuts, we find that
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THE FALSE COIN. 825Hence we can ide
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Index 827Composite magic squares, 1
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Needle problem, 132Negative digits,
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