Mathematical_Recreations-Kraitchik-2e

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PREFACE ...CONTENTSCHAPTER ONE. MATHEMATICS WITHOUTNUMBERSCHAPTER Two. ANCIENT AND CURIOUSPROBLEMS.Arabian Problems - Reapportionment by Decanting .Hindu Problems. . . . . . . . . . . . . .CHAPTER THREE.1. Miscellaneous Problems. . .2. Scales of Notation. . . . .3. To Guess a Selected Number .4. Figurate Numbers. . . . .NUMERICAL PASTIMES5. Mersenne Numbers and Perfect Numbers6. Fermat Numbers . .7. Cyclic Numbers. . .8. Automorphic Numbers9. Prime Numbers.10. MuUigrade. . . .11. Cryptarithmetic. .12. Arithmetical Games13. Josephus' Problem.CHAPTER FOUR. ARITHMETICO-GEOMETRI-CAL QUESTIONS . . . . . . .CHAPTER FIVE.CHAPTER SIX.THE CALENDARPROBABILITIES9PAGE713222732444451586670737577787979839395109117
10 ContentsPAGECHAPTER SEVEN.1. Definitions.MAGIC SQUARES 1421422. Transformations . . . . . . .1443. Magic Squares of the Third Order .4. Elementary Methods of Construction.5. Lattices. . . . . . . . . . . .6. Regular Magic Squares of Odd Order.7. Border Squares. . . . .8. Composite Magic Squares.9. Panmagic Squares.10. MuUimagic Squares . . .14614815215716717017417611. Latin Squares . . . . .17812. Euler (Graeco-Latin Squares)13. Magic Squares of Order 4. .179182CHAPTER EIGHT. GEOMETRIC RECREATIONS 1931. Dissection of Plane Figures2. Mosaics. . .....3. A Mosaic on the Sphere.1931992084. Topology . . . . . .209CHAPTER NINE. PERMUTATIONAL PROBLEMS 2141. Difficult Crossings. 2142. Shunting Problems2223. Distributions. . .2264. Matches and Tournaments.230CHAPTER TEN. THE PROBLEM OF THE QUEENS 2381. The Linear Permutations of n Different Things 2392. The Problem of the Rooks . 2403. The Problem of the Queens 247
Page 2 and 3: MathematicalRecreationsMAURICE KRAI
Page 4: TOMR. DANNIE HEINEMANwho encouraged
Page 7: 8 Mathematical RecreationsMr. W. W.
Page 12 and 13: CHAPTER ONEMATHEMATICS WITHOUT NUMB
Page 14 and 15: 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
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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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Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
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
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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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