Mathematical_Recreations-Kraitchik-2e

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Magic Squares 15101-1 -1-1-1-1 "-1-1-1 1-1-1-1001-1-1-1-1- 1-1101-1-1-1-1-1 1o,1-01 I-I I-I 1-"-1-1-1-1-1- 10'01-01-1-11- -110 1-1-1 10'0'I-I 01-1- 1-1- 1-"- 1-1-1-1-1o,o'0-1-1-01 1-1- 1-1101-1 1-1-10'0'0'1- 1-01 1-1-1 "-I 10'0'0'0-I -I 01-1-1-1101-1-1- 10'0'0'0'1-1-1-01-1 1-"- 1-1-10'0'0'o,o-1-1-1-1-01 1-1101-1 10'0'0'0'0'1-1-1-1- 1-01-1 "-I 10'0'0'0'0'0-1-11-01-110110'0'0'0'0'0'1-1-1- 1-1-1-01 " 10'0'0'0'0'0,o1-1-1-1-1- 011010,0'0'0'0'0'0'1-1-1 1-1-1- 1-[]0'0'0'0,0,0,0,0,-,-, ,-,-, ,- ,-,-, ,-,-,-,-,'0'0'0'0'0'0'0 '0'0'0'0'0'0'0'1-1-1- 1-1-1 1-0"0'0'0'0'0'0'0'0-I 1-1-1-1 1-°' 10,0'0'0'0'0'0'0'I-I 1-1-1-0'0"0'0'0'0'0'0'0'0-I 1-1-0'0'10'0'0'0'0'0'0'0'1-1-1 1- 0'0'0"0'0'0'0'0'0'0'0-I 1-1- 0'0'0'1o,o,o,0'0'0'0'0'1- 1-1-0'0'0'0"0'0'0'0'0'0'0'0-I -I 0'0,o,0'10'0'0'0'0'0'0'0'-I -0'0'0'0'0"0'0'0'0'0'0'0'0-1-0'0'0'0'0, 10'0'0'0'0'0'0'0'1-1-0'0'0'0'0'0"0'0'0'0'0'0'0'0-I 0'0'0'0'0'0' 10'0'0'0'0,o,o,o,1-o,o,o,o,o,o,o"o,o,o,o,o,o,0,0-0'0'0'0'0'0'0'10'0'0'0'0'0'0'0'0'0'0'0'0'0'0'0"0'0'0'0'0'0'0'0FIGURE 31. Generation of Magic Squares of Even Order by theM etlwd of Devedec.ru' '00'00-"U1-"FIGURE 32. FIGURE 33.
15~ Mathematical Recreations- indicates an element that, in the fundamental square, occupiedthe position symmetric to this with respect to the verticalmedian.The center is to be filled in in accordance with Figure 32 orFigure 33, depending on whether the square is of order 4kor 4k+ 2.5. LATTICESMagic squares form only one of many interesting questionsdealing with the disposition of objects in rectangular arraysof cells. Chess and checkers come to mind at once as examples,and we shall find many other games and problemsthat are, or may be, based on the same idea. It will greatlyassist in our treatment of all such problems if we take timenow to develop briefly a concept which underlies them all:the lattice. (The reader should be warned that what we haveto say now has nothing to do with a more recent mathematicaluse of the word lattice to denote a certain type of abstractalgebraic structure. The sense in which we use theword has the claim of priority, and it has never become obsolete.)By a lattice is meant a set of points at a finite distancefrom each other, regularly spaced throughout a space of anynumber of dimensions. Thus a one-dimensional lattice is aset of points equally spaced along a whole line. A twodimensionallattice may be described as the set of points ofintersection of one set of equally spaced parallel lines withanother such set in the same plane. (The spacings of the twosets of parallels need not be the same.) Similarly, a threedimensionallattice can be obtained as the set of points ofintersection of three intersecting sets of equally spaced parallelplanes.The basic configuration, in terms of which all the latticeswith which we shall deal can be described, is the set of points
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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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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
Page 116 and 117: CHAPTER SIXPROBABILITIESTHE serious
Page 118 and 119: Probabilities 119possibilities prod
Page 120 and 121: Probabilitiesl~lone suit will not b
Page 122 and 123: Probabilities 123into some one of w
Page 124 and 125: Probabilities 1~5Finally, the proba
Page 126 and 127: Probabilities 127It is clear that t
Page 128 and 129: Probabilities 129deck is usually no
Page 130 and 131: Probabilitiesunpartments, so that t
Page 132 and 133: Probabilities 133from the pieces. T
Page 134 and 135: x·whence ~ = S + g.Piwins, we must
Page 136 and 137: Probabilities 137with a throw of ta
Page 138 and 139: Probabilities 139do. Does there exi
Page 140 and 141: Probabilities 141staking .1 of what
Page 142 and 143: Magic Squares 143fore we can readil
Page 144 and 145: Magic Squares 145change of rows may
Page 146 and 147: Magic Squares 147b + h = c + g = d
Page 148 and 149: Magic Squares 149fill in the remain
Page 152 and 153: Magic Squares 153in one-, two-, or
Page 154 and 155: Magic Squares 155We shall call the
Page 156 and 157: Magic Squares 157tinct in the modul
Page 158 and 159: Magic Squares 159shows how this met
Page 160 and 161: that is,Magic Squaresb+p-Iu+p-1w ==
Page 162 and 163: Magic Squares 163semimagic if t, u,
Page 164 and 165: Magic Squares 16541, and 14. Hence
Page 166 and 167: Magic Squares 167There are no regul
Page 168 and 169: Magic Squares 169One can also borde
Page 170 and 171: Magic Squares 171element of the giv
Page 172 and 173: Magic Squares173113 124 119 126 1 1
Page 174 and 175: Magic Squares 175possible to form a
Page 176 and 177: Magic Squares 177square of order 12
Page 178 and 179: Magic Squares 17912. EULER (GRAECO-
Page 180 and 181: Magic Squares 181themselves must th
Page 182 and 183: Magic Squares 183Figure 58. If (Fig
Page 184 and 185: Magic Squares 185must take. You can
Page 186 and 187: Magic Squares 187From this one can
Page 188 and 189: Magic Squares 189number of distinct
Page 190 and 191: Magic Squares 191We may remark in p
Page 192 and 193: CHAPTER EIGHTGEOMETRIC RECREATIONS1
Page 194 and 195: Geometric Recreations 1952. Dissect
Page 196 and 197: Geometric Recreations 197SoZ1dion:
Page 198 and 199: 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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