Mathematical_Recreations-Kraitchik-2e

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The Problem of the Queens 251On the ordinary chessboard of 8 2 cells there are 92 solutions,consisting of 11 sets of equivalent ordinary solutionsand one set of equivalent symmetric solutions. There areno doubly symmetric solutions in this case. Following is alist containing one ordinary solution from each set of equivalents:15863724, 16837425, 24683175, 25713864, 25741863,26174835, 26831475, 27368514, 27581463, 35841726,36258174. The set of symmetric solutions may be generatedby 35281746.REGULAR SOLUTIONS. These are defined as before, namelyas solutions generated by a motion (u, v). Now, however,not only must both u and v be prime to n, we must also avoidhaving two queens on the same diagonal. What restrictionthis places on the choice of the generating motion can be seenby the following observation:Any two-dimensional lattice (a, b) + r(u, v), modulo n, inwhich both u and v are prime to n may be obtained by replacing(u, v) by (TU, TV) for any T prime to n. In particularwe may choose T so that ru = +1. When this has been doneit turns out that two elements of the original lattice will beon the same diagonal if and only if TV == ± 1, modulo n.Hence in order to get a regular solution of the problem of thequeens it is necessary and sufficient that the solution be generatedby a motion (1, v) having v, v + 1, and v-I all primeto n. Since these three numbers are consecutive, and one ofany two consecutive numbers is divisible by 2, and one of anythree consecutive numbers is divisible by 3, it results thatno regular solutions exist when n is divisible by 2 or 3.It is clear that if any regular solutions exist they will occurin sets of n obtained from each other by changing the originof the lattice. N one of these n solutions can intersect, sothey may be.superimposed on the same board without overlapping,forming a solution of the problem of coloring eachof the cells of the board with one of n colors in such a way
252 Mathematical Recreationsthat no color appears twice in the same orthogonal or diagonal.Such a juxtaposition of the 11 regular solutions forn = 11 is shown in Figure 124.A regular solution will be symmetric if the central cell isoccupied. This requires that in the iattice (a, b) + r(l, v) wehaven+ln+la + T E:! -2- and b + TV E:! -2-' modulo n.If we take the origin in the first column by setting a = 1, we
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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
Page 200 and 201: Geometric Recreations~OlMosaic ofMo
Page 202 and 203: Geometric Recreations111 1 1 3-+-+-
Page 204 and 205: Geometric Recreations ~05have edges
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Page 208 and 209: Geometric Recreations ~09the 20, an
Page 210 and 211: Geometric Recreations ~11closed --
Page 212 and 213: Geometric Recreations ~13example, s
Page 214 and 215: Permutational Problems ~15child, ar
Page 216 and 217: Permutational Problems 217the minor
Page 218 and 219: Permutational Problems 219The two h
Page 220 and 221: Permutational Problems 221Thus it t
Page 222 and 223: Permutational ProblemsInitially the
Page 224 and 225: Permutational Problemsget onto the
Page 226 and 227: Permutational Problemseter give 2n
Page 228 and 229: Permutational Problems 229go back t
Page 230 and 231: Permutational Problems 231When two
Page 232 and 233: Permutational ProblemaRn = nC2 = n(
Page 234 and 235: Permutational Problems ~35n = 171,1
Page 236 and 237: Permutational Problems 2371,2, 3, 4
Page 238 and 239: The Problem of the Queens~391. THE
Page 240 and 241: The Problem oj the Queens 241tions.
Page 242 and 243: The Problem of the Queens ~43no sol
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Page 246 and 247: The Problem of the Queens 247up int
Page 248 and 249: The Problem of the Queens ~49FIGURE
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Page 254 and 255: The Problem of the Queens '255ular
Page 256 and 257: CHAPTER ELEVENTHE PROBLEM OF THE KN
Page 258 and 259: The Problem of the KnightU92. W ARN
Page 260 and 261: The Problem of the Knight ~61tour i
Page 262 and 263: The Problem of the Knight 263tal me
Page 264 and 265: The Problem of the Knight265There a
Page 266 and 267: CHAPTER TWELVEGAMES1. POSITIONAL GA
Page 268 and 269: Games 269The checker man moves one
Page 270 and 271: Games 271left," namely" no change."
Page 272 and 273: Games 273U- 2 V-1 + u- 2 v + U-1V-2
Page 274 and 275: 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
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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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