Mathematical_Recreations-Kraitchik-2e

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THE FALSE COIN. 825Hence we can identify the group of 3 coins which containsthe false one. If a, b, c are the 3 coins of that groupwe compare in a second balancing the coins a and b.If a>b, a is the false coina<b b" " " "a=b c" " " "The solution is similar if the false coin is lighter thanthe other.In the case of N = 3" coins we need n balancings inorder to identify the false coin. The same number of balancingsis required if 3- 1 <N <3".3. We have 12 coins and we are asked to identify thefalse coin and find its defect by 3 balancings.We distribute the 12 coins, A, B, C ... in 3 groups:ABCD, EFGH. IJKL.FirstbalancingSecondbalancingThirdbalancingConclusionABCD>EFGH ABE>CDF A>B A>OA<B B>OA=B F<OABE<CDF C>D C>OC<D D>OC=D E<OABE=CDF G>H H<OG<H G<OABCD<EFGH ABE>CDF C>D D<OC<D C<OC=D E>OABE<CDF A>B B<OA<B A<OA=B F>OABE=CDF G>H G>OG<H H>OABCD=EFGH. The false coin is in the third group.That is the problem l;which we achieve by 2 balancings.With N = Y2 (3" - 3) coins n balancings are needed.
AAamouli, al, 28Addition tables, magic squares, 182Ages, problems on, 42Alembert, d', 135Alfil, 278Algebraic magic series, 183Algebraic magic squares, 183, 187-192Ancient problems, 22-43Angles, arithmetical, 102-103Anthology, problems from, 23-26Arabian problems, 27-28Arbus, 14Aristodicus, 24Arithmetic, geometry and, 95-108Aubry, 177,214Augeas,24Automorphic numbers, 77-79BBaccara, 129Bachet de Meziriac, 52-55, 84, 149Ballot-box problem, 132Baskerville, chess innovations, 277Battle of numbers, 83-86Becq de Fouquiere, 279Beha Eddin Mohammed ben alRosain al Aamouli, 28Bernoulli, Daniel, 137-138Bernoulli, Nicholas, 137Bertrand, J., 129, 130, 136, 138Betting, systems, 131Bicolored pieces, 313, 315Bimagic, defined, 143Bimagic square, 176-177Binary notation, 52, 56-58Binomial expansion, checkers and,269Boije af Gennas, 44Border squares, 166-170INDEXBridge, hands, 119-121tournaments, 230-237Brooks, 198BufIon, 138CCaesar, Julius, 109-110Caius, 28--29Calendar, 109-116Caligula, 94Capablanca, chess innovations, 277Cards, Euler squares, 181guessing, 60, 313-323problems on, 316-323Cayley, 295Cazalas, 177Centaur, 278--279Central magic square, 189Chancellor, 278Charisius, 24Checkerboard, algebra, 271Checkers, 284-287, 315French, 285-286Japanese, 280Polish,287problems on, 269-271, 315-316Turkish, 287Chess, 280cylindrical, 277fairy, 276-279marked-pawn, 278number of games, 275-276problems on, 13, 238--256, 257-268,272-279supernumerary pieces, 278Chessboard, domination of, 253-256Chicandard, 312-313China, games, 279-280Chinese rings, 58, 89-91Chuquet, 22-23, 29Claus (see Lucas)Clavi us, 22-23Cole, 70Bosnins,14 Color8, eieven, square array, 252:~2(}
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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
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
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 326 and 327: Index 827Composite magic squares, 1
Page 328 and 329: Needle problem, 132Negative digits,
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