Mathematical_Recreations-Kraitchik-2e

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Geometric Recreations~OlMosaic ofModified Triangles.FIGURE 78.FIGURE 79. Mosaic ofModified Triangles.FIGURE 80. Mosaic ofModified Squares.Mosaic ofModified Squares.FIGURE 81.FIGURE 82. Mosaic ofModified Hexagons.FIGURE 83. Mosaic ofModified Hexagons.
Mathematical Recreationsthat is, n = 3 or 4. Both of these cases can be realized, sayby sliding the horizontal rows of black and white triangles inFigure 75 past each other, and similarly for the horizontalrows of squares.Thus we have the following theorem: If the plane is completelycovered by a set of congruent regular polygons, these polygonscan have only 3, 4, or 6 sides. Only in the last case mustthe polygons meet solely at vertices.If the squares be given different colors in the regions determinedby a diagonal they may be arranged so as to givevery beautiful mosaics, as in Figure 77.The shapes of the polygons may be altered in various waysand then put together as before to form many interestingnew designs (Figures 78-83).2. One may also use regular polygons of different kinds ata vertex. Clearly we cannot have fewer than 3 or more than6 polygons at any vertex. The resulting arrangements arecalled ternary, quarternary, quinary, and senary, respectively.Since the sum of the angles around any vertex must be3600 a ternary arrangement of polygons of nl, 1't2, and na sidesrespectively will be possible only ifnl -e2 + 1't2 - 2 + na - 2)1800 = 3600,nl 1't2 nafrom which we find the condition1 1 1 1-+--+-= -.nl ~ na 2In the same manner we find the following conditions forthe other possible arrangements:1 1 1 1-+-+-+-=1.nl 1't2 na n.
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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
Page 150 and 151: Magic Squares 15101-1 -1-1-1-1 "-1-
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
Page 202 and 203: Geometric Recreations111 1 1 3-+-+-
Page 204 and 205: Geometric Recreations ~05have edges
Page 206 and 207: Geometric Recreations 207solutions,
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
Page 244 and 245: The Problem of the Queens 245Such a
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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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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