The Second Book of Mathematical Puzzles and Diversions

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ing and analyzing these four-sided forms, but did not suc- ceed in developing a comprehensive theory that would cover all their discordant variations. Several species of tetraflexagon are nonetheless intensely interesting from the recreational standpoint. Consider first the simplest tetraflexagon, a three-faced structure which can be called the tri-tetraflexagon. It is easily folded from the strip of paper shown in Figure 8 (8a is the front of the strip; 8b, the back). Number the small squares on each side of the strip as indicated, fold both ends inward (8c) and join two edges with a piece of transparent tape (8d). Face 2 is now in front; face 1 is in back. To flex the structure, fold it back along the vertical center line of face 2. Face 1 will fold into the flexagon's interior as face 3 flexes into view. FIG. 8. How to make a tri-tet~aflexagon. Stone and his friends were not the first to discover this interesting structure; it has been used for centuries as a double-action hinge. I have on my desk, for instance, two small picture frames containing photographs. The frames
26 Tetrnflezagons are joined by two tri-tetraflexagon hinges which permit the frames to flex forward or backward with equal ease. The same structure is involved in several children's toys, the most familiar of which is a chain of flat wooden or plas- tic blocks hinged together with crossed tapes. If the toy is manipulated properly, one block seems to tumble down the chain from top to bottom. Actually this is an optical illusion created by the flexing of the tri-tetraflexagon hinges in serial order. The toy was popular in the U. S. during the 1890's, when it was called Jacob's Ladder. (A picture and description of the toy appear in Albert A. Hopkins's Magic: Stage Illusions and Scientific Diversions, 1897.) Two cur- rent models sell under the trade names Klik-Klak Blox and Flip Flop Blocks. There are at least six types of four-faced tetraflexagons, known as tetra-tetraflexagons. A good way to make one is to start with a rectangular piece of thin cardboard ruled into 12 squares. Number the squares on both sides as depicted in Figure 9 (9a and 9b). Cut the rectangle along the broken lines. Start as shown in 9a, then fold the two center squares back and to the left. Fold back the column on the extreme right. The cardboard should now appear as shown in 9c. Again fold back the column on the right. The single square projecting on the left is now folded forward and to the right. This brings all six of the "1" squares to the front. Fasten together the edges of the two middle squares with a piece of transparent tape as shown in 9d. You will find it a simple matter to flex faces 1, 2, and 3 into view, but finding face 4 may take a bit more doing. Naturally you must not tear the cardboard. Higher-order tetraflexagons of this type, if they have an even number of faces, can be constructed from similar rectangular starting patterns; tetraflexagons with an odd number of faces call for patterns analogous to the one used for the tri-tetraflexagon. Actually two rows of small squares are sufficient for
Page 1 and 2: T H E S E C O N D SCIENTIFCC - L ME
Page 3 and 4: The 2nd SCIENTIFIC AMERICAN Book of
Page 5 and 6: Material previously published in Sc
Page 7 and 8: INTRODUCTION CONTENTS The Five Plat
Page 9 and 10: INTRODUCTION SINCE THE APPEARANCE o
Page 11 and 12: Introduction 11 socialism and capit
Page 13 and 14: TETRAHEDRON / The Five Platonic Sol
Page 15 and 16: 16 The Five Platonic Solids both la
Page 17 and 18: 18 The Five Platonic Solids value o
Page 19 and 20: 20 The Five Platonic Solids tance i
Page 21 and 22: 22 Tlre Five Platonic Solids ANSWER
Page 23: CHAPTER TWO w Tetrajlexagons H EXAF
Page 27 and 28: A different variety of tetraflexago
Page 29 and 30: FIG. 11. How to make and flex the f
Page 31 and 32: C H A P T E R THREE Henry Ernest Du
Page 33 and 34: 34 Henry Ernest Dudeney: England's
Page 35 and 36: 36 Henry Ernest Llzcdeney: Englnvd'
Page 43 and 44: I)i,qitnl Roots FIG. 19. Symbols fo
Page 45 and 46: 46 Digital Roots Since the answer i
Page 47 and 48: 48 Digital Roots what cards are at
Page 49 and 50: Digital Roots ANSWERS IN THE GAME p
Page 51 and 52: FIG. 21. The twiddled bolts. planes
Page 53 and 54: 54 Nine Problenzs two units that is
Page 55 and 56: 56 Nine Problems depicted at bottom
Page 57 and 58: Nine Problems FIG. 24. The flight a
Page 59 and 60: 60 Nine Problems men who shook hand
Page 61 and 62: 62 Nine Problems As we saw earlier,
Page 63 and 64: 64 Nine Pro Blems against his own b
Page 65 and 66: 66 The Soma Cube Tix, are discussed
Page 67 and 68: 68 The Soma Cube set of the pieces
Page 69 and 70: WALL SKYSCRAPER DOG PYRAMID FIG. 29
Page 71 and 72: the top illustration of Figure 30.
Page 73 and 74: 74 The Soma Cube
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76 The Soma Cuhe Some of these figu
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CHAPTER SEVEN Recreational Topology
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Recreational Topology FIG. 33. Stew
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Ree~eational Topology FIG. 34. The
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84 Recreational Topology has taken
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86 Rec~eational Topology game is pl
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88 Recreational Topology ANSWERS TH
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90 Phi: The Golden Ratio is commonl
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Phi: The Golden Ratio FIG. 39. The
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94 Phi: The Golden Ratio volved in
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96 Phi: The Golden Ratio There is a
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Phi: The Golden Ratio THE SACRAMENT
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Phi: The Golden Ratio 101 culate ph
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Phi: The Golden Ratio 103 The two e
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The Monkeg and the Coconuts 105 foo
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The Monkey and the Coconuts 107 Dir
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The Monkey and the Coconuts 109 the
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The Monkey and the Coconuts 11 1 le
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Woodstock in the 12th century by Ki
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FIG. 49. A "simply connected" maze
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Mazes 117 tiply connected maze whic
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A CHAPTER E L E V E N rn Recreation
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Recreational Logic 121 man, so we p
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Recreational Logic 123 1. In 1918,
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Recreational Logic 125 truth. Since
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Recreational Logic 127 stand" or "W
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Recrecrtionnl Logic 129 solution is
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Magic Squares 131 order three. An e
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Magic Squares 133 FIG. 54. Albrecht
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Magic Squares 135 can be restored b
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Magic Sqzlnres 137 seen by transfer
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Magic Squares 139 in Rochester, New
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C H A P T E R T H I R T E E N James
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James Hugh Riley Shows, Inc. 143 fe
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James Hugh Riley Shows, Inc. 145 ha
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James Hugh Riley Shou.s, Inc. 147 "
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James Hugh Riley Shows, Inc. 149 ha
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James Hug11 Riley Shows, Inc. 151 e
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Nine More Prohlems 153 Mr. Jones ha
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Nine More Problems 155 ing it the t
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Nine Afore Problems 157 a journey o
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Nine More Prohlerns 159 (This was t
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Nine More Problems 161 (which gives
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Nine More Problems 163 We know that
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C H A P T E R F I F T E E N Eleusis
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Eleusis : The Induction Game 167 pl
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Eleusis : The Induction Game 169 To
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Eleusis: The Indz~ction Game 171 Th
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Eleusis : The Induction Game 173 co
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Origami 175 past 20 years there has
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Paper can also be folded to produce
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Origami 179 At what spot along the
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Origami 181
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ADDENDUM SINCE this chapter appeare
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Origami 185 crease. If the value to
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Squaring the Square 187 H. Stone, n
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FIG. 73. Sqzcnring the Sqziare FIG.
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Squaring the Square 191 gram. But S
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Squaring the Square 193 out any ref
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Squaring the Square 195 and eventua
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Squaring the Sqz~are 112 FIG. 78. 3
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Squaring the Square 199 But I thoug
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Squaring the Square 201 each diagra
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Squaring the Square 203 FIG. 82. si
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Squaring the Square 205 team solved
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Squaring the Square 207 first is to
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Squaring the Square 209 fifth-dimen
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Mechanical Puzzles 21 1 about 2,000
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Mecl~nnicnl Puzzles 213 Dissection
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Mechanical Puzzles 215 is shown in
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Mechanical Puzzles 217 ulated a cer
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Mechanical Puzzles 219 symmetrical
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Probability and Ambiguity 22 1 ple
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Probability and Amhig~iity 223 now
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Probability and Ambiguity 225 study
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Probability and Ambiguity 227 that
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Probability and Ambiguity 229 David
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oric, or 213, a~itl A's arc 113. In
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CHAPTER TWENTY w The Mysterious Dr.
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The Mysterious Dr. Matrix FIG. 91.
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The Mysterious Dr. Matrix 237 the m
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The Mysterious DT; Matrix 239 denti
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The Mysterious Dr. Matrix 241 read
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The Mysterious Dr. Matrix 243 Note
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246 References for Further Reading
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POSTSCRIPT, 1987 No EFFORT has been
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ABOUT THE AUTHOR MARTIN GARDNER has
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