| Preface |
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xi | |
| Topics Explored |
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xiii | |
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Essay 1 Dragons and Poison |
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1 | (4) |
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2 | (3) |
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Essay 2 Folding Tetrahedra |
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5 | (4) |
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5 | (4) |
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9 | (8) |
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10 | (1) |
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3.2 The Area of the Arbelos |
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10 | (1) |
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3.3 The Archimedean Circles |
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11 | (1) |
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3.4 The Other Common Tangent Segment |
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12 | (1) |
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13 | (4) |
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Essay 4 Averages via Distances |
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17 | (6) |
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17 | (1) |
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4.2 Using the Distance Formula: Euclidean Distance |
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17 | (1) |
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4.3 Using the Taxicab Metric |
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18 | (2) |
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20 | (3) |
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23 | (6) |
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23 | (4) |
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5.2 Connections to the Opening Puzzler |
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27 | (2) |
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29 | (8) |
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6.1 Two Observations about Triangles |
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30 | (1) |
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31 | (2) |
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6.3 Solving Feynman's Problem without the Big Guns |
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33 | (4) |
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37 | (8) |
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7.1 The Answer to the First Puzzler |
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39 | (1) |
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7.2 Towards Answering the Second Puzzler |
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40 | (1) |
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41 | (4) |
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45 | (8) |
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8.1 Leibniz's Harmonic Triangle |
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46 | (1) |
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8.2 The Infinite Stocking Property |
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47 | (1) |
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8.3 Variations of Leibniz's Harmonic Triangle |
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48 | (1) |
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8.4 Solving the Opening Puzzle |
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49 | (4) |
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Essay 9 Flipping a Coin for a Year |
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53 | (6) |
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9.1 Solutions to Coin Tossing 1 |
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54 | (2) |
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9.2 Solutions to Coin Tossing 2 |
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56 | (3) |
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Essay 10 Coinciding Digits |
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59 | (4) |
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10.1 The Chinese Remainder Theorem |
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59 | (2) |
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61 | (2) |
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63 | (4) |
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64 | (3) |
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Essay 12 Gauss's Shoelace Formula |
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67 | (14) |
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12.1 Step 1: Nicely Situated Triangles |
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71 | (2) |
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12.2 Step 2: General Triangles |
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73 | (1) |
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12.3 Step 3: Begin Clear of the Effect of Motion |
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74 | (1) |
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12.4 Step 4: Being Clear on Starting Points |
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75 | (1) |
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12.5 Step 5: Steps 1 and 2 Were Unnecessary! |
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76 | (1) |
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12.6 Step 6: Quadrilaterals |
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76 | (2) |
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12.7 Step 7: Beyond Quadrilaterals |
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78 | (3) |
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Essay 13 Subdividing a Square into Triangles |
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81 | (8) |
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82 | (1) |
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13.2 The Impossibility Proof |
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83 | (1) |
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13.3 Case 1: N is an Odd Integer |
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84 | (1) |
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13.4 Case 2: N is an Even Integer |
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85 | (4) |
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Essay 14 Equilateral Lattice Polygons |
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89 | (10) |
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14.1 Areas of Lattice Polygons |
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90 | (1) |
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14.2 Aside: Pick's Theorem |
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91 | (1) |
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14.3 Equilateral Lattice Polygons |
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92 | (1) |
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14.4 The Answer with Cheating |
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93 | (1) |
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94 | (5) |
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Essay 15 Broken Sticks and Viviani's Theorem |
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99 | (10) |
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100 | (1) |
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15.2 Broken Sticks and Triangles |
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101 | (3) |
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15.3 Something Unsettling |
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104 | (1) |
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15.4 Focus on the Left Piece |
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105 | (2) |
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15.5 Summing Probabilities |
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107 | (1) |
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108 | (1) |
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Essay 16 Viviani's Converse? |
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109 | (8) |
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16.1 Planes above Triangles |
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109 | (1) |
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16.2 The Equation of a Plane |
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110 | (1) |
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16.3 The Distance of a Point from a Line |
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111 | (3) |
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16.4 The Converse of Viviani's Theorem |
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114 | (1) |
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115 | (2) |
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Essay 17 Integer Right Triangles |
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117 | (10) |
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17.1 A Cute Way to Find Pythagorean Triples |
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118 | (2) |
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120 | (1) |
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17.3 The Answers to All the Curiosities |
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120 | (7) |
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Essay 18 One More Question about Integer Right Triangles |
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127 | (4) |
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18.1 A Precursor Question |
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127 | (1) |
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18.2 The Answer to the Main Question |
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128 | (3) |
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Essay 19 Intersecting Circles |
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131 | (10) |
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132 | (6) |
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138 | (1) |
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139 | (2) |
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Essay 20 Counting Triangular and Square Numbers |
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141 | (8) |
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20.1 Some Interplay between Square and Triangular Numbers |
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142 | (1) |
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143 | (1) |
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20.3 Counting Figurate Numbers |
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144 | (2) |
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20.4 The Squangular Numbers |
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146 | (1) |
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147 | (2) |
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149 | (10) |
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21.1 On Sums of Consecutive Counting Numbers |
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149 | (5) |
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154 | (2) |
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156 | (3) |
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Essay 22 The Prouhet--Thue--Morse Sequence |
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159 | (10) |
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22.1 The Prouhet--Thue--Morse Sequence |
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159 | (1) |
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160 | (2) |
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22.3 Alternative Constructions |
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162 | (3) |
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165 | (4) |
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Essay 23 Some Partition Numbers |
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169 | (8) |
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23.1 The Partition Numbers |
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169 | (1) |
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23.2 Partitions into a Fixed Number of Parts |
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170 | (1) |
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23.3 Cracking the P3(n) Formula |
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171 | (6) |
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Essay 24 Ordering Colored Fractions |
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177 | (10) |
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24.1 Coloring and Ordering Fractions |
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177 | (2) |
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179 | (1) |
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180 | (1) |
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24.4 Explaining Colored Fractions |
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181 | (4) |
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185 | (1) |
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186 | (1) |
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Essay 25 How Round Is a Cube? |
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187 | (14) |
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25.1 Deficiencies in Surface Circles |
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188 | (2) |
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190 | (2) |
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25.3 A Better Way to Count Total Pointiness |
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192 | (4) |
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25.4 Shaving Corners Does Not Help! |
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196 | (1) |
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25.5 Not All Shapes Are Sphere-like! |
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197 | (1) |
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25.6 Christopher Columbus and Others |
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198 | (3) |
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Essay 26 Base and Exponent Switch |
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201 | (6) |
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26.1 The Graph of y = x1/x for x < 0 |
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201 | (1) |
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26.2 The Graph of xy = yx for x < 0, y < 0 |
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202 | (1) |
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26.3 A Connection to wwww |
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203 | (2) |
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26.4 Appendix: A Tricky Swift Proof |
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205 | (2) |
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Essay 27 Associativity and Commutativity Puzzlers |
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207 | (8) |
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208 | (2) |
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27.2 Explaining the Puzzler and Its Variations |
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210 | (5) |
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Essay 28 Very Triangular and Very Very Triangular Numbers |
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215 | (4) |
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215 | (2) |
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28.2 Very Triangular Numbers |
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217 | (2) |
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219 | (12) |
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29.1 The Equation of a Torus |
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220 | (2) |
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29.2 Slicing an Ideal Bagel: The Puzzler |
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222 | (6) |
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29.3 The Four Circles Property |
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228 | (3) |
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Essay 30 Trapezoidal Numbers |
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231 | (8) |
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231 | (2) |
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30.2 From Rectangles to Trapezoids |
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233 | (1) |
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234 | (2) |
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30.4 Counting Presentations |
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236 | (3) |
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Essay 31 Square Permutations |
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239 | (6) |
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239 | (3) |
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242 | (3) |
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Essay 32 Tupper's Formula |
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245 | (6) |
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32.1 Understanding the Notation |
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245 | (1) |
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32.2 The mod Function in Computer Science |
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246 | (1) |
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247 | (2) |
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32.4 Delightful Recursive Quirkiness |
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249 | (1) |
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32.5 How to Find a Particular Picture |
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250 | (1) |
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250 | (1) |
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Essay 33 Compositional Square Roots |
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251 | (6) |
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33.1 Compositional Square Roots |
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251 | (2) |
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33.2 Constructing Compositional Square Roots |
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253 | (4) |
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Essay 34 Polynomial Permutations |
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257 | |
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34.1 Playing with Polynomials |
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258 | |
