The Arithmetic-Geometric Mean Inequality 57
Positive Polynomials 59
Newton’s Method for Root Finding 60
More Division via Multiplication 63
The Allure of _2/6 64
Jacobian Conjectures 66
Chapter 3 The Number Three 69
The 3x + 1 Problem 69
Triangular Numbers and Bulgarian Solitaire 71
Rock-Paper-Scissors and Borromean Rings 73
Random Walks 74
Trisecting an Angle 77
The Three-Body Problem 78
The Lorenz Attractor and Chaos 81
Period Three Implies Chaos 83
Patterns among the Stars 85
Fermat’s Last Theorem 86
Leftovers Anyone? 89
Egyptian Fractions 90
Arrow’s Impossibility Theorem 93
Mapping Surfaces 95
Guarding an Art Gallery 96
The Poincaré Conjecture 97
Monge’s Three-Circle Theorem 100
Marden’s Theorem 100
The Reuleaux Triangle 103
Sums of Cubes 108
Approximating Decay 109
Chapter 4 The Number Four 111
The Four Color Theorem 111
The Tennis Ball Theorem 114
Sum of Squares Identities 114
Rearranging Four Pieces 115
Ducci Sequences 116
Euler’s Sum of Powers Conjecture 119
Villarceau Circles 122
The Inscribed Square Problem 123
Regular Polygons on a Computer Screen 124
The Four Travelers Problem 125
The Four Exponentials Conjecture 127
Concentric Quadrilaterals 129
The Four Hats Problem 131
Chapter 5 The Number Five 132
The Miquel Five Circles Theorem 132
The Platonic Solids 133
Solving Polynomial Equations 134
Diophantine Approximation 137
The Petersen Graph 138
The Happy Ending Problem 140
Tessellations 141
Of Balls and Sausages 143
Knights Tours on Rectangular Boards 145
Magic with Five Cards 145
Soccer Balls and Domes 148Recycling ad Infinitum 148
The Rogers–Ramanujan Identities 150
Chapter 6 The Number Six 156
Optimal Packing 156
Of Friends and Strangers 161
Six Degrees of Separation 161
A Necklace of Spheres 163
Hexagons in Pascal’s Triangle 164
The Game of Hex 165
TheWendt Determinant 167
Six Lengths in Geometry 168
Chapter 7 The Number Seven 170
The Seven Circles Theorem 170
Digits of 1/7 and Ellipses 171
Strassen’s Matrix Multiplication 173
The Fano Plane 175
Border Patterns 177
The Szilassi Polyhedron and the Heawood Graph 178The Kuratowski Closure–Complement Theorem 179
Can You Hear the Shape of a Drum? 182
Barker Codes 184
Recreational Mathematics 186
Experiments with Integrals 188
Chapter 8 The Number Eight 191
The Pizza Theorem 191
Shuffling Cards 192
The Game of Life 193
Repetition in Pascal’s Triangle 196<
The Sierpi ´nski Carpet 197
Quaternions and Octonions 198
The Summit of E8 202
Chapter 9 The Number nine 205
Nine Points and Collinearity 205
Casting out Nines 207
Primes and Nines 208
The Fifteen Theorem 209
Circle Packings with Two Sizes 210
Catalan’s Conjecture 211
The Heegner Numbers 213
Chapter 10 Solutions 216
Rearranging Four Pieces (Chapter 4) 216
The Four Hats Problem (Chapter 4) 216
The Kuratowski Closure–Complement Theorem
(Chapter 7) 217
Recreational Mathematics (Chapter 7) 217
