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Single Digits: in Praise of Small Numbers March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. ...................................contents preface xi Chapter 1 The Number One 1 Sliced Origami 1 Fibonacci Numbers and the Golden Ratio 2 Representing Numbers Uniquely 5 Factoring Knots 6 Counting and the Stern Sequence 8 Fractals 10 Gilbreath’s Conjecture 13 Benford’s Law 13 The Brouwer Fixed-Point Theorem 16 Inverse Problems 17 Perfect Squares 19 The Bohr–Mollerup Theorem 19 The Picard Theorems 21 Chapter 2 The Number Two 24 The Jordan Curve Theorem and Parity Arguments 24 Aspect Ratio 26 How Symmetric Are You? 27 For general queries, contact [email protected] March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. vi contents The Pythagorean Theorem 29 Beatty Sequences 32 Euler’s Formula 33 Matters of Prime Importance 34 The Ham Sandwich Theorem 38 Power Sets and Powers of Two 39 The Sylvester–Gallai Theorem 42 Formulas for 43 Multiplication 44 The Thue–Morse Sequence 45 Duals 48 Apollonian Circle Packings 51 Perfect Numbers and Mersenne Primes 53 Pythagorean Tuning and the Square Root of 2 54 Inverse Square Laws 56 The Arithmetic-Geometric Mean Inequality 57 Positive Polynomials 59 Newton’s Method for Root Finding 60 More Division via Multiplication 63 The Allure of 2/664 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 For general queries, contact [email protected] March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. contents vii 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 The Third Critical Point 106 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 Knight’s Tours on Rectangular Boards 145 Magic with Five Cards 145 For general queries, contact [email protected] March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. viii contents Soccer Balls and Domes 148 Recycling 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 The Wendt 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 178 The 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 Sierpinski´ Carpet 197 Quaternions and Octonions 198 The Summit of E8 202 For general queries, contact [email protected] March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. contents ix 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 further reading 219 credits for illustrations 223 index 225 For general queries, contact [email protected].
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