This page intentionally left blank the mathematical ideas that animate Great magic Tricks Persi Diaconis and Ron Graham With a foreword by Martin Gardner PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright © 2012 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Diaconis, Persi. Magical mathematics : the mathematical ideas that animate great magic tricks / Persi Diaconis, Ron Graham ; with a foreword by Martin Gardner. p. cm. Includes bibliographical references and index. ISBN 978-0-691-15164-9 (hardback) 1. Card tricks—Mathematics. I. Graham, Ron, 1950– II. Title. GV1549.D53 2011 793.8'5—dc23 2011014755 British Library Cataloging-in-Publication Data is available This book has been composed in ITC New Baskerville Printed on acid-free paper. ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 To our wives, Susan and Fan This page intentionally left blank Contents Foreword ix Preface xi Mathematics in the Air 1 1 Royal Hummer 8 Back to Magic 15 2 in cycles 17 The Magic of de Bruijn Sequences 18 Going Further 25 3 Is This StuFF Actually Good For Anything? 30 Robotic Vision 30 Making Codes 34 To the Core of Our Being 38 This de Bruijn Stuff Is Cool but Can It Get You a Job? 42 4 universal cycles 47 Order Matters 47 A Mind-reading Effect 52 Universal Cycles Again 55 5 From the Gilbreath Principle to the Mandelbrot Set 61 The Gilbreath Principle 61 The Mandelbrot Set 72 6 Neat Shuffles 84 A Mind-reading Computer 85 A Look Inside Perfect Shuffl es 92 viii Contents A Look Inside Monge and Milk Shuffl es 96 A Look Inside Down-and-Under Shuffl es 98 All the Shuffl es Are Related 99 7 The Oldest Mathematical Entertainment? 103 The Miracle Divination 105 How Many Magic Tricks Are There? 114 8 Magic in the Book of Changes 119 Introduction to the Book of Changes 121 Using the I Ching for Divination 122 Probability and the Book of Changes 125 Some Magic (Tricks) 127 Probability and the I Ching 136 9 What Goes Up Must Come Down 137 Writing It Down 138 Getting Started in Juggling 145 10 Stars of Mathematical Magic (and some of the best tricks in the book) 153 Alex Elmsley 156 Bob Neale 160 Henry Christ 173 Stewart James 181 Charles Thornton Jordan 189 Bob Hummer 201 Martin Gardner 211 11 Going further 220 12 on secrets 225 Notes 231 Index 239 foreword If you are not familiar with the strange, semisecret world of modern conjuring you may be surprised to know that there are thousands of entertaining tricks with cards, dice, coins, and other objects that re- quire no sleight of hand. They work because they are based on math- ematical principles. Consider, for example, what mathematicians call the Gilbreath Principle, named after Norman Gilbreath, its magician discoverer. Ar- range a deck so the colors alternate, red, black, red, black, and so on. Deal the cards to form a pile about equal to half the deck, then riffl e shuffl e the piles together. You’ll be amazed to fi nd that every pair of cards, taken from the top of the shuffl ed deck, consists of a red card and a black card! Dozens of beautiful card tricks—the best are ex- plained in this marvelous book—exploit the Gilbreath Principle and its generalizations. Although you can astound friends with tricks based on this princi- ple, they are in this book for another reason. The principle turned out to have applications far beyond trivial math. For example, it is closely related to the famous Mandelbrot set, an infi nite fractal pattern gener- ated on a computer screen by a simple formula. But that is not all. The Dutch mathematician N. G. de Bruijn dis- covered that the Gilbreath principle applies to the theory of Penrose tiles (two shapes that tile the plane only in a nonperiodic way) as well as to the solid form of Penrose tiles, which underlies what are called quasicrystals. Still another application of the principle, carefully ex- plained in this book, is to the design of computer algorithms for sort- ing procedures. The authors are eminent mathematicians. Ron Graham, retired from Bell Labs and now a professor at the University of California, San Diego, is an expert on combinatorial math. Persi Diaconis is an equally x Fo re w o rd famous statistician at Stanford University. Each man has a hobby. Ron is a top juggler. Persi is a skilled card magician. You will learn from their book the math properties of unusual shuf- fl es: the faro, the milk shuffl e, the Monge shuffl e, and the Austra- lian or down-and-under shuffl e. You will learn some tricks using the I Ching, an ancient Chinese fortune-telling volume. You will learn how parity (odd or even) can play a roll in magic as well as provide power- ful shortcut proofs. Not only is this book a superb, informally written introduction to mathematical magic, but near the book’s end the authors supply pic- tures and biographical sketches of magicians who have made the great- est contributions to mathematical magic, from the reclusive Charles Jordan to the eccentric Bob Hummer. Best of all, you will be introduced to many little-known theorems of advanced mathematics. The authors lead you from delightful self- working magic tricks to serious math, then back again to magic. It will be a long time before another book so clearly and entertainingly surveys the vast fi eld of mathematical hocus-pocus. Martin Gardner Norman, OK April, 2010 Preface The two of us have been mixing entertainment with mathematics for most of our lives. We started off on the entertainment side, one as a magician, the other as a juggler and trampolinist. We were seduced into studying mathematics by . well, the stories that are told in this book. Both of us now make a living doing mathematics; teaching, proving, and conjecturing. The two fi elds have been shuffl ed together for us, with frequently performed talks on mathematics and magic tricks and the mathemat- ics of juggling. The connections go deeper. Some magic tricks use “real mathematics” and lead to questions beyond the limits of mod- ern mathematics (see our chapter on shuffl ing cards). Sometimes, we have been able to solve the math problems and create new magic tricks (see chapter 2). Both of our worlds have a dense social structure; thousands of play- ers turning ideas over and over. Some of this wisdom of the ages is woven through our book. In addition to hundreds of friends and col- leagues, dozens of people have made sustained contributions to this book. On the magic front, Steve Freeman, Ricky Jay, Bob Neale, and Ron- ald Wohl have been coworkers, selfl essly contributing their brilliance and wisdom. The students in our Magic and Mathematics classes at Harvard and Stanford have all helped. We particularly thank Joe Fen- del. We had the benefi t of amazing, insightful readings of our text by Art Benjamin, Steve Butler, Colm Mulcahy, and Barry Mazur. Their combined comments rivaled the length of our book. Laurie Beckett, Michael Christ, Jerry Ferrell, Albrecht Heeffer, Bill Kalush, Mitsunobu Matsuyama, and Sherry Wood went out of their way to help us out. Our editors at Princeton University Press, Ed Tenner, Vickie Kearn, and Mark Bellis have been crucial allies. xii Preface Our families, Fan Chung Graham, Ché Graham, and Susan Holmes, have helped in so many ways that we can’t fi nd a number system rich enough to list them. Fan’s mathematical work appears in chapters 2–4, and Ché and Susan shot (and reshot) numerous photos. Susan also contributed to the history and many other chapters. We hope that our book will shine a friendly light on the corners of the world that are our homes. Thanks and welcome. Persi Diaconis and Ron Graham This page intentionally left blank Chapter 1 Mathematics in the Air Most mathematical tricks make for poor magic and in fact have very little mathematics in them. The phrase “mathematical card trick” con- jures up visions of endless dealing into piles and audience members wondering how long they will have to sit politely. Our charge is to present entertaining tricks that are easy to perform and yet have in- teresting mathematics inside them. We cannot do this without your help. To get started, please go fi nd four playing cards. They can be any four cards, all different or the four aces. It doesn’t matter. Let us begin by performing the trick for you. Since we can do it without being present, you’ll be able to do it for a friend on the Figure 1. Four cards Figure 2. Four cards in a packet phone. After practicing, try call- ing your kid brother or your mom and perform the following. Have a look at the bottom card of the packet. That’s your card and you have to remember it. Figure 3. Looking at bottom card 2 Chapter 1 Next, the cards are going to be mixed by some simple in- structions. Put the top card on the bottom of the packet. Turn the current top card face-up and place it back on top. Figure 4. Top card placed on bottom Figure 5. Current top card turned face-up Now, give the packet a cut.