SECRE+S

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SECRE+S

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MEN+AL MA+H

The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks

Arthur Benjamin and Michael Shermer

Copyright © 2006 by Arthur Benjamin and Michael Shermer All rights reserved. Published in the United States by Three Rivers Press, an imprint of the Crown Publishing Group, a division of Random House, Inc., New York. www.crownpublishing.com Originally published in different form as Mathemagics by Lowell House, Los Angeles, in 1993. Three Rivers Press and the Tugboat design are registered trademarks of Random House, Inc. Library of Congress Cataloging-in-Publication Data Benjamin, Arthur. Secrets of mental math : the mathemagician’s guide to lightning calculation and amazing math tricks / Arthur Benjamin and Michael Shermer.— 1st ed. p. cm. Includes bibliographical references and index. 1. Mental arithmetic—Study and teaching. 2. Magic tricks in mathematics education. 3. Mental calculators. I. Shermer, Michael. II. Title. QA111.B44 2006 510—dc22 2005037289 eISBN: 978-0-307-34746-6 v1.0

I dedicate this book to my wife, Deena, and daughters, Laurel and Ariel. —Arthur Benjamin

My dedication is to my wife, Kim, for being my most trusted conﬁdante and personal counselor. —Michael Shermer

Acknowledgments

The authors wish to thank Steve Ross and Katie McHugh at Random House for their support of this book. Special thanks to Natalya St. Clair for typesetting the initial draft, which was partly supported by a grant from the Mellon Foundation. Arthur Benjamin especially wants to acknowledge those who inspired him to become both a mathematician and a magician— cognitive psychologist William G. Chase, magicians Paul Gertner and James Randi, and mathematicians Alan J. Goldman and Edward R. Scheinerman. Finally, thanks to all of my colleagues and students at Harvey Mudd College, and to my wife, Deena, and daughters, Laurel and Ariel, for constant inspiration.

Contents

Foreword by Bill Nye (the Science Guy®) Foreword by James Randi Prologue by Michael Shermer Introduction by Arthur Benjamin Chapter 0 Quick Tricks:

xi xvii xix xxiii

Easy (and Impressive) Calculations

Chapter 1 A Little Give and Take:

1

Mental Addition and Subtraction

Chapter 2 Products of a Misspent Youth:

11

Basic Multiplication

Chapter 3 New and Improved Products:

29

Intermediate Multiplication

Chapter 4 Divide and Conquer:

53

Mental Division

Chapter 5 Good Enough:

80

The Art of “Guesstimation”

108

x Contents

Chapter 6 Math for the Board:

Pencil-and-Paper Math

Chapter 7 A Memorable Chapter:

131

Memorizing Numbers

Chapter 8 The Tough Stuff Made Easy:

151

Advanced Multiplication

Chapter 9 Presto-digit-ation:

163

The Art of Mathematical Magic

Chapter

199

Epilogue by Michael Shermer: How Math Helps Us Think About Weird Things Answers Bibliography Index

222 233 271 273

Foreword

by Bill Nye (the Science Guy®)

I like to think about the ﬁrst humans, the people who came up with the idea to count things. They must have noticed right away that ﬁguring on your ﬁngertips works great. Perhaps Og (a typical ancient cave guy) or one of his pals or associates said, “There are one, two, three, four, ﬁve of us here, so we need ﬁve pieces of fruit.” Later, “Hey, look,” someone must have said (or grunted), “you can count the number of people at the campﬁre, the number of birds on a tree, stones in a row, logs for a ﬁre, or grapes in a bunch, just with your ﬁngers.” It was a great start. It’s probably also how you came to ﬁrst know numbers. You’ve probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called...