Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page ii This book has been optimized for viewing at a monitor setting of 1024 x 768 pixels. Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page i SECRE+S OF + + MEN AL MA H Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page ii Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page iii + SECRE S OF + + MEN AL MA H The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks Arthur Benjamin andMichael Shermer Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page iv 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 Mathemagicsby 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 Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page v 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 Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page vi Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page vii 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— cognitivepsychologistWilliamG.Chase,magiciansPaulGertner 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, anddaughters,LaurelandAriel,forconstantinspiration. Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page viii