<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w0 h1" alt="" src="https://files.passeidireto.com/0c97285c-937f-4d86-ae93-bd07dc808b15/bg1.png"></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w1 h2" data-page-no="2"><div class="pc pc2 w1 h2"><div class="t m0 x1 h3 y1 ff1 fs0 fc0 sc0 ls0 ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls0">5/4/06 <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls0">1:37 <span class="_0 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls0">Page <span class="_0 blank"></span>ii</span></span></span></span></span></span></div><div class="t m0 x2 h4 y2 ff2 fs1 fc0 sc0 ls2 ws1">This book has been optimized for viewing<span class="fs2"> </span></div><div class="t m0 x3 h4 y3 ff2 fs1 fc0 sc0 ls3 ws2">at a monitor setting <span class="ls4 ws3">of 1024 x 768 pixels<span class="fs2">. </span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf3" class="pf w1 h2" data-page-no="3"><div class="pc pc3 w1 h2"><img fetchpriority="low" loading="lazy" class="bi x4 y4 w2 h5" alt="" src="https://files.passeidireto.com/0c97285c-937f-4d86-ae93-bd07dc808b15/bg3.png"><div class="t m1 x5 h6 y5 ff3 fs3 fc0 sc0 ls5">SECRE</div><div class="t m0 x6 h7 y6 ff3 fs4 fc1 sc0 ls1">+</div><div class="t m1 x7 h6 y5 ff3 fs3 fc0 sc0 ls1">S</div><div class="t m0 x8 h8 y7 ff3 fs5 fc0 sc0 ls6">OF</div><div class="t m1 x9 h6 y8 ff3 fs3 fc0 sc0 ls7">MEN</div><div class="t m0 xa h7 y9 ff3 fs4 fc1 sc0 ls1">+</div><div class="t m1 xb h6 y8 ff3 fs3 fc0 sc0 ls8 ws4">AL MA</div><div class="t m0 xc h7 y9 ff3 fs4 fc1 sc0 ls1">+</div><div class="t m1 xd h6 y8 ff3 fs3 fc0 sc0 ls1">H</div><div class="t m0 x1 h3 y1 ff4 fs0 fc2 sc0 ls9 ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls9">5/4/06 <span class="_2 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls9">1:37 <span class="_0 blank"></span>PM <span class="_2 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls9">Page <span class="_0 blank"></span><span class="ls1">i</span></span></span></span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf4" class="pf w1 h2" data-page-no="4"><div class="pc pc4 w1 h2"><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls0 ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls0">5/4/06 <span class="_1 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls0">1:37 <span class="_0 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls0">Page <span class="_0 blank"></span>ii</span></span></span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf5" class="pf w1 h2" data-page-no="5"><div class="pc pc5 w1 h2"><img fetchpriority="low" loading="lazy" class="bi xe ya w3 h9" alt="" src="https://files.passeidireto.com/0c97285c-937f-4d86-ae93-bd07dc808b15/bg5.png"><div class="t m0 xf ha yb ff3 fs6 fc0 sc0 lsa">SECRE</div><div class="t m0 x7 hb yc ff3 fs7 fc1 sc0 ls1">+</div><div class="t m0 x10 ha yb ff3 fs6 fc0 sc0 ls1">S</div><div class="t m0 x11 hc yd ff3 fs8 fc0 sc0 lsb">OF</div><div class="t m0 xe ha ye ff3 fs6 fc0 sc0 lsc">MEN</div><div class="t m0 x12 hb yf ff3 fs7 fc1 sc0 ls1">+</div><div class="t m0 x13 ha ye ff3 fs6 fc0 sc0 lsd ws6">AL MA</div><div class="t m0 x14 hb yf ff3 fs7 fc1 sc0 ls1">+</div><div class="t m0 x15 ha ye ff3 fs6 fc0 sc0 ls1">H</div><div class="t m0 x16 hd y10 ff5 fs9 fc0 sc0 lse ws0">The Mathemagician\u2019<span class="_3 blank"></span>s Guide to Lightning Calculation </div><div class="t m0 x17 hd y11 ff5 fs9 fc0 sc0 lse ws7">and Amazing <span class="_4 blank"> </span>Math <span class="_5 blank"></span>T<span class="_2 blank"></span>ricks</div><div class="t m0 x12 he y12 ff3 fsa fc0 sc0 lsf ws0">Ar<span class="_6 blank"> </span>thur Benjamin </div><div class="t m0 x18 hf y13 ff5 fsb fc0 sc0 ls10 ws5">and <span class="ff3 fsa lsf ws0 v1">Michael Shermer</span></div><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls11 ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls11">5/4/06 <span class="_1 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls11">1:37 <span class="_0 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls11">Page <span class="_1 blank"></span>iii</span></span></span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf6" class="pf w1 h2" data-page-no="6"><div class="pc pc6 w1 h2"><div class="t m0 x19 h10 y14 ff6 fs0 fc2 sc0 ls12 ws0">Copyright © 2006 by Arthur Benjamin and Michael Shermer</div><div class="t m0 x19 h10 y15 ff6 fs0 fc2 sc0 ls12 ws0">All rights reserved.</div><div class="t m0 x19 h10 y16 ff6 fs0 fc2 sc0 ls12 ws0">Published in the United States by Three Rivers Press, an imprint of the Crown</div><div class="t m0 x19 h10 y17 ff6 fs0 fc2 sc0 ls12 ws0">Publishing Group, a division of Random House, Inc., New Y<span class="_7 blank"></span>ork.</div><div class="t m0 x19 h10 y18 ff6 fs0 fc3 sc0 ls12 ws8">www<span class="_7 blank"></span>.crownpublishing.com</div><div class="t m0 x19 h10 y19 ff6 fs0 fc2 sc0 ls12 ws0">Originally published in different form as <span class="ff7 ws9">Mathemagics </span>by Lowell House, </div><div class="t m0 x19 h10 y1a ff6 fs0 fc2 sc0 ls12 ws0">Los Angeles, in 1993.</div><div class="t m0 x19 h10 y1b ff6 fs0 fc2 sc0 ls12 ws0">Three Rivers Press and the T<span class="_7 blank"></span>ugboat design are registered trademarks of Random</div><div class="t m0 x19 h10 y1c ff6 fs0 fc2 sc0 ls12 ws0">House, Inc.</div><div class="t m0 x19 h10 y1d ff6 fs0 fc2 sc0 ls12 ws0">Library of Congress Cataloging-in-Publication Data</div><div class="t m0 x19 h10 y1e ff6 fs0 fc2 sc0 ls12 ws0">Benjamin, Arthur<span class="_3 blank"></span>.</div><div class="t m0 x2 h10 y1f ff6 fs0 fc2 sc0 ls12 ws0">Secrets of mental math : the mathemagician\u2019<span class="_7 blank"></span>s guide to lightning calculation and</div><div class="t m0 x19 h10 y20 ff6 fs0 fc2 sc0 ls12 ws0">amazing math tricks /<span class="_8 blank"> </span>Arthur Benjamin and Michael Shermer<span class="_3 blank"></span>.<span class="_6 blank"> </span>\u2014<span class="_9 blank"> </span>1st ed.</div><div class="t m0 x1a h10 y21 ff6 fs0 fc2 sc0 ls13 ws0">p. cm.</div><div class="t m0 x3 h10 y22 ff6 fs0 fc2 sc0 ls12 ws0">Includes bibliographical references and index.</div><div class="t m0 x2 h10 y23 ff6 fs0 fc2 sc0 ls12 ws0">1.<span class="_a blank"> </span>Mental arithmetic<span class="_6 blank"> </span>\u2014<span class="_6 blank"> </span>Study and teaching.<span class="_b blank"> </span>2.<span class="_a blank"> </span>Magic tricks in mathematics</div><div class="t m0 x19 h10 y24 ff6 fs0 fc2 sc0 ls12 ws0">education.<span class="_b blank"> </span>3.<span class="_a blank"> </span>Mental calculators.<span class="_c blank"> </span>I.<span class="_a blank"> </span>Shermer<span class="_3 blank"></span>, Michael.<span class="_b blank"> </span>II.<span class="_a blank"> </span>T<span class="_5 blank"></span>itle.</div><div class="t m0 x3 h10 y25 ff6 fs0 fc2 sc0 ls12 wsa">QA111.B44 2006</div><div class="t m0 x3 h10 y26 ff6 fs0 fc2 sc0 ls12 wsb">510 \u2014 dc22</div><div class="t m0 x19 h10 y27 ff6 fs0 fc2 sc0 ls12">2005037289</div><div class="t m0 x19 h11 y28 ff8 fs0 fc2 sc0 ls12">eI<span class="ff9">SBN<span class="ffa">:</span><span class="ws0"> 978-0-307-<span class="ffb">34746-6</span></span></span></div><div class="t m0 x19 h11 y29 ffb fs0 fc2 sc0 ls12 ws0">v1.0 </div><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls14 ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls14">5/4/06 <span class="_2 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls14">1:37 <span class="_2 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls14">Page <span class="_2 blank"></span>iv</span></span></span></span></span></span></div><a class="l"><div class="d m2" style="border-style:none;position:absolute;left:46.347000px;bottom:358.793000px;width:96.240000px;height:8.767000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf7" class="pf w1 h2" data-page-no="7"><div class="pc pc7 w1 h2"><img fetchpriority="low" loading="lazy" class="bi xe y2a w4 h12" alt="" src="https://files.passeidireto.com/0c97285c-937f-4d86-ae93-bd07dc808b15/bg7.png"><div class="t m0 x4 h13 y2b ff5 fsc fc0 sc0 ls15 wsc">I dedicate this book to m<span class="_5 blank"></span>y wife,<span class="_8 blank"> </span>Deena,</div><div class="t m0 x1b h13 y2c ff5 fsc fc0 sc0 ls15 ws0">and daughters,<span class="_8 blank"> </span>Laurel and <span class="_3 blank"></span>Ariel.</div><div class="t m0 x1c h14 y2d ff5 fsd fc0 sc0 ls16 ws0">\u2014Arthur Benjamin</div><div class="t m0 x1d h13 y2e ff5 fsc fc0 sc0 ls15 wsc">My dedication is to m<span class="_5 blank"></span>y wife,<span class="_8 blank"> </span>Kim,</div><div class="t m0 x1e h13 y2f ff5 fsc fc0 sc0 ls17 ws0">for being m<span class="_7 blank"></span>y most trusted con\ufb01dante </div><div class="t m0 x1f h13 y30 ff5 fsc fc0 sc0 ls17 ws0">and personal counselor<span class="_3 blank"></span>.</div><div class="t m0 x1c h14 y31 ff5 fsd fc0 sc0 ls16 ws0">\u2014Michael Shermer</div><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls18 wsd">Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page v</div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf8" class="pf w1 h2" data-page-no="8"><div class="pc pc8 w1 h2"><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls14 ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls14">5/4/06 <span class="_2 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls14">1:37 <span class="_2 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls14">Page <span class="_2 blank"></span>vi</span></span></span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pf9" class="pf w1 h2" data-page-no="9"><div class="pc pc9 w1 h2"><img fetchpriority="low" loading="lazy" class="bi xe y32 w4 h15" alt="" src="https://files.passeidireto.com/0c97285c-937f-4d86-ae93-bd07dc808b15/bg9.png"><div class="t m0 xe h16 y33 ff3 fse fc0 sc0 ls19 wse">Ackno<span class="_7 blank"></span>wledgments</div><div class="t m0 xe h17 y34 ff6 fsc fc0 sc0 ls17 ws16">The authors wish to thank Steve Ross and Katie McHugh at</div><div class="t m0 xe h17 y35 ff6 fsc fc0 sc0 ls17 ws17">Random House for their support of this book. Special thanks to</div><div class="t m0 xe h17 y36 ff6 fsc fc0 sc0 ls17 ws18">Natalya St. Clair for typesetting the initial draft, which was</div><div class="t m0 xe h17 y37 ff6 fsc fc0 sc0 ls17 ws0">partly supported by a grant from the Mellon Foundation.</div><div class="t m0 x2 h17 y38 ff6 fsc fc0 sc0 ls1a wsf">Arthur Benjamin especially wants to acknowledge those who</div><div class="t m0 xe h17 y39 ff6 fsc fc0 sc0 ls1a ws10">inspired him to become both a mathematician and a magician<span class="_6 blank"> </span>\u2014</div><div class="t m0 xe h17 y3a ff6 fsc fc0 sc0 ls1a ws11">cognitive psychologist William G. Chase, magicians Paul Gertner</div><div class="t m0 xe h17 y3b ff6 fsc fc0 sc0 ls1a ws12">and James Randi, and mathematicians Alan J. Goldman and</div><div class="t m0 xe h17 y3c ff6 fsc fc0 sc0 ls1a ws13">Edward R. Scheinerman. Finally<span class="_3 blank"></span>, thanks to all of my colleagues</div><div class="t m0 xe h17 y3d ff6 fsc fc0 sc0 ls1a ws14">and students at Harvey Mudd College, and to my wife, Deena,</div><div class="t m0 xe h17 y3e ff6 fsc fc0 sc0 ls1a ws15">and daughters, Laurel and Ariel, for constant inspiration.</div><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls1b ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls1c">5/4/06 <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls1c">1:37 <span class="_0 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_0 blank"></span><span class="ls1b">Page <span class="_0 blank"></span>vii</span></span></span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div> <div id="pfa" class="pf w1 h2" data-page-no="a"><div class="pc pca w1 h2"><div class="t m0 x1 h3 y1 ff4 fs0 fc0 sc0 ls1d ws0">Benj_0307338401_4p_fm_r1.r.qxd <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls1d">5/4/06 <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls1d">1:37 <span class="_1 blank"></span>PM <span class="_0 blank"></span><span class="ls1"> <span class="_1 blank"></span><span class="ls1d">Page <span class="_0 blank"></span>viii</span></span></span></span></span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-119.016000,-108.000000]}'></div></div>
