Math Secrets

The Secrets of Mental Math
Arthur T. Benjamin, Ph.D.

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Arthur T. Benjamin, Ph.D.
Professor of Mathematics
Harvey Mudd CollegeP

rofessor Arthur T. Benjamin is a Professor of
Mathematics at Harvey Mudd College. He
graduated from Carnegie Mellon University
in 1983, where he earned a B.S. in Applied
Mathematics with university honors. He received
his Ph.D. in Mathematical Sciences in 1989 from
Johns Hopkins University, where he was supported
by a National Science Foundation graduate fellowship and a Rufus P.Isaacs
fellowship. Since 1989, Professor Benjamin has been a faculty member of
the Mathematics Department at Harvey Mudd College, where he has served
as department chair. He has spent sabbatical visits at Caltech, Brandeis
University, and the University of New South Wales in Sydney, Australia.
In 1999, Professor Benjamin received the Southern California Section of
the Mathematical Association ofAmerica (MAA) Award for Distinguished
College or University Teaching of Mathematics, and in 2000, he received the
MAA Deborah and Franklin Tepper Haimo National Award for Distinguished
College or University Teaching of Mathematics. He was also named the
2006–2008 George Pólya Lecturer by the MAA.
Professor Benjamin’s research interests include combinatorics, game theory,
and number theory,with a special fondness for Fibonacci numbers. Many
of these ideas appear in his book (coauthored with Jennifer Quinn) Proofs
That Really Count: The Art of Combinatorial Proof, published by the MAA.
In 2006, that book received the MAA’s Beckenbach Book Prize. From 2004
to 2008, Professors Benjamin and Quinn served as the coeditors of Math
Horizons magazine, which is published by the MAA andenjoyed by more
than 20,000 readers, mostly undergraduate math students and their teachers.
In 2009, the MAA published Professor Benjamin’s latest book, Biscuits of
Number Theory, coedited with Ezra Brown.

i

Professor Benjamin is also a professional magician. He has given more than
1000 “mathemagics” shows to audiences all over the world (from primary
schools to scienti c conferences),in which he demonstrates and explains
his calculating talents. His techniques are explained in his book Secrets of
Mental Math: The Mathemagician’s Guide to Lightning Calculation and
Amazing Math Tricks. Proli c math and science writer Martin Gardner calls
it “the clearest, simplest, most entertaining, and best book yet on the art of
calculating in your head.” An avid game player, ProfessorBenjamin was
winner of the American Backgammon Tour in 1997.
Professor Benjamin has appeared on dozens of television and radio programs,
including the Today show, The Colbert Report, CNN, and National Public
Radio. He has been featured in Scienti c American, Omni, Discover, People,
Esquire, The New York Times, the Los Angeles Times, and Reader’s Digest.
In 2005, Reader’s Digest called him“America’s Best Math Whiz.”

ii

Table of Contents

INTRODUCTION
Professor Biography ............................................................................i
Course Scope .....................................................................................1
Acknowledgments ..............................................................................3
LECTURE GUIDES
LECTURE 1
Math...