Proofs that Really Count: The Art of Combinatorial Proof

Dolciani Mathematical Expositions

by Arthur T. Benjamin, Jennifer Quinn

Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.

Genres: Mathematics

206 pages, Hardcover

First published August 1, 2003

About the author

Arthur Benjamin holds a PhD from Johns Hopkins University and is a professor of mathematics at Harvey Mudd College, where he has taught since 1989. He is a noted “mathemagician,” known for being able to perform complicated computations in his head. He is the author, most recently, of The Secrets of Mental Math, and has appeared on The Today Show and The Colbert Report. Benjamin has been profiled in such publications as the New York Times, the Los Angeles Times, USA Today, Scientific American, Discover, and Wired.

Reviews

[Mark Moon]

This was tons of fun! Lots of cool ideas about how to understand Fibonacci numbers, Lucas numbers, continued fraction expansions, and harmonic numbers in terms of "concrete" things you can count (generally configurations of different kinds of "blocks", with constraints on allowable configurations) I read this book fairly lightly (i.e. didn't do many exercises), because I was just reading it for fun. I'll come back and do more exercises if I ever teach these ideas, or if I really need to count something important.

[Mangzilla]

Starts with Fibonacci as domino fill problem. Shows how to conceive proofs (of identities) as counting the same set in two different ways. Equivalently, establishing 1-to-1 (or 1-to-n) correspondence.
Also emphasizes the "conditional" technique of dividing into subsets and counting each subset.

[Jeff Yoak]

This is one of my favorite math books ever, but it is hard enough that I can't make it through in the small bites I have time for. I plan on revisiting when I have time to spend hours at a time focused on it.