Customer Reviews

Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions)

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I was introduced to this book by a talk that one of the authors (Arthur Benjamin) gave at the MAA Mathfest in Albuquerque in August of 2005. The talk was one of the very best mathematics talks that I've ever attended. Everyone in the audience could follow what was going on, and we all left with an understanding of the basic approach to combinatorial identities used in this book. The authors' approach is to prove combinatorial identities by defining a quantity and then obtaining different formulas for that quantity. One formula becomes the left hand side of an identity while another formula becomes the right hand side.

When I read the book I found that it was just as clearly written, with lots of beautiful examples.

The proofs in this book are easy enough for a bright high schooler or even an exceptional middle schooler to understand, while still making use of insightful tricks that keep the solutions far from being obvious.

"Thoroughly engaging... Accessible to a very broad audience... While the theorems covered may not be new to research mathematicians, I would wager that very few of us have seen them proven in quite this way." -- American Mathematical Monthly [http://www.maa.org/reviews/reallycount.html]

I am not a mathematician and I learn something cool and useful from this book every few paragraphs. Highly recommended.

A Review of the book "Proofs that Really Count"
by Gerald T. Westbrook.

1. Background.
This is my second review of a special book on mathematics. The first book was entitled: "A history of Pi." It was posted on Amazon on November 24, 2011, entitled: "Pi defined via a Polygon of ever increasing order, inside a circle."

2. Comments from Amazon

(1) Book Description
The award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns can be understood by simple counting arguments. Numerous hints and references are given for all exercises and the extensive appendix of identities will be a valuable resource.

(2) Biography
Arthur Benjamin is a professor of mathematics at Harvey Mudd College in Claremont, California. He is also a professional magician and performs his mixture of math and magic all over the world.

(3) Related Book
The book "Secrets of Mental Math is co-authored by Benjamin and Michael Shermer. He is host of the Caltech public lecture series, a contributing editor to and monthly columnist of Scientific American, the publisher of Skeptic magazine, and the author of several science books.

(4) Customer Reviews.
A math prof was introduced to this book by a talk that Arthur Benjamin gave in Albuquerque in August of 2005. He reported the talk was one of the very best mathematics talks that he had ever attended. While I I found the book was clearly written, some of the mathematical notation was, at times, overwhelming.
Another reviewer argued that the proofs in this book are easy enough for a bright high school student to understand.
A third reviewer called it: "Thoroughly engaging... Accessible to a very broad audience... While the theorems covered may not be new to research mathematicians, I would wager that very few of us have seen them proven in quite this way. I am not a mathematician and I learn something cool and useful from this book every few paragraphs. Highly recommended."

3. Comments on the names Fibonacci and Gibonacci
Chapter 1 is about the Fibonacci Identities. As noted above I have written on the Fibonacci Series (FS), which is divergent, and the Ratio Series (RS), which is convergent. The authors definition of FS is identical to what I have used. I do not see the RS in this book. My definition is:

Rn = Fn/Fn-1 for n greater than zero.

The first five terms are 1, 2, 1.5, 1.6666..., 1.60. Term 6 is 1.625. Term 10 is 1.6181.... In the limit, as n goes to infinity, Rn is the Golden Mean. Let the Golden Mean be represented by x. It has a most interesting and probably unique property, first discovered by the Ancient Greeks, that:
x = [1/(x-1)]. This expands to a quadratic equation: (x)sqd - x -1 = 0. A solution yields, for the positive number, x = ½ + ½ * SqRt[(5)], or 1.6180339.... etc, etc, etc. This is what I have called the first of the three key mathematical constants: the Golden Mean, Pi and Epsilon, e, the base of the natural logarithms.
Chapter 2 is, in part, about the Gibonacci Identities I had not heard of these identities and numbers before. The Gibonacci Number, Gn, is defined, for n greater than 2, by

Gn = Gn-1 + Gn-2.

This chapter is interested in identities involving Gibonacci Numbers, which is short-hand for the Generalized Fibonacci Numbers.
Chapter 2 is also about the Lucas Numbers. Again I had not heard of this number before. The Lucas numbers are defined by L0 = 2, L1 = 1, and for greater than 1, by

Ln = Ln-1 + Ln-2

The first numbers in this sequence are: 2, 1, 3, 4, 7, 11, 18 , 29, 47, 76, 123, 199 .... etc, etc, etc.

The ratio of these Lucas Numbers also turns out to be the Golden mean, although I can't see where the authors state this.

4. Conclusions
The authors have indeed produced a fascinating book, with many intriguing equtions, sketches, pictures and cartoons. I particularly liked the cartoon on page 96. And on pages 24 and 25, the authors get into "dominoes." Indeed the number of identies and figures is almost overwhelming. I liked the book, and I would recommend it, but the buyer should note that the authors needed to develop a less mathematical organization or flow of concepts. They also should have had a much better Index, especially for a book with so many complexities. And finally, the price is very high.

This book is a pleasure to read and find new combinatorial gems. I'd say its accessible to anyone who's done basic counting - either in high school or a first discrete math course. I found it a bit odd that it starts with the fiibonacci numbers instead of simpler identities like Pascal, Chairperson, etc but the proofs are fairly illustrative though not the very through (the authors seem to believe that the student is already familiar with the ideas of disjoint sets and unions). Overall though, if you know a few counting identities and know the basic idea behind counting in two ways, this book is a wonderful extension!

It also makes for really good casual reading because unlike most math, reading a combinatorial proof doesn't usually require rewriting with pen and paper to understand well and have that "Aha!" moment.

I haven't read this book yet, but I have a signed copy after seeing Jenny Quinn speak at the 2005 meeting of the Northwest chapter of the Mathmatics Association of America. If her written work is anything like her speaking, then this should be a great book. Her combinatorial proofs are an interesting approach to old equations, and she presents them in a very clear manner. A most enthusiastic lady.