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Review: Charming Proofs: A Journey Into Elegant Mathematics

Charming Proofs: A Journey Into Elegant Mathematics is a delightful book, published by the Mathematical Association of America (MAA), that lives up to its name.

Given my joyful experiences of exploring challenging problems in middle school and in high school I have a soft spot for elegant problems that are accessible to motivated students who don't have any background in advanced mathematics. And, I have a soft spot for MAA books because they were among the first math books I devoured, specifically their MAA contest prep books.

Here's a brief description, from the publisher's page, of the structure of the book.

Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges.

What I particularly enjoy about Charming Proofs is its mix of excellent writing, great illustrations, and interesting yet accessible problems. To be honest, this is the case with every MAA problem-solving book I can think of. I may be biased but I suspect that many would agree with my overall assessment of MAA books of this type.

Here are some interesting challenges from the book:

Prove that the vertex angles of any star pentagon sum to 180 degrees.

Is it possible to construct an equilateral lattice triangle?

Does a version of Pick's Theorem hold for three-dimensional lattice polyhedra?

Prove that Heron's formula and the Pythagorean theorem are equivalent.

Is it true that three times the sum of three squares is always a sum of four squares?

Prove there are infinitely many dissection proofs of the Pythagorean theorem.

Charming Proofs is one of those great books that you can pick up, choose a chapter that strikes your fancy, work through the chapter, then be rewarded with a number of challenging problems to explore. I should warn you that the book is biased towards geometric and visual problems rather than analytical ones. This makes sense given that visual problems lend themselves better to charming (visual) solutions.

I'd like to address the concern that some people have with the price of MAA books. This book, for example, list for nearly $60 USD. This is certainly more expensive that any popular math book written for the general public. From that perspective the book is indeed expensive. But, if you consider the other extreme, math text books that can retail for $100 or more, Charming Proofs is inexpensive. Whether the book is expensive or not, I believe, depends on how you're going to use the book. If you're going to use the book as a text and teach yourself the material, and Charming Proofs is of textbook quality, then you're looking at a $60 textbook. If you're looking for a more casual involvement with a book then you can see if your library can get you a copy. Many libraries have partnerships with libraries throughout the country and can get you many books that they don't hold. In this particular case, Amazon.com sells the book for as little as $21.99 (new) from their partners so the cost is not prohibitive.

For those of you interested in a review of Charming Proofs from a mathematician's perspective I heartily recommend Alex Bogomolny's review at Cut the Knot.

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