What is a charming proof? Alsina (Polytechnic Univ. of Catalonia, Spain) and Nelsen (Lewis and Clark College) cite D. Schattschneider ("Beauty and Truth in Mathematics") in Mathematics and the Aesthetic, edited by N. Sinclair, D. Pimm, and W. Higginson (CH, Jul'07, 44-6282): such a proof should exhibit elegance, ingenuity, and insight, and it should provide connections and paradigms. A charming proof should be eligible for inclusion in Erdos's mythical "book," which contains the most perfect proofs possible of all mathematical results. M. Aigner and G. Ziegler's Proofs from the Book (4th ed., 2010) contains a sampling of such proofs, and the book under review, even though there is some overlap, provides more. The overall flavor here is more geometric and visual, and less analytic, than Aigner and Ziegler's work. Each chapter provides challenge exercises with solutions. A brief listing of section topics includes triangulation of convex polygons, the Erdos-Mordell inequality, an angle-trisecting cone, the quadratrix of Hippias, and the Wallis product. Not surprisingly, the charm of the contents means that in sum this is a charming book. It would be a good supplement to introduction-to-proof courses, as well as topics for discussions in mathematics clubs. Summing Up: Highly recommended. Lower- and upper-division undergraduates and general readers. --D. Robbins Trinity College, CHOICE Magazine
This is a collection of remarkable proofs, all using elementary mathematical or geometrical arguments, and all very simple but often extraordinarily powerful. While some will be well known, I imagine that almost every reader will find material here that they have not encountered before.
Although the book is a mathematics book, I feel sure that some of the theorems would have direct relevance to statistics. For example, how about: for any even number of different points distributed inside a circle it is always possible to draw a line across the circle missing every point and such that exactly half lie on each side of the line. Surely this can find application in segmentation analysis, for applications in marketing and other areas.
In addition to the proofs themselves, there are over 130 "challenges" aimed at stimulating the reader to create similar such "charming proofs." Solutions to these challenges appear at the end of the book.
I cannot help but feel that working carefully through the proofs in this book would materially improve one's creative powers and ability to think laterally. --David J. Hand, International Statistical Review
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. 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? Prover 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, lists for nearly $60 USD. This is certainly more expensive than 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 Bogomolyny's review at Cut the Knot. --Sol, Making Math fun and accessible