As the authors write in the Preface,

The aim of this book is to present a collection of remarkable proofs in elementary mathematics (numbers, geometry, inequalities, functions, origami, tilings…) that are exceptionally elegant, full of ingenuity, and succinct. By means of surprising argument or a powerful visual representation, we hope the charming proofs in our collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs.

I can confidently confirm that the authors have achieved the stated goal admirably. They put together more than 100 mostly elementary mathematics facts — quite curious in their own right — proved by elegant and often surprising arguments.

The idea of beauty in mathematics, when expressed in general terms of, say, ingenuity and concise reasoning, is undoubtedly shared by all mathematicians. Views may differ, though, when it comes to specifics of a particular fact or a proof. Tastes among mathematicians also differ as they probably do among other artists and artisans. I remember being amazed to discover that the late G.-C. Rota admitted surprise at the statement of Morley’s theorem but did not see the beauty in either the theorem itself or any of its numerous proofs. (Morley’s theorem is included in the book and the reader will be able to form his or her own opinion.)

It is clear that the authors set out to pursue a difficult task of selecting from thousands of elementary mathematical facts a reasonable number of representatives that individually and in combination could convey the spirit of excitement experienced by “those who know” at the sight of “you know what.” The authors — C. Alsina and R. B. Nelsen — are well known to the American (and probably broader) math community by their joint books (*Math Made Visual* and *When Less Is More: Visualizing Basic Inequalities*) and by their frequent submissions of proofs without words to the MAA magazines. Roger Nelsen is also the author of two earlier collections, *Proofs Without Words* and *Proofs Without Words II*. Trendsetters in the visual presentation of mathematics and mathematical proofs, the authors have put out another tasteful collection that will be appreciated by many in the mathematics community.

That said, I have a couple of reservations. One is in the spirit of the previous paragraph: there are a few proofs in the book that I would do differently. I’ll give just one example: Section 5.11 (*Are most triangles obtuse?*). Lewis Carroll, the famous author of *Alice in Wonderland* and *Through the Looking Glass*, discussed this as one of his *Pillow Problems*. His solution was to distinguish between three cases: a selected side may be the longest, middle, or shortest in a triangle. Each case leads to a different probability estimate without actually addressing the problem as a whole. The authors explain the differences in the three formulas by the ambiguity inherent in the task of selecting random points in the plane. But this ambiguity is quite irrelevant here. The differences are more naturally explained by the different roles played by the chosen side. More importantly, in my view, there is indeed an elegant and absolutely surprising solution (by R. Guy) included in one of the references. This solution is entirely independent of the manner by which three vertices of a triangle are picked in the plane. The three vertices of a triangle and its orthocenter form an orthocentric system, meaning the set of four points such that when three of them are taken at a time, the remaining fourth point serves as the orthocenter of the corresponding triangle. The punch line is the observation that of the four triangles formed by four points in the orthocentric system only one is acute!

My second reservation concerns occasional omission of sources. The book contains a comprehensive index and extensive bibliography. However, quite a few of the sections in the book lack proper attribution. For example, Sections 2.3 (*The golden ratio*), 5.3 (*The inradius of the right triangle* attributed to Li Hui in *PWW II*), 5.5 (*The incircle and Heron’s formula*), 5.6 (*The circumcircle and Euler’s triangle inequality*) point to no source, although the material they cover is not original. Whenever a result is well known, a reference could allow for a follow up and would be appreciated by an interested reader.

The Pythagorean theorem (Section 5.1), with hundreds of proofs to choose from, was bound to be a point of contention. A minor peeve of mine concerns less the selection of a proof and more the manner in which the theorem was introduced. The authors mention that the theorem appears as Proposition 47 in Book I of Euclid’s *Elements*. Euclid, however, proved the Pythagorean theorem twice: the second time as Proposition 31 in Book VI. There must to be a reason why Euclid felt it necessary to include a second proof of a theorem. One possibility is that, due to the importance of the Pythagorean theorem, he wanted to include it as early as possible — regardless of the effort or esthetics of the proof. The second proof depends on the notions of similarity and areas, and therefore had to wait until book VI. It is a little more general than the Pythagorean theorem proper, by allowing arbitrary rectilinear figures on the sides of a right triangle. However, to my knowledge, the theorem in the general form is used nowhere in the *Elements*. So it stands to reason that Euclid’s intention was to include the second proof for esthetic, not pragmatic reasons. The proof is wonderfully simple and needs only one extra line — the altitude to the hypotenuse. This altitude splits a right triangle into two smaller ones both similar to the base triangle and whose areas add up to that of the latter. The areas of the three triangles are proportional to the squares on the sides of the base triangle which proves the theorem.

However, the Pythagorean theorem is in no way central to the book. The book is really a collection of more than 100 curious mathematical facts accompanied by elegant proofs. A good many of these are proofs without words, as might have been expected. If I were compelled to choose a favorite I would reluctantly pick a statement from Section 5.8. *Given a convex n-gon P*_{n}, with n ≥ 4, there is a convex (n–1)-gon with the same area that can be constructed with straight edge and compass. The proof is just a diagram. Beautiful.

Of course not all proofs are pww. Euler’s polyhedron formula (Sections 11.8–11.11) is proved twice in the book. Remarkably, the second proof establishes equivalence of Euler’s formula and Descartes’ Angular Defect Theorem. It is curious to observe how close Descartes came to discovering this formula. But he did not.

The book covers a wide range of topics, from Number Theory and questions of rationality through combinatorial geometry, geometry of triangles, quadrilaterals, then specifically squares and equilateral triangles and further from stereometry and set theory. Many appeared in other publications, but together they make a coherent collection that would please many a mathematics fan. As with their previous books, the authors supply each of the 12 chapters with challenge problems and the latter with solutions found at the end of the book. There are more than 130 solved challenges. This makes the book of special interest to the middle and high school teachers of talented youth and the students themselves.

Alex Bogomolny is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math.