To combine the quotes from the authors’ Introduction and the Foreword by Robert L. Bryant, Director of the Mathematical Science Research Institute (MSRI),

**Math circles** are weekly math programs that attract middle and high school students to mathematics by exposing them to intriguing and intellectually stimulating topics, rarely encountered in classrooms.

Math circles are voluntary, extracurricular, after-school programs. The students who are there are much less likely to be motivated by the need to satisfy an academic requirement, prepare to a career, or enhance resumé. They are, for the most part, there because they love mathematics. The teachers encounter students who are willing and hungry to learn, while students encounter teachers with expertise and enthusiasm far beyond the usual classroom experience. Teachers and students look forward with anticipation to the next meeting.

Math circles vary in their organizations, styles of sessions, and goals. But they all have one thing in common: *to inspire in students an understanding of and a lifelong love for mathematics.*

The book, based on the notes from several sessions of the Berkeley Math Circle (BMC), reaches that goal beautifully. It is written with enthusiasm and flair; the book breathes love for mathematics and the desire to convey and share it with the reader. I’d say the authors and the editors had a (legitimate) love affair with the book.

The editors deserve praise for producing a coherent style that is followed throughout. The book is adorned with marginal icons to highlight (among other things) exercises, problems, warm-ups, problem solving techniques, applications of the Pigeonhole principle; separate icons for the two steps of math induction, contradiction… At first surprised, I ended up finding them very useful when browsing the book in search of a particular item that I had noticed on first reading. On the whole, the book is thoughtfully organized and well written. The text is often accompanied by light-hearted illustrations.

The book consists of twelve chapters (naturally referred to as Sessions) covering (roughly speaking) inversion in the plane, combinatorics, groups structure of permutations in the case of Rubik’s cube, congruences, mathematical induction, mass point geometry, complex numbers, games with invariants, a collection of favorite problems, monovariants (monotone quantities associated with the states of the problem), and two chapters on proofs. In the Epilogue the authors give the background of math circles in general and a short history of the Berkeley math circle in particular.

There are a great many problems and exercises. Problems are solved in the text, as are some of the exercises; most of the exercises have hints or solutions at the end of the chapters. Many of the problems are drawn from various mathematical olympiads, but their difficulty only seldom is designated as advanced. Problems are often used to motivate development of a theory, with solutions coming out eventually as the result. For example, the first chapter opens with the statement of Ptolemy’s theorem followed by an intriguing remark that the theorem stems from the straight line identity AB + BC = AC (with B between A and C) by an inversion transform. A proof of the theorem appears towards the end of the chapter after the theory of inversion has been sufficiently developed.

It is hard to recommend a book like this as a “good read.” Much of the book is intended for a guided work out. But even experienced instructors, who should not find it difficult to solve the sequences of problems and exercises, would find many refreshing turns in the discussions that make the book a worth while read.

It is hard to choose favorites; if made to, I would point to the Chapter 10 on using invariants and Chapter 12 on monovariants in problem solving.

The *Escape of the Clones* (Chapter 10) game is played on a quarter of an infinite grid with a small region comprising a few grid squares designated as a prison. In the corner square a prisoner is placed whose the only allowed move is to morph into two clones located on the squares to the right and up from its location. Each of the clones follows the same rule. No two may be located in the same square. For what shapes is escape from the prison possible? Although the problem dates back to 1981, it is not very well known. While the complete solution requires some advanced mathematics, the problem lends itself to fascinating experimentation and to getting non-trivial conclusions.

The discussion on monovariants focuses around the problem *Walking around a Mansion*: 2000 people reside in the rooms of a 123-room mansion. Each minute, as long as not everyone is in the same room, somebody walks from one room into a different room with at least as many people in it. Prove that eventually all the people will gather in one room.

The key to the solution is setting up the measure of concentration of the mansion’s inhabitants. If n_{j} is the number of people in the room j, then the coefficient of concentration is defined as ∑n_{j}^{2}, with the sum taken over all 123 rooms. When a fellow strolls from a room with n people to a room with m people (which is only possible when n ≤ m) then the change in concentration is equal to

(n – 1)² + (m + 1)² – n² – m² = 2(m – n + 1) ≥ 2,

because, as was stipulated, n ≤ m. The concentration keeps growing with every stroll. However, since the number of possible distributions of a finite number of people among a finite number of rooms is finite, the concentration cannot grow forever, and is bound to reach its maximum. From the above inequality, the maximum is achieved only when all people have been gathered in the same room.

Every now and then the authors offer tips (called *Problem Solving Techniques* — PST, for short) to share tactical advice for handling various problems. There are 97 of them in all. For example, the *Walking around a Mansion* problem opens with the PST 88: Although mathematics requires rigor, *don’t be afraid of intuition*. You can be guided by vague concepts at first; then look for concrete ways to express them mathematically. Why should the concentration use the squares? The PST 89 explains: Find some measure that gives disproportionately larger weight to rooms with more people. One way to do this is to add *squares* of the number of people in each room. Having demonstrated the power of the monovariants, the PST 96 sums up the experience: To prove that some repeated process must eventually terminate, *use a monovariant*. Well, not to spoil the good impression I’d like to leave of the book, I’d be less categorical about that. Here is a problem where a solution is obtained more easily by other means:

Pancakes are uniquely ordered from 1 through N (N > 1) according to their size. They are stacked in a random order. One is allowed to flip any substack with the tip at the top. The size of the substack to flip is determined by the number of the top pancake. Prove that eventually the smallest pancake (i.e., pancake 1) will pop up at the top, thus terminating the process.

I have yet to find a monovariant that solves this problem, while with a different approach the solution takes a few sentences.

The BMC was born in 1998, but its organizers and instructors brought along the many decades tradition from the former USSR and the Eastern Europe. These experiences have been documented in the AMS’s *Mathematical Circles (Russian Experience)* and in several translations by the MAA (*Geometric Transformations* by I. Yaglom, now in 4 volumes, and many problem books). In the USSR books in the *Library of the Mathematical Circles* series began appearing in the late 1950s. The present book is announced as volume I of the forthcoming series. This is a good sign for all math loving students and insructors. The book is a welcome beginning for the emerging tradition in the US math education.

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.