I help administer the American Mathematics Competitions (formerly the American High School Math Exam) for high school students who live near our college, SUNY Potsdam. It’s a great way to meet promote mathematics and promote our department — I’d highly encourage anyone to do the same. In addition to the AMC awards our department buys book prizes for the top students and I have regularly given the first edition of *A Mathematical Mosaic: Patterns and Problem Solving* as a prize. I was well aware of Ravi Vakil’s incredible record of successes in the IMO and Putnam contest, not to mention his reputation for eloquence and generally for being the nicest guy you could ever meet. How could his book not be good? Strangely enough I had never read it — so when asked to review the second edition I jumped at the opportunity. I read every line of every page and worked through many, though not all, of the problems. It was a thoroughly enjoyable experience.

The marketing for this book relies heavily on Vakil’s phenomenal success in high level math contests. As a result, I expected a master class in problem solving techniques. I was expecting too much. When it comes to mathematics, the book is more of a potpourri than a mosaic. This book is aimed at (exceedingly bright) high school students. It contains a collection of problems that are for the most part familiar to any mathematician. To be fair, I learned something new about every one one of those topics and sometimes very surprising things indeed.

A good example of what I mean is the discussion of divisibility rules. I knew many divisibility rules but I didn’t know Vakil’s rules for divisibility by 7, much less how to extend that idea to divisibility rules for 17, 19, 23, 27, 29, etc. The proof that these rules work was fun and interesting. This is a perfect example of what I liked about this book. It reminded me of something familiar then extended it to something I didn’t know and then challenged me to extend it even further. It usually leaves things open for considerable exploration.

But who needs such esoteric divisibility rules? Any handy computer with Maple or Pari can determine if a 70,000 digit number is divisible by 17 in a flash. There is no reason for anyone to know such rules — is there? Well, a top problem solver like Vakil would want to know such rules since contests do not permit calculators. And then it dawned on me — top contest solvers commit to memory many different facts and techniques that the average mathematician, much less math student, does not know. That’s one way they solve difficult problems — by drawing together disparate ideas and facts and with a flash of inspiration welding them into a solution. Although it is not overt — by reading between the lines and scrutinizing the topics and ideas covered you can get an idea of what happens in the mind of a top contest problem solver.

This book is really fun to read and I would recommend it to anyone — though it is ideal for the budding mathematician who is trapped in high school and looking for more mathematical knowledge and challenge. But I think it is ideal not only for the math content but the messages it implicitly delivers, too. When it comes to describing mathematical culture this book is a wonderful mosaic. Using biographies of other young math contest stars and many historical and cultural references Vakil delivers a compelling portrait of math culture and mathematicians. I think many young bright minds would be very attracted to that community.

In summary, this is an excellent book for top high school students, contest problem solvers or anyone who enjoys a good (elementary) mathematical challenge.

Full Disclosure: I started my Ph.D. at the University of Toronto several months after Ravi Vakil finished his undergraduate degree there. He had made quite an impact on people there — I was living among a legion of Vakil fans. I even met Ravi once or twice (though I doubt he remembers me) and met several of the characters mentioned in this book.

Blair Madore is Associate Professor of Mathematics at SUNY Potsdam.