This book is a follow up to Havil’s very well received Nonplussed, which I have yet to read. After reading the first half of Impossible, I came to the conclusion that Havil had used all his best stuff in Nonplussed. I was wrong. Yes, almost all the topics Havil discusses in this book were familiar to me, but that is fair — this book is aimed at those interested in recreational mathematics, not professional mathematicians.
Early in the book I couldn’t get past the author’s penchant to dally whimsically with wit and quotation and curious asides. I wanted to get to the answer! Yet somewhere in the middle of the book I suddenly began to appreciate the detailed mathematical answers to interesting problems. Havil is not afraid to use (a considerable amount of) algebra, conditional probability, or other mathematical techniques that may be foreign to the average reader of recreational mathematics. To assuage those readers he provides an appendix with all the necessities.
The problems themselves are interesting — especially if you haven’t seen them before. And even if you have seen them, Havil often has a twist you may not have heard of, or a reference to an original source, or an alternative proof that will be new to you. For me the surprises included a multi-stage Monty Hall problem, a proof that the leading segments of powers of 2 include all natural numbers, and a proof of Benford’s law (the distribution of the leading digit of numbers) without using measure theory.
The best part of this book was that it was full of historical detail and had references to the literature. I give many public talks about interesting math problems, including several in this book, but I have never researched them as thoroughly as Havil. He has provided me with a lot of mathematical and cultural references that will make my future talk more interesting and complete.
I would highly recommend this book as a reference for the mathematician who likes recreational mathematics, or as a good read for the recreational enthusiast with a penchant for more rigor. It is not as easy to read as a Martin Gardner book, but it is just as rewarding.
Blair Madore is Associate Professor of Mathematics at SUNY Potsdam.