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Impossible? Surprising Solutions to Counterintuitive Conundrums

Julian Havil
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Blair Madore
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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.

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akirak's picture

This is not really a book for the layman. I found it heavy going. Many problems lack sufficient explanation.

However, there are a number of gems which are definitely worth exploring. For example, there is Simpson's Paradox. This is an example where (a/b) > (c/d) and (p/q) > (r/s) but (a+p)/(b+q) is less than (c+r)/(d+s). Other interesting topics are the connection between the continued fraction of 1/e and the optimal number of r out of n, a discussion of why the infinite sum 1/n is called the harmonic series, and an explanation of why order is lost in complex numbers.

Moreover, there are some fantastic quotes by various mathematicians, such as the following, by De Morgan: "The ratio of log of -1 to square root of -1 is the same as circumference to diameter of a circle."

This book must be read in conjunction with the author's other title "Nonplussed!' because he refers to it in a number of places. I have not read "Nonplussed!' and I think a book like this should be totally independent of any other text.

A major problem with the book is progression is too fast. One has no time to digest an idea before the author moves on to higher dimensions. A good example of this is the "Monty Hall' problem: within a couple of pages of describing the problem, the author has moved on to various extensions and generalisations of the problem. It would have been better to progress at a slower rate so that the reader understands the initial problem and then is able to follow the extensions on this problem.

In general I found myself taking a lot of time to get through the book, because I had to keep going back and looking at things again and again in a bid to understand. There are a number of typos; in particular the brief appendix at the end seems to be full of them. The infinite series for sin and cosine is wrong. It should have alternating signs. There is no fig 4 which relates to subintervals. The proof that log(2) is irrational is incorrect.

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