NB. I was sent this book as a review copy.

This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way.

The level of mathematics needed is generally only up to relatively basic calculus, though there is the occasional diversion into a slightly more complex area, though anyone with basic first year university mathematics, or even a keen high school student who has done a little reading ahead, would be able to get a lot from the questions.

I found that there were a number of ways of going through the questions. Some of them are enjoyable to read, and simply ponder. For me, occasionally figuring out what should be done, without writing anything down, was enough to be pretty confident that I saw the ingenuity in the puzzle and the solution and I was happy to leave it at that. Some puzzles got me scribbling, and calculating quite explicitly and really wanting to get the final result which we had bene guided towards. And a third category of problem really needs one to sit down at a computer and get simulating. There are some lovely computational-based puzzles which give a general sense of what a Monte-Carlo simulation entails, particularly those which ask for the probability of some interesting set of events occurring. In these examples, Matlab code is even given to get the reader going, and in the solutions, the full code is provided.

Each puzzle is given a few pages, first to set the scene and ask the basic question, then to guide the intuition as to how to think about the problem, then perhaps some more hints, and often at the end some extensions of the problem that take it from a good brain-teaser to a real conundrum.

A few of my favourite problems were:

- The mathematics of NASTYGLASS – a glass which acts like an audio filter which makes the viewer feel sick just to look at it, just as such an audio filter can bring on a sense of unease for any audio file sent through it.
- The seemingly innocuous, but ultimately rather fascinating puzzle of a point mass falling off a sphere of ice. This is a question often put into undergraduate physics courses, but it turns out that the answer is much more subtle than is generally explained.
- The rate of acceleration of a raindrop through fog – starting with the most basic assumptions and going on to a very thorough investigation of the possible behaviour, leading indeed to the title of the book.
- Extensions to the Birthday paradox and some wonderful thoughts on combinatorics and probabilities.
- and many more…

The second half of the book is then the solutions to the problems, again written colloquially, but extremely understandably and without shirking any technical necessities.

All of the above being said, I have one major, and important contention with the book, and that is its very premise. Nahin quotes in the introduction of the book a line from a letter to the Boston Globe (Alain and Patricia Jehlen in 2015) which reads:

It’s fun knowing how to solve quadratic equations involving the square root of minus 1. And that knowledge is needed in a few lines of work. But why should it be required for taking entry-level college courses, the gateway to so many good jobs? […]

Advanced algebra is the new Latin: It has some uses, but mostly it’s a hurdle that trips up young people who could be successful in challenging occupations but don’t get the chance because they haven’t learned the quadratic formula.

Many of those who learn the quadratic formula, if they did it just to pass a test, could have used their time more productively learning life skills and important knowledge, such as how to reason, communicate, understand history and society, or play a musical instrument.

Many of those who learn the quadratic formula, if they did it just to pass a test, could have used their time more productively learning life skills and important knowledge, such as how to reason, communicate, understand history and society, or play a musical instrument.

He then claims that this is simply not true, and that this book serves as a series of examples as to how misguided the idea is.

However, I don’t believe that, however fascinating the puzzles are, this gives any good reason why a person should believe that learning the formula for quadratic equations is suddenly useful. I would guess that anyone who believes in the general pursuit of science would not contend for a moment that intermediate and advanced mathematics are not useful subjects and indeed vital for scientific progress, and that even if you are not going to continue with science, the type of thinking that is learnt in a science course is very useful in many other pursuits. However, showing the puzzles in the book, none of which are truly everyday questions that most people would ask, or be too worried about the actual answers of, should not, in my opinion, sway anyone into suddenly thinking that learning about quadratic equations would improve their quality of life.

It’s a pity that the book is based on what I consider to be a shaky foundation, because everything else in the book is truly wonderful, and the fact that I am not convinced by its raison d’être doesn’t mean that I don’t think that all keen mathematical physicists shouldn’t have this in their arsenal of puzzle books.

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