Foolproof - Brian Hayes *****

The last time I enjoyed a popular maths book as much as this one was reading Martin Gardner’s Mathematical Puzzles and Diversions as a teenager. The trouble with a lot of ‘fun’ maths books is that they cover material that mathematicians consider fascinating, such as pairs of primes that are only two apart, which fail to raise much excitement in normal human beings.

Here, all the articles have something a little more to them. So, even though Brian Hayes may be dealing with something fairly abstruse-sounding like the ratio of the volume of an n-dimensional hypersphere to the smallest hypercube that contains it, the article always has an interesting edge - in this case that although the ‘volume’ of the hypersphere grows up to the fifth dimension it gets smaller and smaller thereafter, becoming an almost undetectable part of the hypercube.

If that doesn’t grab you, many articles in this collection aren’t as abstruse, covering everything from random walks to a strange betting game. What's more, an extra delight for me is that Hayes throws in a lot of computing reflections, even including snippets of code as a way of explaining some processes. I particularly loved the exploration of pseudorandom and quasirandom numbers (not the same thing) and their implications for Monte Carlo methods.

The only times I felt Hayes loses it a bit is when he gets too heavily into research mode and gives us more detail than we need. For example, he digs into the origins of the story of the young Gauss adding up 1 to 100 almost instantaneously at school. His exploration of this mathematical legend is impressive, but he enumerates every possible source and route for the various versions of the legend to have originated, taking us to a level that feels unnecessarily complete. Similarly he lost me a bit when he tries to forensically examine why a Victorian mathematician who calculated pi to 707 places went wrong from the errors that he made in his calculations. But this kind of over-detailed analysis is rare.

I suspect the ideal reader is someone who has an aged physics, maths or computer science degree, who is still aware of (say) what Monte Carlo methods or eigenvalues are in a vague sense, but needs some gentle reminders. The essential, however, is to have a sense of wonder in discovery. For people like us it’s a brilliant book.

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Review by Brian Clegg

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