*(2016) O'Shea, Prometheus Books, £16.99 / Can$20.00 / US$19, trdpbk, 330pp, ISBN 978-1-633-88148-8*

Given this is maths, this is a surprisingly absorbing book to which I warmed as I traversed each chapter.

You may have gathered that I began reading this anticipating something of a struggle, and indeed there was as I had pre-judged this book by – not so much its cover as – its title. I anticipated that this book would be all about prime numbers, their discovery, their irrationality, their importance in cryptography, and the various different types of primes through to pseudoprimes. In fact I was so convinced myself of this that I kept putting off picking this up and starting to read.

As it happens, 'The Call of the Primes' is an early chapter in this book but other chapters are on Pi, 'E', the square root of minus 1, and other mathematical concepts, all very intriguing. What this book is not (as its title initially suggested to me) are surprising patterns, peculiar puzzles and marvels arising out of primes; what this book is, is a quick tour de force of surprising patterns, peculiar puzzles and marvels *including* primes!

Confusion over, I began to enjoy the book and became quite engrossed. My favourite chapter was 'The Monty Hall problem and other deceptive Puzzles in Probability Theory'. Indeed it contains one of the best explanations of the Monty Hall Problem I have encountered since Postmentier's Magnificent Mistakes in Mathematics (also published by Prometheus). If you are not familiar with this problem, it stems from a game played on a 1960s,US TV show hosted by one Monty Hall. The problem consists of a contestant being presented with three boxes and being told that one box contains a million dollars put there by the host. The contestant then is asked to chose a box. before the box is opened, the show's host then decides to give the contestant a clue as to which box contains the million by opening one box he knows does not contain the million and so now the contestant has only two boxes to choose from, not three. The question is should the contestant change their decision or stick with their original choice? Or will this make no difference?

Now it has to be said (I cannot leave you in too much suspense) that there is a specific answer – that O'Shea explains – to this problem that doubles the chance of the contestant winning. This problem also (though the author does not say it) neatly illustrate how an observer's act of observing impacts on perceptions of reality. And the answer, when you discover it you may feel is controversial and indeed when, in the 1960s it was explained even some (fortunately a minority but still a significant one) of university maths lecturers got it wrong!

The level of maths required to enjoy this book is (in British terms) that of mid-secondary school: for pupils aged around 15 or 16. There are some equations but none-are more advanced than this O-level (GCSE) level of maths understanding. (Things might be different in the USA as they don't study maths; they do something called 'math' instead, but I'm guessing my 15 or 16 year-old's comprehension estimate still applies.)

And if like me your maths school days are nearly half a century ago don't worry, this book is still perfectly understandable.

So to whom will this book appeal? The curious of all ages, 15 or 16 year-old pupils thinking of engaging in more advanced studies will find this both informative and entertaining, and then oldies like myself can use it working through some of the formulae to keep our brains active. Even without all of this it is an entertaining read. Those into numerology will find the e-mails the author cites, that he received from his friend Ming Cong, a treasure trove of mathematical coincidences. For example, if you create a code converting letters of the alphabet into numbers starting with A = 1, B = 2, etc., then add up the corresponding numbers of the words TWO HUNDRED AND FIFTY ONE you get 251.

If only my maths teachers taught us some of what O'Shea gives us in The Call of the Primes, then who knows, I might have gone into physics rather than the natural sciences. (Leave that thought alone; perhaps it's best not to go there.)

I'll give The Call of the Primes a decided thumbs up, thankful that my digit's proportions possibly being close to that of a Golden Ratio is most likely coincidental. I hope so, because reading this one can't help but feel that God is a mathematician.

Jonathan Cowie

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