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鈥淭WO identical coins of equal radius are placed side by side with one of them fixed. Starting head up and without slipping, rotate one about the other until it is on the other side of the fixed coin. Is the rotated coin now head up or down?鈥� (See Diagram).

So begins Julian Havil鈥檚 romp through the mathematical world of mind-boggling implausibility. If you don鈥檛 know the answer, guess: you鈥檝e got a 50:50 chance of being right. Then try it for real with a couple of pennies, and you鈥檒l soon see. But do you see why? That鈥檚 trickier. Shouldn鈥檛 the coin be upside down if you rotate it through 180 degrees?

鈥淭he mathematical world is filled with implausible ideas鈥�

Sometimes intuition lets you down. You know the situation: you鈥檙e faced with a problem for which you think you have an answer. Experience tells you that you鈥檙e right, every nerve in your body screams that you鈥檙e right, but you鈥檙e wrong. Miserably, painfully, embarrassingly wrong. Perhaps you鈥檝e misjudged the width of a car or the guilt of a suspect. The universe is filled with dark alleys in which the unsuspecting visitor can be fleeced by the laws of nature.

Which is why it is a relief to turn to the world of mathematics, where every properly formed problem eventually cedes to the appropriate mathematical attack 鈥� or so we鈥檇 like to think. Cold, hard reasoning makes mincemeat of implausibility. There鈥檚 no place for the counter-intuitive when you have logic to guide the way, right?

Actually, no. The uncomfortable truth is rather different: the mathematical world is filled with implausible ideas that turn out to be not just plausible, but provably, beautifully, eternally true. Nonplussed is a flotilla of such ideas collected by Havil, a teacher at Winchester College private boys鈥� school in Hampshire, UK.

Many of Havil鈥檚 examples originate in the world of probability, where counter-intuitive ideas abound. For example, imagine the final of the men鈥檚 singles tennis championship at Wimbledon. The world鈥檚 top-ranked player is serving, with the score at 40-15. Would you be surprised to learn that the chances of him winning the game from this position are actually lower than they were at the start of the game?

If so, you might be shocked to read about the birthday paradox. In a room full of people, there must be at least 366 to ensure that two share a birthday (ignoring leap years). But how many people do you need for the odds to be merely better than 50:50? If you鈥檙e thinking you should divide 366 by 2, you鈥檙e on the wrong track. The answer is superbly at odds with common sense and not at all easy to prove.

Havil introduces these and other hard-to-believe ideas with an often fascinating discussion of their history. He then tackles each conundrum with the necessary mathematical tools and illuminates them with quotes that are often surprising and occasionally amusing.

He assumes, perhaps optimistically, that readers will require no more than a high school education in mathematics to follow the book. If it has been a while since your high school years, you may need to remind yourself of one or two concepts. Nevertheless, there is much charm in Havil鈥檚 combination of the anecdotal introduction of a problem followed by its mathematical conclusion.

For the most part, Havil is a good guide to this landscape. Nowhere does he shirk his mathematical responsibilities 鈥� you will be immersed in Taylor expansions, recurrence relations and logarithmic spirals. If these phrases sound like gobbledegook, this book isn鈥檛 for you. But if they stir the faintest of mathematical memories, take heart. Havil offers a helping hand up many of the steepest climbs. You may need to stop and catch your breath, but you鈥檒l reach the summit in the end.

Occasionally, and perhaps inevitably in a book of this kind, author and reader will part company. It matters not. Each chapter is self-contained so if you get lost in one, try another. You鈥檒l be constantly amazed 鈥� perhaps never more so than by Havil鈥檚 description of a goblet that has finite surface area but infinite volume. If you can, ask a barman to fill it with beer. You鈥檒l never have to buy another.