Suppose you walk past a barber's shop one day, and see a sign that says

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself?

If you watch a steam engine, you may not know how it works but you can soon get a fairly good idea of its behaviour, and you can predict its future behaviour accurately. Even though you don't understand its workings, you can see it's a pretty simple machine, so you can trust it to behave in a simple way: you have confidence in your predictions based on a short sample of its behaviour.

As Tony Gardiner says in at the beginning of this book, "the last ten years or so has seen a remarkable blossoming of public interest in mathematics [but] most of the books produced have been for adults, rather than for students. Moreover, most are in prose format - for those who want to 'read about' mathematics, rather than those who want to get their hands dirty solving problems."

This book is built on an extended metaphor, which casts equations as the poetry of science. According to the editor Graham Farmelo (head of Science Communication at the Science Museum in London), great equations and great poems are alike in a number of ways. Both suffer if anything is added, changed, or taken away, both are a rich stimulus to the prepared imagination, and both draw much of their power from their conciseness.

If "How to solve it" really contained an infallible recipe for doing so, mathematics would not be mathematics and the world would be quite different. Of course it doesn't - it can't - but it can - and does - contain a great deal of food for thought for the budding mathematician. Like many other Central Europeans, Pólya relocated to the US at the beginning of the Second World War. There he worked at Stanford University and wrote this immensely successful book (more than a million copies sold) in 1945.

Which fits best - a square peg in a round hole, or a round peg in a square hole?

Science writer and exhibition researcher Alison Boyle tells Plus about her work creating up-to-the-minute news exhibits at the Science Museum in London.

Clearly the modern electronic computer couldn't have been built before electronics existed, but it's not clear why computers powered by steam or clockwork weren't invented earlier. Tom Körner speculates on the historical reasons why computers were invented when they were.

Fluid mechanics is the study of flows in both liquids and gases, and is therefore enormously important in understanding many natural phenomena, as well as in industrial applications. Geophysicist Herbert Huppert tells us what happens when two fluids of different densities meet, for example when volcanos erupt and hot ash-laden air is poured out into the atmosphere.

If your team scores first in a football match, how likely is it to win? And when is it worth committing a professional foul? John Haigh shows us how to use probability to answer these and other questions, and explains the implications for the rules of the game.