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 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.

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.

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.

Plus is very proud to present Professor Stephen Hawking's own Birthday Symposium address.

Will we ever be able to make computers that think and feel? If not, why not? And what has all this got to do with tiles? Plus talks to Sir Roger Penrose about all this and more.

Astronomer Royal Sir Martin Rees gives Plus a whistlestop tour of some of the more extraordinary features of our cosmos, and explains how lucky we are that the universe is the way it is.

What happens when one black hole meets another? Professor Kip Thorne shows us how to eavesdrop on these cosmic events by watching for telltale gravitational waves.

Neuropsychologist Brian Butterworth tells us about research showing that even newborn babies have a basic understanding of number. It seems we are all mathematicians!

Chemists John Watling and Allen Thomas talk to Plus about the vital role of maths in presenting criminal evidence.

The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in - that is, to describe the geometric rules which govern it. Based on these axioms, he proved theorems - some of the earliest uses of proof in the history of mathematics.