Computers can do many things, but there are some things they can't do. They certainly can't play tennis or the violin, but those aren't the kinds of thing we're concerned with. There are computational questions, questions of the kind that we would naturally turn to a computer to help us with, that, in fact, they cannot answer (and nor, therefore, can we).

There are many sorts of games played in a "bunco booth", where a trickster or sleight-of-hand expert tries to relieve you of your money by getting you to place bets - on which cup the ball is under, for instance, or where the queen of spades is. Lots of these games can be analysed using probability theory, and it soon becomes obvious that the games are tipped heavily in favour of the trickster!

Last October, two mathematicians won £1m when it was revealed that they were the first to solve the Eternity jigsaw puzzle. It had taken them six months and a generous helping of mathematical analysis. Mark Wainwright meets the pair and finds out how they did it.

The idea is this. To start with, you will choose an envelope at random, say by tossing a coin, and look at its contents, which is a cheque for some number - say n. (By randomising like this, you can be sure I haven't subconsciously induced you to prefer one envelope or the other.) You want to make sure that the bigger the number is, the more likely you are to keep it, in other words, the less likely you are to swap.

Money is peculiar stuff. It has no use of any kind apart from its value in exchange for something else, and this grows over time as it earns interest, or shrinks as inflation overtakes it. If you have money to invest, there are a bewildering array of different kinds of financial instrument available: interest-bearing accounts, bonds, pension funds, stocks and shares, options ...

This is a game played between a team of 3 people (Ann, Bob and Chris, say), and a TV game show host. The team enters the room, and the host places a hat on each of their heads. Each hat is either red or blue at random (the host tosses a coin for each team-member to decide which colour of hat to give them). The players can see each others' hats, but no-one can see their own hat.