## Mathematical mysteries

One of the most puzzling aspects of human behaviour is cooperation, in situations where backstabbing and selfishness would seem to be more rewarding. From the point of view of evolutionary theory, the very existence of altruism and cooperation appear mysterious.

Just over 220 years have passed since the death of one of the most distinguished mathematicians in history: Daniel Bernoulli, who died on March 17th, 1782. The name of Bernoulli asks for precision since the family from Basle produced no fewer than eight outstanding mathematicians within three generations.

Nineteenth-century German mathematician Leopold Kronecker once said

God created the integers, all the rest is the work of man.

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?

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.

Suppose you have an infinitely large sheet of paper (mathematicians refer to this hypothetical object as the plane). You also have a number of different colours - pots of paint, perhaps. Your aim is to colour every point on the plane using the colours available. That is, each point must be assigned one colour.

Chomp is a simple two-dimensional game, played as follows.

Cookies are set out on a rectangular grid. The bottom left cookie is poisoned.

Two players take it in turn to "chomp" - that is, to eat one of the remaining cookies, plus all the cookies above and to the right of that cookie.

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!

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.

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.