Matches of a certain popular brand are contained in a box designed to hold 100 matches. This box has two compartments so that if you open the box casually and without thought as I do, then you will select a match at random from one compartment or the other with equal probability. At some stage you will empty one of the compartments; and having more statistical insight, perhaps you will be less surprised than me by the large number of matches often left unused in the other compartment.

The way this problem is stated tempts one to think of it as an isolated puzzle. But there are a number of reasons why the problem may be worth investigating more closely. Perhaps you found it so mind-boggling that you couldn’t do it at all. Or you may have succeeded in finding one such list, but could not find a convincing proof that your list was the only possible solution. In either case it would have been worth having a go at a simpler version of the same problem first. But even if you solved the puzzle completely, the mathematician in you should be mildly inquisitive as to why it asks for a list of eight whole numbers, rather than say seven, or nine, or even six hundred and thirty four.

The dramatic increase in the speed of numerical calculations made possible by electronic wizardry has opened up new vistas in school mathematics in the last few years. One application is the solution of equations by iterative procedures. Interest so far has been only on the speed of convergence of the iteration and not on how a given iteration will succeed or fail—an equally interesting area. At school level this is a relatively uncharted area, and one where a sixth-former with a calculator could produce results of interest.

An interesting and important problem in probability theory is known as ‘Bertrand’s paradox’. Consider a chord chosen ‘at random’ inside a circle: what is the probability that the length of the chord exceeds the length of side of the inscribed equilateral triangle? Three different answers have been given to this problem p = ½, ⅓ or ¼.

The traditional method of teaching group theory, based on the axioms closure, associativity, identity, and inversion, leads to a large measure of abstraction in the learning procedure. The reason for this is the nature of group theory itself, with its abstract sets and binary operations defined within them. This traditional method of learning group theory could be compared to the learning of geometry without drawing triangles, something quite unreasonable, at least at the high school level. A further difficulty in studying group theory is the large number of concepts which are required from the beginning.

In the summer at Avignon a little Disneyland train conveys tourists from the Palais des Papes to the summit of a little hill behind the cathedral called the Rocher des Doms. In the garden on this hill there is a most remarkable sundial; the visitor himself provides the gnomon, and a linear scale on the flagstone pavement shows him where to stand at different dates. His shadow then falls towards one or other of a ring of fixed pyramidal markers which indicate the hours of the day.