A familiar result is that in any group of six people there are three who either all know each other or are all strangers to each other. This can be reformulated to say that, if six points are marked on a piece of paper and every pair of points is joined by a red or blue line, then there will be a monochromatic triangle, i.e. one which is all red or all blue. This result is related to a theorem which the Cambridge philosopher Frank Ramsey proved in 1930.

I, in common with many others, dislike pretentious titles, particularly ones which disguise human problems in the jargon of computer hardware, so I sympathise if mine leaves you cold. The difficulties which I want to address are the emotional and academic ones so often encountered in the transition from secondary education to a degree course containing a major component of mathematics. They are difficulties of which I have been aware for many years, but on which my attention has been focused more sharply by two recent conferences: the first was the Mathematical Association annual conference in Manchester at which I was asked to lead splinter group discussions on this topic, and the second was the latest University of Nottingham Undergraduate Mathematics Teaching Conference at which I participated in a group discussing “Perennial difficulties” (and which will issue a report later this year). The very title used at the latter meeting expresses clearly the intractability of many of these problems.

A bag contains m > 0 black balls and n > 0 white balls. A sequence of balls from the bag is discarded in the following manner:

• (i) A ball is chosen at random and discarded.

• (ii) Another ball is chosen at random from the remainder. If its colour is different from the last it is replaced in the bag and the process repeated from the beginning (i.e. (i)). If the second ball is the same colour as the first it is discarded and we proceed from (ii). Thus the balls are sampled and discarded until a change in colour occurs, at which point the last ball is replaced and the process starts afresh.