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.

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.

The editor of this journal recently deplored the inability of members of the government to make correct use of the simple mathematics that they encounter in their various capacities. The purpose of this article is to call attention to a common mathematical failing of those who are “authorities” in questions such as the life expectancies of non-renewable resources. My particular example comes from Britain, although failures such as this one are common among “experts” in the U.S.

Dr. C. D. Langford recently sent me some numerical work he had done on automorphic numbers. He found automorphic numbers in the scales of 6, 10, 12, 14, and 15 and he discovered what proved to be a general method of finding an automorphic number of r + 1 digits from one of r digits. Dr. Langford’s notes stimulated me to look for explanations of his results.