As an industrial mathematician, who has done some teaching and has had some connections with the educational establishment, I feel extremely honoured to have been asked to serve the Association, this year, as its President. I feel it is a culmination of the growing links and dialogue between education and employment. I am happy to have been a part of that process.

A dance is to be arranged for n married couples in which all the women’s names are put into a hat enabling the men to select their partners at random. What we are going to consider is the number of married couples who end up dancing together.

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.

1. Introduction

Nearly forty years ago (and I barely knowing the three times table), my father taught me the twenty-one card trick, amid background murmuring from mother as to its mysterious mathematical nature. Now, with a rather more questioning six year old, I have been made to look at how it works, and see what variations a budding Paul Daniels might introduce.

About four years ago after consulting my copy of Cundy and Rollett [1] and then leaving it on my desk, I noticed that it had fallen open at a diagram of an approximation to Von Koch’s ‘snowflake’ curve. Interest in such curves has increased of late but for most mathematics students they make a fleeting appearance in the analysis course as examples of closed, continuous curves of infinite length and differentiable nowhere.

MacNab [1] has recently considered the Euler line of a triangle, and derived expressions for the position vectors of certain important points like the orthocentre, the circumcentre and the incentre. In this note I show how these results, and some other ones, can be derived by a more direct use of vector methods than MacNab employed. I believe that such applications provide excellent material for use in a first course on vectors since only elementary properties are needed. Nevertheless, the power of vector methods is well illustrated and, in addition, the results are interesting and the symmetry of the algebra is very appealing. While I suspect that much of the material is not original–for example some of the results are in the well-known book by Weatherburn [2]–I am not aware of the existence of a single concise treatment, and I believe this has potential value.