A simple lock contains a bolt and one or more levers which are lifted the correct amount by the appropriate key, thus allowing the bolt to be moved in and out. In addition there are often circular obstructions or “wards” to stop some keys being turned in the lock.

A very appropriate use of a computer in a mathematics class is to provide investigation vehicles for formulating conjectures to be verified theoretically. In other words, first we notice something and then we try to prove it. In this paper we shall reverse the process, first arguing the plausibility of a very fascinating result and then obtaining confirmation by looking at some computer generated data. The argument is based on the work of the eighteenth century mathematical giant, Leonhard Euler (1707-1783).

As a mathematician involved in teaching students whose abilities range from barely numerate to MSc level, I am frequently concerned by the lack of basic understanding exhibited by pupils regarding the subject of integration. Invariably the most practical way of introducing the subject of integration is by thinking of it as “anti-differentiation”, so that the problem is to pick a function which “when differentiated will give what you first thought of”. This of course may be allied to the idea of the area under a curve to give the student a conceptual feel for the subject. Usually when simple rules for the integration of powers, trigonometric functions, exponentials, logarithms and so on have been mastered, the next step is to introduce the standard methods for proceeding.

It is my purpose to establish a set of inequalities between symmetric homogeneous polynomials in several real variables. That may sound a bit grand, but the simplest example of such an inequality is the familiar

A more complicated inequality of the same general type is the inequality

Again increasing the degree of complexity we have the pair of inequalities

At a very early age we are taught such theorems as “the angle on the circumference of a circle subtended by a diameter is a right angle”. This is part of our Euclidean heritage. But from Euclid's terse deductive style it is often difficult to draw links between the geometrical arguments which he presents and more modern mathematics. The purpose of this article is to step off the Euclidean path, dispense with the obligation to present formal deductive arguments and explore. The treatment is elementary and we make use of computer graphics. The graphs in this article were produced by the Picaso graph plotting package using the Calcomp 960 plotter on a Vax 11/785 computer.

In a previous article in the March 1979 Gazette I looked at a population of 589 students entering a particular university department between 1971 and 1975 to read for a degree in mathematics, or in mathematics-with-another-subject. These students graduated between 1974 and 1978.

The present article provides updated information for students entering between 1976 and 1983 (graduating 1979 to 1986 respectively). Several factors have changed since the earlier survey. A significant change is the increase in the number of entrants offering single-subject mathematics at A-level. A second important factor has been the consequences flowing from the UGC cuts in universities in 1981; admission targets are now much more closely controlled and the level of offer to intending entrants has often tended to rise (certainly the case at this institution). There has been, as a result, a decrease in the number of students entering with poorish grades.

This article could be subtitled ‘Thoughts on contemplating a model of five tetrahedra inscribed in a dodecahedron’; it is an attempt to communicate the pleasure that ensues when logical reasoning is combined with a visual delight in geometrical figures.