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.

I would like to offer the following contribution to the discussion on the Choice of programming language for schools which will (I hope) ensue following F. R. Watson's and David Tali's article in the December 1987 Gazette. (lam happy to consider other points of view for publication: Ed.) As the authors observe, if the schools do not have a policy of their own, they may well have one they don't like foisted onto them.

While I worked at Teeside Polytechnic I had some difficulty in arriving in the morning by 9.00. Not being an early bird my desire was to pull into the carpark at 8.58, but this I did not seem able to achieve. If I left home at 8.15 the journey was smooth and trouble-free, and I was at my parking place by 8.50. Whereas if I left a moment later there were a couple more cars at the estate exit, each succeeding roundabout and set of traffic lights was busier, and eventually I drove into the car-park at 9.05—not only five minutes late, but also quite likely finding the spaces all full and having to resort to street parking. It seemed that no matter how I varied my start time the arrival interval of 8.50 to 9.05 was somehow inaccessible to me. Of course some people did arrive in this interval—if I did the sensible thing and left early I enviously watched them arrive at the time I would have wished. But they did not start from where I started. So I was drawn to the belief that the relationship between start and arrival time was a genuine discontinuous function, and not simply a continuous function with a rather steep critical section. Maybe the catastrophe theorists could analyse this further (but to call my misfortune a catastrophe would be a bit strong).

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.

There is at the moment considerable pressure to change the course of A-level mathematics, pressure which is now so intense that fairly radical change is both inevitable and desirable. A variety of causes have brought us to this situation. In the era of post-Cockcroft mathematics we have seen many of its recommendations implemented in the 11-16 age range. Most notable perhaps are;

1. (i) the provision of differentiated courses and examinations,

2. (ii) the encouragement of greater pupil participation in their own learning of mathematics,

3. (iii) 0 the construction of courses using a ‘bottom up’ approach.

A tree is a network of vertices connected by edges such that the removal of any edge disconnects the network. In the language of graph theory a tree is a connected simple graph in which every edge is a bridge. Here are all the trees with 4 vertices.

In 1742 Goldbach sent a letter to Euler which contained the conjecture that:

Every even number greater than 2 is the sum of two prime numbers.

It is easy to verify Goldbach's conjecture numerically up to some quite high limits but no proof has yet been found. I have long had the uneasy feeling that the conjecture is not about the prime numbers at all but would equally apply to any ‘sufficiently dense’ sequence of odd numbers. However a recent attempt to generalise the sieve of Eratosthenes has led me to regard Goldbach's conjecture with a good deal more respect.

This perennial problem has led to many articles in the Gazette (listed by Sullivan in the June 1987 edition) many of them geometric or combinatorial in nature. If we look at some older algebra texts, published before the 1960s say, we see that the method of differences is one of the standard methods for dealing with the problem. With the current interest in discrete mathematics this method merits re-examination. A feature of the method is its effectiveness in finding the solution to the problem for all powers and it is not limited to the sums of squares and cubes. In this article we hope to show that it can be treated more transparently using the calculus of finite differences.

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.

For a given positive integer n, the nth triangular number, Tn, is defined by

it is consistent to extend this definition by writing T0 = 0 and T-n-1 = Tn, so that the above formula holds for all integers n, whether positive, zero or negative.

Here is an interesting result on sequences amenable to investigation and proof in the classroom. Consider the sequence

where, as usual, [x] denotes the integer part of x. You will find that you get the sequence

which consists of the natural numbers with the perfect squares omitted. This result is due to H. Halberstam [1] and in this article we verify it and look for generalisations to sequences which omit mth powers.

Some mathematics problems, while easy to pose and visualise, can be deceptively difficult to solve. For example, suppose we want to find the area of a three foot wide footpath around the edge of an elliptical shaped swimming pool. The problem is trivial if the pool is circular, in which case we simply subtract the areas of two concentric circles. At first blush, the elliptical pool problem does not appear to be much more difficult, especially if we can recall that the formula for the area of an ellipse is A = πab where a and b are respectively the lengths of the semi-major and semi-minor axes. In any case, we might think that it would be a straightforward calculus problem involving the area between two curves.