Some articles (like the previous one) are concerned with the construction of magic squares. Here we observe a simple matrix property of some 3 × 3 magic squares.

Finding myself one day involved in a Beetle drive, and somewhat deficient in the “will to win” essential for success in such competitive circumstances, I began idly to speculate on some of the probability problems involved—they proved to be more absorbing, and more complicated, than I had originally envisaged.

If we draw a lot of circles centred on the x-axis and all having radius 1, as in the diagram, then the eye picks out the horizontal lines y = ± 1 running along the tops and bottoms of the circles.

This is the story of a problem that began in an innocent way and in the course of time spread out in unexpected directions involving some interesting side-issues. It all began with the following problem which I constructed for the Mathematical Challenge competition run by the Scottish Mathematical Council.

It is a popular pastime amongst mathematicians to search for primes. The latest ‘largest known prime’ is often quoted in the popular press. A common place to search for primes is amongst numbers of the form 2p − 1, where p is itself prime. The purpose of this article is to present two interesting sequences which provide rewarding classroom exercises. A list of these exercises is given and some conclusions regarding the prime-ness of the so-called Mersenne number 2p − 1 are drawn. The final test for prime-ness which we deduce was devised by Edouard Lucas in France at the end of the nineteenth century. The proof, which we have tried to break down into a sequence of elementary exercises, is due to D. H. Lehmer, who in the 1930s used his remarkable electro-mechanical machine to find the largest prime then known.

If asked to give the origins of the calculus, most pupils who know anything about the subject would answer that Newton and Leibniz invented the calculus in the seventeenth century. It is interesting then to be able to demonstrate the method by which Archimedes found the volume of various solid objects in the third century B.C.

In a recently published book by Greenspan classical dynamics is modelled by a set of difference equations. There are two distinct reasons why this approach could be useful. First, since all our measurements are made at specific times and are thus not continuous it may be that the model is a “better” representation than the usual one using the calculus. Secondly, since most practical problems are now solved by numerical techniques, and one of the most used of these is the method of finite difference, a formulation of the basic equations in difference form makes them directly amenable to numerical solution. There are numerous ways of modelling the derivatives as differences and Greenspan chooses a way which preserves the energy equation. This is aesthetically pleasing and also gives rise generally to stable numerical schemes.