Problems have been important in the history of mathematics, for the attempts at solution, whether successful or not, have often given results much wider than the original questions. The more obdurate have become notorious. Though circle-squaring and duplicating the cube are obsolete, angle trisection survives. This problem, Fermat’s last theorem and the map-colouring question still appear occasionally in the Problem Bureau’s correspondence.

I want to deal with one piece of applied mathematics which is a good example of the way in which—in nearly all applications of mathematics—the important steps are those of tentative model-building, and the testing of such models against reality. It is this feature that makes mechanics such an atypical case; there the model, Newton’s laws, is made for one beforehand. In the present instance the initial problem, whose partial solution leads, as usual, to other problems, can be stated very simply: in most cities the population density, σ, tends to fall off with distance, r, from the centre.

An amusing exercise to give to a mathematical sixth form is:

If the roots of the equation

F1(x) ≡ x3 + x2 − 2x − 1 = 0 (1)

are α, β, γ, form the equation whose roots are α2 − 2, β2 − 2, γ2 − 2.

The usual methods used will be, either to find the coefficients of the new equation by forming the appropriate symmetric functions of α, β, γ; or else to eliminate x from the relation y = x2 – 2 and the given cubic equation.

Many teachers are now familiar with sets of non-singular matrices which form groups under matrix multiplication. It is not so commonly known that multiplicative groups of singular matrices exist. The purpose of this article is to give some examples of such groups, to discuss their general features, and to indicate a method for the reader to construct further examples when required.

Venn diagrams are usually considered to be a method for visualising set-theoretic identities and suggesting true theorems. It is frowned upon to prove a set-theoretic identity using them, such proofs being considered non-rigorous. One of the better kept secrets of the trade is that they can be made rigorous by proving the right kind of ‘metatheorem’. A proof is quite instructive, and might be of interest to set-theoretically minded sixth formers.

Imagine six drawing pins stuck in the wall. If we stretch threads, either red or blue, between every pair of pins, we inevitably create at least one triangle all of one colour. This we shall call Theorem 1: the proof follows.

Consider any pin. From this emanate five threads (since there are five other pins), of which three must be of one colour. We may without loss of generality assume that they are three red threads.

In many situations, for example in teaching convergence to students not specialising in mathematics, an approach to the idea of a limit is required which is easily motivated, mathematically acceptable, and yet not technically terrifying. The usual outcome is a compromise which lies uneasily between these aims. Here, I try to present an introduction to convergent sequences (of real numbers) which meets these objectives; it can easily be adapted to accommodate limits of functions. The approach is based on the sandwich rule (or squeeze principle) and the intuitively straightforward idea of the limit of a monotone sequence.

At some time in the eighties or nineties of the last century the Rev. William Done Bushell, a mathematics master at Harrow School, bought a portrait painting at an auction sale at a house in Harrow. It does not seem to have had any pedigree but bore the inscription “Robt. Recorde M.D. 1556”. It was thus presumed to depict the Tudor physician and mathematician Robert Recorde (1510?–1558) who wrote some of the earliest books in English on mathematics and was the first to use the symbol “=” for equality. No other portrait was (or is) known.