During the past 20 years the notion of set has been introduced into school mathematics courses not only at the secondary but also at the primary level. When one considers that neither Newton nor Gauss made any explicit use of the concept, it is rather remarkable that teachers have thought it desirable to teach the concept so early in their school courses. I should like to consider briefly some of the many possible reasons, historical and pedagogic, why “set” has become a school topic. One reason is, of course, the belief which Russell and Whitehead fostered in their Principia mathematica in the first decade of the century, that all mathematical concepts can be reduced to the concept of set. I don’t intend to discuss here the extent to which they failed to justify that belief, although I shall have occasion to mention some of the difficulties which their programme ran into. A second reason is the important part which sets have played in mathematical research this century.

The term measuring number is acquiring currency as a popular synonym for non-negative real number. This is good. Yet the belief persists that the real numbers, by whatever name, must remain fundamentally inaccessible to school children, obscured behind a veil of Dedekind cuts or Cauchy sequences. (Thus Griffiths and Howson, in what is generally a humane book [1], devote several pages to an ingenious, but ultimately condescending, attempt to foreshadow the mysteries of real numbers in terms of points of a line.)

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 would like to present a method for calculating determinants based on the notion of a graph (or network), a notion that is familiar to many pupils taking a modern mathematics course. The method involves four steps: (i) drawing the graph of the matrix; (ii) identifying certain subgraphs called Δ subgraphs; (iii) associating a number called a weight to each of these subgraphs; (iv) adding the weights to obtain the determinant. Let us take each of these steps in turn.

This article is an exploration of some common ground between network theory in mathematics and isomerism in chemistry. As with all interdisciplinary studies, it is necessary to begin by identifying technical terms across the boundary between the two subjects. Once that has been done, the diagrams drawn by the chemist are interpreted in the language of geometry, and links are established between results in the two subjects.

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.

Many communist countries hold internal olympiads in order to identify and thereafter foster special talents. Since 1959 they have held an annual International Mathematical Olympiad (IMO) for all European communist countries (except Albania) and subsequently Mongolia, Cuba and Vietnam. Non-communist countries were first invited in 1967 and France, Sweden, Italy and Great Britain accepted the invitation. Italy has dropped out and Austria, Netherlands and USA now regularly participate. Belgium, Finland and, last year, Greece have also taken part. Each country sends a maximum of eight competitors and two adults.

The Alexandrian Greek mathematician Heron—whom we associate with the formula √{s(s − a)(s − b)(s − c)} for the area of a triangle—got involved in a calculation about a pyramid design which led to the evaluation of √(81 − 144). This occurs in his book Stereometria (C. A.D. 75) and to ‘solve’ it the numbers are turned round thus: √(144 − 81), to give √63 which is taken to be . (Is this a reasonable approximation for √63?) It is not known whether Heron made this transpositional error or whether a copier of his work was responsible. This seems to be the first occasion in which the square root of a negative number was stumbled across—a concept not properly understood for another 1750 years!

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.

I like an active Saturday afternoon, so it comes about that I often end up watching sport on television. The other day it was speedway racing. I gradually became aware that I was watching 16 riders riding four at a time in a complicated arrangement of heats, the object of which was to ensure that each rider rode against every other rider exactly once. Naturally (?) there were 20 heats; but the question arises as to what numbers of riders may be arranged into a tournament, given the firm speedway requirement that four riders are to be on the track at a time, and the requirement that each meets each just once. It is also a problem to arrange the riders into heats suitably.

Have you ever looked at a dartboard and wondered why it is numbered in its distinctive order?

If we ignore the bull, the inner, and the double and treble rings, the standard dartboard takes the form of a circle divided into 20 equal sectors numbered as in Fig. 1. (There are boards to be found in the Manchester area—and perhaps elsewhere—with a different arrangement, but the one shown is by far the commonest.)

A frieze pattern is an array of numbers arranged in staggered rows, like the bricks in a wall. In the case of a multiplicative frieze pattern the top and bottom rows consist entirely of ones, and for each ‘diamond’ of four elements

the unimodular rule

ad = bc + 1 (2)

holds. These frieze patterns have many curious properties, and for further details the reader is referred to the article Triangulated polygons and frieze patterns by J. H. Conway and H. S. M. Coxeter which appeared in Gazette 57, 87–94 and 175–183 (1973).

“A fair coin is tossed successively until either two heads occur in a row or three tails occur in a row. What is the probability that the sequence ends with two heads?”

Thus the problem was stated on the Cambridge Scholarship paper. My problem on being asked for help was to produce a solution using only sixth form methods, and hopefully a solution which would prove interesting and instructive. Certainly the problem had captured the imagination of my questioners, although the method for its solution was proving to be elusive.

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.

It is easy to design simple gearing enabling a clock or instrument to indicate hours, minutes and seconds, since the ratios are ‘man-made’ integers. If an instrument is required to indicate astronomical data we find that nature is not so considerate, and the gearing presents an interesting problem.

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.

The construction of curves of constant width using circular arcs is well known; the procedure may be found, for example, in [1]. This article describes a different method for constructing a family of ‘smooth’ curves of constant width. Basic properties of such curves may be found in [1].

Let C be a regular, smooth, and convex curve in the euclidean plane. Regularity implies that each point of C lies on only one support line and each support line contains only one point of C, smoothness implies the existence of derivatives at each point of C, and convexity implies the curve is a simple closed curve whose interior points form a convex set. Select a point O on C as origin, use the support line to C at O as the x-axis, and give the curve a counter-clockwise orientation (see Fig. 1).