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 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.

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.

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).

Today, “sets” are taught in schools at a variety of levels and the basics of set theory are reviewed in many college mathematics courses. After students have gained some facility with the rudiments of set theory, it is instructive for them to see what would happen if our set theory were a little different. This not only reinforces their knowledge of the standard theory, but exposes them to other possible mathematical structures. This article shows one way in which this might be accomplished.

The article Problems, solved and unsolved in the Gazette for last March (60 (411), 1–11) drew reactions from a number of readers. There is not sufficient space to show in full all the proofs sent in, but I will try to indicate the methods used. For statements of the problems themselves I must ask readers to refer to the earlier article.