1.1 It might be supposed that I had chosen the title of this address to fit in with the great national debate on education which has taken place since the Prime Minister’s speech at Ruskin College in October last. However, this is not so (and thus this is not a trendy genuflexion to these recent events). It is entirely fortuitous that the theme does enable me to make references to the debate and to some of the issues raised by it. Before and at the Conference last year I kept wondering what approach I should take in this address. I knew that I wished to choose a theme which fitted in with my personal and recent experiences.

For some mechanical puzzles there exists a complete mathematical analysis. The purpose of this note is to describe one such puzzle, together with a number of variations of it. In the classroom the puzzle could be used as a teaching aid to introduce children to binary notation, mathematical induction, congruences, as well as some of the concepts of group theory. My own ten-year-old son was delighted to find that counting in binary notation, an idea to which he had been introduced in a purely theoretical way, leads to an easy method of solving the puzzle.

The problem discussed in this article is to find the number of patterns that can be formed by colouring the faces of a cube, each face to be of one colour, colourings being considered to be of the same pattern if they can be made to coincide by rotation of the cube. To make the method clear, we shall begin by considering the analogous problem of colouring a hexagon divided into six equilateral triangles.

1. Introduction. Recently I was looking for a simple but interesting model which a class of students could complete in about twenty minutes, which would illustrate groups of symmetries in three-dimensional space. My thoughts went back to the lampshade described in the Gazette for June 1971 (No. 393, pp. 310–312) but I felt that it would take too long to make, even if each student merely cut out a strip which I had marked out beforehand. Forced in this way to reconsider the matter, I found that the last sentence of my article was false—there are in fact four other such structures, which I now describe.

In his interesting article on Redesigning the dartboard, published in Gazette No. 413 (October 1976, pp. 171–178), Keith Selkirk refers to a dartboard he has designed by attempting to minimise the magnitude of the vector resultant obtained when vectors of magnitude equal to the score and in the appropriate direction are added.

The sequence 0,1, a, aa, a(aa), . . . has certain unexpected features which may easily be demonstrated with the aid of a programmable pocket calculator. The investigation of the convergence of this sequence provides an exercise which is both interesting and informative.

The discussion of symmetry in modern mathematics syllabuses can be taken a stage further in the sixth form to show how symmetry can often be used to simplify the solution of certain types of problems. This article gives just one example, that of determining the eigenvalues and eigenvectors of an n × n matrix. The theory involves two steps. In the first we show how the eigenvalues of a matrix M can under certain circumstances be found from those of a commuting matrix P. In the second step we show how to construct, from the symmetry properties of matrix M, commuting matrices P whose eigenvalues are readily found.

Several writers have drawn attention to the fact that there is a connection between determinants and sets of loops in directed graphs. This fact is sometimes useful in evaluating the determinant of a matrix M, as was explained by Greenman in his recent article [1]. Towards the end of his article he introduces Δij-subgraphs and relates them to the cofactors Cij of the matrix M. Now if M is invertible, then M−1 = adj M/det M, where the (j, i)-entry of the adjoint is just the cofactor Cij. Hence one would expect to be able to give a graphical explanation for the fact that M(adj M/det M) = I, and this will be done in the present article.

There are various ways of introducing a discussion of the completeness of the system of real numbers. In this article we take the unusual approach suggested by the provocative question of the title.

In any elementary treatment of the convergence of sequences and series, a discussion of series of positive and negative terms will lead to the theorem that an absolutely convergent series is convergent.

Have you heard of the game of Amida? It seems to be unknown in Europe, but in Japan is very commonly used as a method for drawing lots. It also provides an engaging example of a one-one correspondence which may be useful to teachers in several contexts.

The procedure is as follows. The adjudicator draws on a sheet of paper a sort of compound ladder consisting of a number of uprights, one for each contestant, joined by horizontal rungs, in such a way that each rung joins a pair of adjacent uprights, and no two rungs have a common endpoint on an upright (Fig. 1). Subject to these rules, the adjudicator may place the rungs as he pleases, and as many or as few as he likes, even none at all.

An artist friend, Mr Graham Day of Corsham, Wiltshire, recently sent me the following, asking whether he had accidentally discovered a ruler-and-compass construction for a regular pentagon. His own main interest in the configuration stemmed from a similarity to one in Indian mystical art.

A favourite exercise on Simpson’s rule is to evaluate π as the area of a quadrant of a circle of radius 2, i.e. as the area under y = √(4 – x2) from x = 0 to x = 2. It is somewhat discouraging, and a poor advertisement for the technique, to arrive at the answer 3·127 008 with intervals of width 0·2.

Let π be the plane spanned by the vectors b and c. The product vector b × c is normal to this plane and hence a × (b × c), being perpendicular to this vector, will be in π.

In a recent Note [1] a nomogram was described for calculating the times of sunrise, sunset and twilight in terms of the latitude of the observer and the time of year.

I found the article Chemically, the same or different? by G. T. Woods [1] to be very interesting. On p. 253 of that article the problem of providing a formula for the number of isomers, as a function of the number n of carbon atoms, of certain molecular structures is suggested.