Of all the changes that have taken place in the mathematical curriculum, both in schools and universities, nothing is more striking than the decline in the central role of geometry. Euclidean geometry, together with the allied subject of projective geometry, has been dethroned and in some places almost banished from the scene. While educational reform was certainly needed there is always the danger that the pendulum may swing too far the other way and that insufficient attention may be paid to geometry in its various forms. Much of the difficulty here centres round the elusive nature of the subject: What is geometry? I would like to examine this question in a very general way in the hope that this may clarify the educational reasons for teaching geometry, and for deciding what is appropriate material at different levels.

The ‘digital root’ of 73219 is 4, since 7 + 3 + 2 +1 + 9 = 22 and 2 + 2 = 4. This idea features in some popular puzzles (and in a recent Gazette note 65.26). Section 1 of this article concerns some basic properties of digital roots, while Section 2 gives a characterisation of functions like the ‘digital root’ function. The former provides interesting exercises in modular arithmetic; the latter in elementary ring theory.

The Voyager missions to Saturn in November 1980 and August 1981 produced a wealth of new information and many spectacular pictures. Naturally attention focused on the physical structure of the planet and its major satellites and on the intricate fine structure of its rings. In the course of both missions many less spectacular observations were made, including the study of a number of small satellites. One of these tiny moons, Dione B (also known as S12), whose discovery from Earth was confirmed by Voyager 1, is the subject of this article.

The following note was prompted by the introductory remarks on magic squares in the stimulating source-book on linear algebra by T. J. Fletcher.

Let M(n) denote the vector space (over the rationals Q) of all nxn matrices with rational entries. Then a matrix A ∈ M(n) is said to be magic if all rows, all columns and both main diagonals of A have the same sum. Thus, as a trivial example, if B denotes the matrix all of whose entries are 1, then qB is magic for any q ∈ Q. We will denote the set of all nxn ‘magic-matrices’ by Mag(n); it is a straightforward exercise to check that Mag(n) is a subspace of M(n), and the purpose of this note is to give a reasonably efficient computation of its dimension.

I am unable to answer Ian Cook’s query (Gazette, note 65.25) concerning the original discoverer of the construction which he quotes from Nicholson’s Carpenter’s new guide, but readers may be interested in learning more about Peter Nicholson, who is variously described in his own books and in biographies as: architect, carpenter, surveyor, practical builder and private teacher of the mathematics.

Mazes and axe-shaped labyrinths have been with us since man discovered how to create problems. We think of them as being associated with Theseus and a certain bullheaded gentleman, religion, marriage rites, transport routes and eventually street maps. Our own maze anti-hero of note was probably ‘Capability’ Brown, who delighted in grubbing them up and landscaping the remains.

It is well known that if two people wish to share a cake fairly then one cuts it into two pieces and the other person chooses which of the pieces to have. This article considers the problem of sharing the cake fairly when there are more than two people. We are looking for procedures which ensure justice for an honest person even when there is dishonest collusion by the other people.

Spherical trigonometry has fallen somewhat out of fashion and so it was a salutary experience to be consulted recently by a would-be circumnavigator of the world.

A great circle route can be analysed graphically by first using a gnomonic chart. Such a chart is obtained by projection from the centre of the earth on to any plane. The distance of the plane from the centre determines the scale of the chart. On a gnomonic chart every great circle appears as a straight line and every straight line on the chart represents a great circle. Of course chart can only represent one hemisphere and even then the whole hemisphere would need an infinite chart.