Given a straight line L in the plane, how can we find all lines parallel to L? Here is one way: at each point P of L draw a straight line perpendicular to L. Now choose a number d and measure a distance d down these lines, always in the same direction. Joining up the points P′ so obtained gives the parallel to L at distance d.

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.

The problem of expressing numbers as the sum of two cubes of non-zero rational numbers has a long history, some results being given by Diophantus. Several mathematicians have since contributed to our knowledge of the problem, in particular in the discovery of numbers which cannot be expressed as the sum of two cubes. Fermat discovered that 1, 18 and 36 belong to this class of numbers, Euler and Legendre demonstrated that so do 3, 4 and 5 and some infinite sets of such numbers were discovered by Sylvester. However, much of interest still remains to be discovered in this area. In this article I hope to both summarise the existing results on this problem and to produce a new class of numbers which cannot be expressed as the sum of two cubes. Most importantly, a general method of tackling the problem will be described which should stimulate further investigation.

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.

We will consider two related maximisation problems relative to an arbitrarily given triangle. Their solutions will illustrate the power of cyclic quadrilateral constructions and the utility of Ptolemy’s theorem for obtaining symmetric formulae.

I have just received my latest transfusion of fresh ideas. What teacher can be enthusiastic day after day, year after year, without pausing to renew his strength, revitalise and remotivate himself? We need occasionally to be fired by contact with better minds than our own, to observe excellent teaching and thinking to remind ourselves of what they are like. For such excellence is the ideal we all should seek. It is so easy to have one’s spark attenuated year by year by the dull monotony of ritual in a school’s monastic life—so that one hardly notices.

It has always seemed to me that one of the most fascinating parts of learning mathematics is the recognition of an inter-relation between what seemed before to be totally unconnected branches of mathematics. These relationships, which often help to cement ideas in both branches of the subject, are very often not discussed in school text books, perhaps because authors worry that they may be assuming too much background on the part of their readers.

The great mathematician Henri Poincare [1] described the method of proof by mathematical induction as mathematical reasoning par excellence. Induction is, undoubtedly, a method of great significance for mathematics. Unfortunately, induction is also a rather complicated form of proof. Furthermore, it is taught in schools at sixth form level. In fact, it is probably the only method of proof explicitly taught in British schools. Perhaps it is not surprising that some students find the method difficult to apply. In view of the importance and difficulty of the method, mathematical induction is worthy of further attention. In this article, the history and equivalents of mathematical induction are discussed.

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.

It is now widely accepted that the first or “kinematic” part of Einstein’s 1905 paper on Relativity [1] can be made accessible to interested non-specialist students. This part is mathematically straightforward, although its first readers found it conceptually puzzling. “From this”, remarks Einstein, as he follows one of the simpler results of applying the Lorentz transformation to our measures of space and time, “there ensues the following peculiar consequence…” It was these “peculiar consequences” that repelled many physicists of his own day. Later, the challenge to intuition caught public imagination and led to the many popular and semi-popular introductions to the Special Theory. Today there are also several accounts available at a mathematical level appropriate to sixth form or college students.

The widespread availability of electronic computers has widened the domain of possible investigations at all levels in mathematics, in the classroom no less than in advanced research. For instance, provided convergence is not exceptionally slow, series may be summed quite accurately and very quickly using a pocket calculator. This enables us to check theoretical work like the evaluation of certain integrals which can be expressed as series.

The cracking or tearing of thin sheets of material is of considerable interest in modern-day engineering. A recent investigation into the behaviour of concrete indicated that it would be of interest to consider a probabilistic model in which a crack is regarded as some form of non-self-intersecting random walk in the plane. This paper considers some simple models and in the Appendix are included some results on paper tearing.

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.