As usual we define the p-multiplied L.T. of f(t) to be (1) and we adopt the abbreviation (2) introduced in 1938. The theorem in § 2 was obtained by the author in 1940. It would have been published, together with other work on L.T., some years ago, but for war-time restrictions.

In this article, we discuss a combinatorial problem arising from the National Lottery. The question considered is the following: Given a works (or any other) National Lottery syndicate of say n persons, how small can n be and still allow to ensure that at least one person in the syndicate has a winning ticket? Of course we will only be able to ensure that the required ticket wins the minimum prize of £10. However, we will show that a system to achieve this end can be constructed using some remarkable combinatorics which are well-known parts of modern mathematics. In fact, the system we propose for this problem has n = 290 and at least two tickets will win £10 prizes. It will be perfectly possible that the system will result in some members of the syndicate winning a fourth, third, second or even first prize. Although the combinatorics are quite interesting, it is not possible to ‘beat’ the National Lottery using our system. We have applied our system to the actual results of the National Lottery since its inception. Our experimental evidence shows that even using our seemingly efficient scheme, payout averages only about 28% of expenditure (showing an average loss of about 72%). However, in a typical draw, our system would actually have produced 5 or so ‘winning’ tickets for the syndicate.

Following on from Nick MacKinnon’s work in the June 1989 Gazette concerning four different methods of square-rooting 2 × 2 matrices, we may ask if it is possible to compute all the square roots of any given 2 × 2 matrix, and if so, how? From now on by a “matrix” we mean a real or complex 2 ×2 matrix. We know that every non-zero number has precisely two square roots, but the situation is rather more complicated for matrices. In fact we shall see that the zero matrix and any other multiple of the identity matrix has an infinite number of square roots. All other matrices have only a finite number of square roots, with some (rather surprisingly) having none at all! Our method of enquiry is based on Nick MacKinnon's first method, namely matrix diagonalisation.

We shall try to show that, in some well-defined sense, every right-angled triangle is ‘approximately similar’ to a Pythagorean triangle: and then we’ll ask how large the Pythagorean triangle must be.

It follows at once from Pythagoras’ theorem about a right-angled triangle that the sum of the squares of the lengths of the diagonals of a rectangle is equal to the sum of the squares of the lengths of the four sides. Apollonius showed that the assertion holds for a parallelogram and, more recently, Amir-Moez and Hamilton gave a generalization to quadrilaterals by introducing a correction term which depends on the distance between the mid-points of the diagonals.

“What is the probability that a randomly drawn triangle is acute-angled?” The problem sounds simple enough. However, what is interesting about it is the interpretation given to “random”. We investigate three possibilities

(a) that the angles of the triangle are chosen;

(b) that the lengths of the sides are chosen; and

(c) that the coordinates of the vertices of the triangle are chosen.

In my 1982 Presidential Address to the Mathematical Association I tried to explain the general role of Geometry in Mathematics so it seems appropriate, on this centenary occasion, that I should move beyond the confines of Mathematics and discuss the interrelation of Geometry and Physics. There are two very good reasons for doing this. One is historical and arises from the close ties between the two subjects in their early evolution. A second and more topical reason is that, over the past two decades, there has been a remarkable burst of interaction of a quite unexpected kind between Geometry and Physics.

When asked to give this lecture I was given to understand that this Society would like a change from the discussion of pedagogical methods and school problems, and could do with something that had nothing to do with teaching at all. In other words, I was to talk about stars and not to mention schools. Well, I promise to do that, though I must confess that the fact that members of the Society are teachers was at the back of my mind when choosing the subject. The schoolboy and schoolgirl and the man in the street know that celestial distances are very large; in fact the adjective “astronomical” has been sadly degraded by being applied to such mundane affairs as the figures of the national debt—and a schoolboy of enquiring mind may ask how the astronomers measure such distances. Such an enquiry is often tinged with a certain scepticism—sometimes politely veiled, sometimes not. When you tell a member of the public that a certain nebula is many million light years away and that a light year is 6 billion miles, he is apt to adopt the attitude of mind indicated by the phrase, “You’re telling me”, and to wonder what meaning such a monstrous distance can have: and how does the astronomer know, anyway?

A School time-table is one of the means by which we give opportunities to all pupil, a member of a class in a school, to study during a definite time (a school period) and place (a desk in a room: see below) a particular lesson, one of a prescribed syllabus in a subject, with the help of a particular teacher, member of the school staff. It thus involves a process of grouping the five variables—period, teacher, pupil, room and subject—according to certain desirable objectives and rules.