The purpose of this paper is to consider the question: if p is the probability that a player P will win a given rally against a player Q, what is the probability that P will win a given game? The answer, of course, depends upon the method of scoring; and the methods considered here will be those used in squash and, for comparison table tennis.

In table tennis P earns a point each time that he wins a rally. But in squash he must serve the rally that he wins in order to gain a point; and he relinquishes the service when he loses a rally that he serves.

Have you ever looked at a dartboard and wondered why it is numbered in its distinctive order?

If we ignore the bull, the inner, and the double and treble rings, the standard dartboard takes the form of a circle divided into 20 equal sectors numbered as in Fig. 1. (There are boards to be found in the Manchester area—and perhaps elsewhere—with a different arrangement, but the one shown is by far the commonest.)

This article provides a brief introduction to Evangelista Torricelli and to one aspect his work. It is divided into three parts. Part 1 gives some biographical details and a short survey of his work and achievements. Part 2 is concerned with a geometrical approach towards the study of projectiles as seen in Torricelli's early work De motu. Part 3 shows how Torricelli applied his projectile studies to practical ballistic problems.

A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.

When I started teaching real analysis to first-year college of education students, I thought that at least I ought to be consistent in what I said. When I told the students what a word meant, I thought they ought to hear the same story as they heard from my colleagues who were teaching other branches of mathematics. I even thought I ought to tell them something like what they would find in a consensus of the real analysis books they might look at. I knew that I ought not to use different meanings for the same word on different occasions. Alas . . .

We have developed a novel diagrammatic approach for understanding and teaching probability theory — Probability Space diagrams [1]. Our studies of learning and instruction with Probability Space (PS) diagrams have demonstrated that they can significantly enhance students' conceptual understanding. This article illustrates the utility of PS diagrams by applying them to the explanation of some difficult concepts and notoriously counterintuitive problems in probability. We first outline the nature of the system.

By a nearly diagonal matrix is meant a matrix in which the off-diagonal elements are small compared with those on the diagonal. Such matrices occur, for instance, in connection with non-orthogonal experimental designs. Inversion may be performed by methods of general applicability such as the Doolittle or Cholesky procedures (direct methods) or relaxation (an iterative method); many such procedures are described in the literature. Those already familiar with any of these methods may well prefer them to the one described below because of their wider applicability, but anyone not so placed who is often concerned with the inversion of nearly diagonal matrices should find the method described here useful.

The offer of real money is a powerful inducement to work at a problem. So when Sir Bryan Thwaites put £100 behind his conjecture about what happens when we repeatedly take positive differences of a sequence of rational numbers, I could not resist his challenge. My solution may not have been the first to reach him, nor the most elegant, but it was fun to do. In showing that any sequence of 2n such numbers would eventually reach a string of zeros, my argument (and the one given in) depended heavily on the fact that they were rational numbers. I couldn’t help wondering what would happen if we started with a sequence of real numbers.

The study of convex polyhedra from a mathematical point of view began in ancient Greece well over two thousand years ago. It is believed that Euclid’s “Elements” was written to inspire a wider appreciation of the properties of the five regular polyhedra or “Platonic solids”, as well as to place their study on a secure mathematical foundation. The discovery of convex polytopes (the analogous figures in four or more dimensions) and their investigation by the Swiss mathematician Ludwig Schläfli just over a century ago, led to a renewed interest in polyhedra, and since that time hundreds of papers on this subject have been published. However, there are still many unsolved problems, and the purpose of this article is to state a few of these. These problems have widespread appeal because they can be readily understood with very little previous mathematical knowledge, and although they may be quite difficult, there is always the possibility that someone with a very minimum of mathematical background will devise a method of approach that leads to a solution.