This extremum problem is really a classical beauty. It has had a long and interesting history since it was formulated by Fermat in the 17th century. Given three points A, B and C, the task is to find a point P such that the sum of distances PA + PB + PC is minimal; (see Figure 1). After a few years Torricelli found the solution: P should be situated so that the angles between the half-lines PA, PB and PC are all 120° (except when one angle of the triangle ABC is greater than or equal to 120°). The solution has then been rediscovered many times in new and interesting ways.

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.

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.

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.

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.)

The reader will be familiar with the golden rectangle, i.e. that rectangle which leaves another rectangle of the same shape when the largest possible square is cut from it (Fig. 1). In this article the effect of successively cutting squares from rectangles of other shapes will be investigated.

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.