Thank you for the honour of being your president. Being such was not an ambition of mine – it was not even considered as a possibility. However, I have enjoyed the experience, and will for the few more hours which are allowed to me.

In the course of preparing this address, I looked at the article about presidential addresses written by Mary Bradburn in the 1996 Centenary Issue of the Mathematical Gazette. That was a mistake – too many different models to follow, each sounding more interesting than my own ideas. One odd thing, to my mind, and something not special to the MA, is having a President give the presidential address at the very end of the term of office. I always feel that hearing someone give a lecture helps me to find out a bit about them; so a presidential address is a chance for all to learn about their President – but too late, too late!

At school I used to love the geometry of the circle, but in my own career I had little opportunity to teach it. Now in semi-retirement my work with Teasers in the Sunday Times brings me into contact with many entertaining aspects of mathematics, not least being the occasional piece of geometry. In this short article I will quote four potted versions of recent Teasers, each of which relies on some interesting piece of mathematics, and I shall sketch in each case how that property can be deduced neatly using circles.

Every year, thousands of students around the world encounter Ferrers graphs. Most of these students probably assume that Ferrers was a mathematician and that his name is among the hundreds at a well-known website specialising in mathematical biography. The first assumption is correct.

Norman Macleod Ferrers was born 11 August 1829 in Gloucester. In 1851, he received a BA degree from Cambridge University, with honours. He was Senior Wrangler and winner of first prize in the Smith mathematics competition. He studied law and was called to the bar in 1855, but returned to Caius College in 1856 as a mathematical lecturer, and was ordained apriest in 1861.

Many professional mathematicians wish to contribute to making mathematics more attractive to gifted young people. One way to pursue this goal is to create of problems that require a great deal of common sense, imagination, and, very often, a specific problem solving strategy. This article introduces such a strategy, the principle of the extreme element. Although the name is not widely used, the principle may help you to solve some non-standard mathematical problems. The material is based on my personal experience gained in extracurricular work with mathematically gifted students, and in my participation in mathematics competitions at different levels as a contestant and organiser.

An ellipse is ellipse-shaped, of course! Every mathematician knows what an ellipse looks like – at least, this is generally true in the Cartesian plane, with the standard distance function or metric. However, there are other metrics which can be applied to the plane ℝ2 and it might be useful to know what an ellipse looks like when distances are defined by these metrics. The set of points under consideration will be ℝ2, the real plane, and the metrics under consideration will be the standard metric, the Manhattan (or taxi-cab) metric and the maximum modulus metric.

The Independent [1], prior to the 1996 Wimbledon tournament, carried a story to the effect that Tennis had a problem. Rackets were getting longer and, so the International Tennis Federation (ITF) claimed, rallies were getting shorter as the serve dominated. The story provided some surprising figures. The current limit to the length of a racket is 32 inches, but the longer rackets which are the cause of the problem are 29 inches, 2 inches longer than the standard racket. The article claimed that such rackets 'give players up to 14 per cent more power than standard models'. The article went further,

Prodigiously talented with a remarkable flair for anticipating correct results and initiating fruitful areas of research, Stanislaw Ulam (‘Stan’ to his friends) was an unusual mathematician. The considerable breadth of his preoccupations, even in the halcyon days of his youth in Poland, distinguished him from his peers. Sustained by a supreme self-confidence, Ulam preferred pioneering new fields of mathematics to elaborating the ideas of others. Indeed, impatient with detail he tended to leave technicalities to those with whom he collaborated. Uprooted from his native Poland in his twenties, Ulam was to spend two thirds of his working life associated with or employed by the Los Alamos National Laboratory.

My interest in this fascinating area of mathematics was aroused by a recent article [1] in which it was shown that

has an infinity of non-trivial solutions (i.e. u ≠ v) in positive integers, rationals and algebraic irrationals. Replacing u and v by their reciprocals, x and y, permits the equation to be written as

In a recent issue of the Gazette S. Parameśwaran defined a positive integer n to be an S•P number if it is the product of the sum of its digits and the product of its digits; for example, 135 and 144 are S•P numbers because

His conjecture that there are only finitely many such numbers was proved in where it was shown that any S•P number is less than 1060. Here, we prove the following result.

Ann Hirst and Keith Lloyd have derived many interesting properties of the Cassini ovals rr′ = a2, r and r′ being the distances of a variable point P from two fixed points A and B, which we shall find it convenient to take a unit distance apart. r, r′ are often referred to as bipolar coordinates and by relating them in different ways, a variety of interesting families of curves can be defined; such families can be associated with other families which are their orthogonal trajectories. The purpose of this article is to draw attention to a number of such cases.

A positive integer, n, which is equal to the sum, S, of its digits multiplied by their product, P, is called an S·P number. S. Parameśwaran gave three examples of such numbers in:

and he conjectured that there are only a finite number of them. This was confirmed when several of us proved that each S·P number has at most 60 digits. There the matter rested for a time. Even if it were possible to test all numbers from 1 to 1060 at the rate of 1040 per second (far beyond the capacity of today’s fastest computer) it would take much longer than the estimated age of the universe to complete the task.

Consider the motion of a particle of unit mass acted upon by a homogeneous, central, conservative force f(r), where r is the distance between the centre of force and the particle. In this connection, homogeneity means the following property of the force: there exists a constant α such that for every c

Arithmetical properties of numbers have been a source of fascination for mathematics students and professionals throughout the ages. In this paper we consider a puzzle of sorts in which we move the first digit of a positive integer A to the last position and establish conditions under which A will be equal to a multiple of the new integer. For instance, if A equals 820512 then A = 4A’where A’ = 205128.

In Isaac Asimov's novel Foundation and Earth (Grafton 1987) the inhabitants of the planet Solaria have evolved from early Earth space explorers. They have developed a society with many robots and few people. The small population of Solarians live as far from each other as possible. These solitary people are confined to their estates, which are distributed evenly across the planet. In this article I investigate the pretty problem of even distribution of points on a sphere. With the exception of one piece of spherical trigonometry the work needed to derive my (admittedly restricted) set of results is routine for sixth-form mathematicians but has the interesting pay-off announced in the title.

This article shows a link between Pell’s equation and the right-angled triangles for 60° and 45° via the trigonometric ratios for 15° and the extraction of square roots.

It is well known that the trigonometric ratios of 60°, 30° and 45° are easily obtained from the triangles ABC and RST which are shown in Figure 1.

The genesis of this problem, which was communicated to the first author by Robert Vertes, is in a proposed ‘ice-breaking’ activity at a party attended by an even number of guests. Each guest is assigned a unique number from the set {1, 2, …, n}, where n is even, and their task is to form themselves into ½n pairs so that the sum of the numbers in each pair is a perfect square. It is not difficult to verify that this cannot be accomplished with fewer than eight people, while eight people can be split into the four pairs {1,8}, {2,7}, {3,6} and {4,5}, each summing to the square 9. Sixteen people can also be split into eight pairs with the required property, viz. the four pairs listed above together with {9, 16}, {10,15}, {11, 14} and {12, 13}. There is no requirement that every pair must sum to the same square.