I came to the Mathematical Association through our local branch and being involved in the Plymouth Annual Conference. After three years as Chair of Branches Committee I thought that it was time to setde back to the lazy golfing life of the south-west! Thus it came as a surprise and an honour to be invited to be President of our Association. I believe that one of the important features of the Mathematical Association is the branch structure and I am saddened to see the decline in active branches (including the West Devon and Cornwall Branch). Thus one of my wishes as your President was to visit as many branches as possible to support and thank the active members who keep their branch alive. Now I realise the reality of the travelling salesman problem and the challenge of travelling from Plymouth to the far flung corners of the UK (and Ireland) by public transport. But it has been possible and I have enjoyed visits to Dublin, Belfast, Exeter, Liverpool, Cheltenham, Leeds, Cardiff, London, Norwich, Sheffield and Hull as well as temporarily resurrecting our local branch in the south-west. I had the pleasure of visiting Glasgow two years ago and I look forward to Newcastle in May.

By analysing the forces acting on an oar during the power stroke it is possible to estimate the mean velocity given to a racing shell. Use is made of oarlock force data, obtained experimentally by the Australian Institute of Sport, to obtain accurate predictions in the case of a racing pair. Similar conclusions would hold for boats with any number of oars. The effect of altering the angle between oar shaft and blade is investigated, leading to a suggestion for a design change to improve oar efficiency. Predictions are made of the resulting improvement in boat position in a 2000-metre race.

For several years, the notion of ‘mathematical proof’ has been at the centre of much discussion, writing and educational research. To a very large extent the discussions have focused on the nature of the acceptability of the reasons for a mathematical conclusion. Admittedly there have been contributions concerning the role of proof in mathematics and also the mechanics of pupils‘ proof behaviour, including works which discuss both affective and pedagogical aspects, but the dominant context has usually been one of validation. That is to say, discussions have largely been in the domain of what ‘counts as a proof’ and for whom. However, I claim that there is another dimension to proof that is of central importance in the intellectual development of students at school and university. This relates to the phenomenon in mathematical thinking in which a section of mathematical structure is mentally compressed into a single unit, small enough to fit into the conscious focus of attention at a given time, and possessing an interiority which is able to both guide manipulation of the unit and also be subsequently expanded without loss of detail. This article discusses the role of this phenomenon in the manipulation of mathematical statements involved in proof. An analysis in these terms of the dynamics of certain types of mathematical proof suggests both an explanation for students’ difficulties and a possible means of help in some cases.

The publication of [1], which contains all, or almost all, relevant results from all over the world since Euclid’s time up to the middle 1960s, confirmed that this field of study has good traditions, is alive and in good shape. Two decades have passed and the appearance of [2] proved that this part of mathematics is flourishing more than ever. This field (which we will call GI) is a peculiar one: firstly because the subject matter doesn’t presuppose any great mathematical knowledge or technique; and secondly because it is so attractive that the best achievements in this field suit the refined taste of professional mathematicians, the unaffected taste of amateurs, and the yet unsophisticated taste of inexperienced intelligent students. Moreover, due to the specific features of this field of study, with some luck, all of them, as well as the enthusiastic mathematics teachers, have more or less equal chances to make remarkable contributions. As in any other branch of mathematics, a talent, good luck and/or hard work, can produce real gems, but in GI the achievements are more visible, more tangible for simple mathematical folk’s appreciation, and have undoubted educational value. They are well ready for easy usage by mathematical educators both at school and university level, due to their main advantage: many of them are simple, but by no means trivial.

How might we define x!? This is the next in my series of articles [1-3]; by now we know quite a bit about the factorial function, except that we have no real definition of it, nothing that would allow us to evaluate, say, ½!. So how do we move on to the next stage?

There is a range of strategies. We could find or manufacture a suitable function f that satisfies the basic relations

and adopt it as our definition.

Eperson's conjecture is that if n ⩾ 3 is odd then 3n2 can be written in at least two ways as a sum of three squares. We shall give a fairly elementary proof of this, from scratch.

Let q be a real or complex number with

Define for n ⩾ 1.

With one exception, namely {2, 3, 5, 7}, it is impossible to have four consecutive primes p1, p2, p3, p4 with p4 - p1 < 8. An interval of seven or less cannot contain more than three odd numbers unless one of them is a multiple of three. On the other hand, groups of four primes p, p + 2, p + 6, p + 8, usually called prime quadruplets, are fairly common. The first is {5, 7, 11, 13}, followed by {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829} and so on. Just as with prime twins, pairs of primes p, p + 2, it is conjectured that the sequence of prime quadruplets goes on for ever. Indeed, the apparently simpler prime twin conjecture is currently an unsolved problem of mathematics although in 1973, Jing-Run Chen proved a weaker form: There are infinitely many primes p such that p + 2 is either prime or the product of two primes (See Halberstam & Richert [1]). A similar result holds for quadruplets [1, Theorem 10.6]: There exist infinitely many primes p such that (p + 2) (p + 6) (p + 8) has at most 14 prime factors. The prime quadruplet conjecture would then follow if we could reduce ‘14’ to ‘3’ but this seems be a problem of extreme difficulty.

In his thought-provoking Note 82.45 (November 1998) about products of vectors, Bob Ardler asks (p. 454) “Why does dot thrive in n dimensions, but cross only in 3?” and (p. 456) “Which, if any, generalisations of the cross product have been pursued?”

Both of these matters have been treated fairly recently in several journals, including the Gazette, but the subject was first discussed and completely settled in 1942 by B. Eckmann in [1] (see especially pp. 338-339); cf. also [2].

Let us first consider the 2 x 2 matrices and Their sum and product are given by

Here the entries a, b, c, d, e, f, g, h can come from a field, such as the real numbers, or more generally from a ring, commutative or not.

Recursively defined sequences (in other words difference equations) appear in modelling systems changing in discrete time steps. The asymptotic behaviour of such sequences tells us information about the state of the modelled system after a long time. The simplest difference equations are linear, but the standard solving method of even these simple equations involves eigenvalue and eigenvector computation, complex numbers and linear algebra.

One of the more interesting uses of mathematics and statistics is their application to everyday situations in assisting people to make informed choices about their courses of action. Nowhere is this more evident than in the application of basic mathematical principles to the range of gambling opportunities. Indeed, a new university statistics course that I give in Australia on Gambling and Sport has attracted an enrolment of over 400 students in its first year alone. In this subject we discuss the intricacies of games such as Craps, Keno, Roulette, poker machines, horse racing, and a wide variety of lotteries. The sports content of the course relates largely to using mathematics to formulate optimal playing strategies.

The Belgian Jesuit, Gregory of St Vincent (1584 - 1667), along with Cavaleri, Fermat and Descartes was amongst those who prepared the way for the calculus of Newton and Leibniz. He was the author of one long book Opus Geometricum (1647) in which he gave a more precise account of progressions and limits than had been previously available and which Leibniz particularly valued. The term ‘exhaustion’, for the limiting process of the Greeks, is due to Gregory of St Vincent. This book contains a section on the hyperbola in which Gregory studied the area between the curve and an asymptote and this section provided the basis for Anthony de Sarasa's claim in 1649 that the area here was like a logarithm.