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.

David Hilbert, one of the giants of mathematics, delivered a lecture at the International Congress of Mathematics at Paris in 1900. The first part of the lecture, a preamble to his announcement of the now-famous 23 problems, began with the words:

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.

It is a trite but true remark to say that there are two fundamental random processes, one being the Poisson process and the other being Brownian motion. They are each fundamental in two ways; first, they do seem to describe much of the natural world very well, and second, the student is well-advised to study and master these two before going on to look at harder and more recondite processes. This note gives a brief glimpse of some aspects of the Poisson process, which is technically by far the more tractable of the two.

It is well known that Lewis Carroll enjoyed the company of children and entertained them with fantastic stories, whilst on boating expeditions or on beaches, but his diaries also reveal that he enjoyed teaching children mathematics and other school subjects. On April 16th 1855, he recorded his concern with the education of his younger sister Louisa, who had an aptitude for mathematics: ‘Went into Darlington – bought Swale’s Chamber’s Euclid for Louisa. I had to scratch out a good deal he had interpolated, (e.g. definitions of words of his own) and put in some he had left out. An author has no right to mangle the original writer whom he employs: all additional matter should be carefully distinguished from the genuine text. N. B. Pott’s Euclid is the only edition worth getting – both Capell and Chamber’s are mangled editions.’ Three days later he recorded: ‘Advanced Louisa’s mathematics to simple Equations (third day of Algebra), and the first 12 propositions of Euclid.’ On the next day, he left his home at the rectory of Croft, and returned to Oxford for the Easter Term.

I have a soft spot for geometric problems which lead to diophantine equations, that is, equations for which only positive integer solutions are sought. I have not seen the following naturally occurring problem discussed anywhere.

Problem A

A closed rectangular box has its volume numerically equal to its total surface area. If the length, breadth and height of the box are all positive integers, find all possible sets of dimensions of the box.

The solution of this pretty problem is completely elementary and can be fully understood by those with a basic knowledge of algebra.

We remark that to solve any three-dimensional problem it is often useful to step down a dimension and to consider the corresponding problem for plane figures.

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.

Some remarkable new series for π, in which the convergence is exponentially rapid, have been discovered by Bailey et al. [1, 2] using machine subroutines; see also Wagon [3]. An example is equation (17) below. Although the authors also provide algebraic proofs, these seem at first sight to be very special. The purpose of this note is to provide a more general demonstration and to deduce other instances of such series. It will be shown that all the results follow naturally from one or other of two simple lemmas.

In the recent Gazette [1], John Sharp has given us a most entertaining account of the number η (eta) which is the only real root of the equation x3 - x2 - x - 1 = 0 , and its connection with the snub cube. I have been investigating this a bit further, with some results that may be of interest.

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.

For convenience, if P, P′ are points in the Euclidean plane R2, [PP′] will denote the closed segment . Excluding the endpoints, we get the open segment ] PP′ [. Further, if Px is a half line, ]Px will stand for Px - {P}.

The Farey sequence of order n(Fn) is the sequence of all reduced fractions between 0 and 1, whose denominator does not exceed n, arranged in increasing order of magnitude.

For example, F6 is .

The Fibonacci sequence is the sequence for which u1 = 1, u2 = 1 and un = un-1 + un-2,namely, 1, 1, 2, 3, 5, 8, 13, … , un

At first sight, there is little connection between Farey’s fractions and Fibonacci’s integers. The purpose of this note is to show and explore such a connection and hence derive some properties of the Fibonacci sequence directly from previously proved properties of the Farey sequences. The former properties are well-known, but their rather unusual derivation from Farey properties may have some interest.

Consider the vulgar fraction 257/47. We can convert it to an integer plus proper fraction. If we now invert the fractional part and repeat the process on it, we get

This note describes a possible approach to the direct calculation of both derivatives and integrals of power functions using rather simple algebra and limits that could be easily accessible to students beginning a calculus course. The calculations provide a powerful example of the fundamental theorem, and could be used to allow students to discover the fundamental theorem from the example. The main calculation, which is not found in many calculus textbooks and is due to Descartes, is described by Maor [2]. While this approach may seem like the popular American ‘reform’ calculus, one may find a somewhat similar approach (presenting integral calculus from first principles) in Tom Apostol’s classic and lovely text [1].

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.