One of the problems that will vex any President of the Mathematical Association is the topic of the address with which he or she closes his or her year of office. This occupied me, on and off, for more than a year. In my case, in addition to my desire to acknowledge the honour of the invitation made to me, I was deeply conscious of the fact that I would be the 100th individual to serve as President. I dabbled with some pet themes, typically concerning the lack of genuine problem-solving or proof in English school mathematics, before concluding that the most sensible thing would be to talk on the topic about which I know most. My research interests are in comparative mathematics education. I have been fortunate, over the last twenty years or so, to have been able to visit and videotape mathematics classrooms in several European countries. In so doing I have had my understanding of mathematics teaching transformed in ways that led, almost inevitably, to the theme of both this talk and the conference which brought my Presidency to an end: Learning from Others.

Suppose that we have a computing machine that can only deal with the operations of addition, subtraction and multiplication, but not division, of integers. Can such a machine be used to find the decimal expansion of a given real number α to an arbitrary length? The answer is ‘yes’, at least in the sense that α is the limit of a sequence of rational numbers, and one can obtain the decimal expansion of a positive rational number a/b without division. For example, for any positive k, we try all non-negative c < b and d < 10ka, and see if 10ka = db + c. There will be success because all we are doing is reducing 10ka modulo b in a particularly moronic way. This guarantees the existence and uniqueness of d, and d/10k is then the desired expansion. Take, for example, = and k = 12; then, after a tedious search, we should find that, for c = 3, there is d = 230 769 230 769 because 3 × 1012 = 230769230769 × 13 + 3, so that = 0.230769230769 + × 10-12. Such a procedure of searching for c, d is hopelessly inefficient, of course, and a more efficient method is given in the next section.

The game ‘toss up’, manufactured by Patch Products (http://www.patchproducts.com/). is played with ten fair, six-sided dice whose sides are coloured green, yellow and red. In particular, each die has 3 green sides, 2 yellow sides and I red side. Ordinary fair six-sided dice may be used instead, of course, provided three sides are identified as ‘green’ (e.g. 1, 2, 3), two sides are identified as ‘yellow’ (e.g. 4,5) and one side is identified as ‘red’ (e.g. 6). Players take turns with each turn consisting of (potentially) several rolls of dice. All the dice are used on the first roll of a turn. Players accumulate points as they play and the first player to reach 100 points wins.

When William Wynne Willson died in the summer of 2010 he left behind an incomplete text covering a number of geometrical topics, including the so-called ‘Simson line’. Geoff Wain, who was completing and editing the text with a view to possible publication, sent me a copy. One chapter, entitled ‘Whose line is it anyway?’ contains the following paragraph.

‘In 1799 the Scottish mathematician William Wallace … published a geometric theorem. Nowadays this result is known as ‘Simson's line’. It is not clear why it should have this name as Robert Simson, another Scottish mathematician, had died aged 80 in 1768, the year in which Wallace was born. Although Simson published a considerable body of mathematical writings … there seems to be no trace of this theorem among them.

In an article entitled ‘A Mathematical Theorem about Golf’ [1] G.H. Hardy introduced a simple model of golfing. He assumed, that, at one hole, a golfer has probability p of gaining a stroke with a single shot, and probability q that his shot costs him a stroke. Such strokes will be described as good (G) or bad (B), respectively, leaving probability 1 − p − q for an ordinary (O) stroke (see also [2]). For example, on a par four hole, successive strokes OGO will result in a birdie (a score which is one stroke less than par) and BBGOO in a bogey (a score which is one stroke more than par). In this paper the probability distribution P(Tk = n) will be derived for the number of strokes T a player may take on a hole of par k.

Some years ago I was invited to take part in a fund-raising competition, [1]. The rules had an interesting feature that led me to analyse the situation mathematically. This resulted in some interesting formulae and one inequality, which I will describe here.

The competition offered a single prize and the entry form looked similar to the partially-completed example illustrated below.

The form listed 40 categories similar to the ones shown. Participants were required to write down an ‘answer’ for each category, e.g. for ‘an animal’, you might choose DOG. No other indications were given, so each answer was a completely free choice.

This article concerns the evaluation of the ‘logsine’ integral

We shall encounter it in several guises. Indeed, standard integration techniques used below readily show that (1) has the same value as the following integrals:

En passant, it is worth noting that forms (6) and (7) are the best behaved for numerical integration.

I first met the logsine integral as a callow youth in that strange hinterland of results that you may not have met at school but were not guaranteed to meet later on either.

Traditionally ‘Applied Mathematics’ meant mathematics applied to mechanics, physics and the hard sciences, leaving the application of mathematics to social sciences fairly neglected. This is both a challenge and an opportunity.

An instance of this is in the science of voting.

Let's go down to Chuddlehampton-upon-Severn to see how mathematics can be applied in a ‘real-life situation’. The good folk of this small village have to elect their local councillor, the person to represent them at the Regional Council. “Go and elect someone” was the only instruction they were given. So how were they to go about it?

In the midst of reading Robert Graves' I, Claudius [1], I was surprised to encounter a description of an interesting cipher used by the fictional Augustus. I therefore thought it would be worthwhile to write a short article to draw attention to what is perhaps not a particularly well-known cipher appearing in fiction. I will assume that the reader is familiar with the principle behind the famous Vigenère cipher (see [2] or [3]).

In the novel, Claudius, as narrator, relates how, upon Augustus' death, he is succeeded somewhat reluctantly by his stepson Tiberius, due to the machinations of Livia, who was Augustus' widow and Tiberius' mother. As her son's imperial reign begins, it is clear that Livia intends to hold on to the influence that she wielded during her husband's.

In n-dimensional Euclidean space no reflection with respect to a hyperplane can be realised by a rigid motion. But this is possible if we allow rigid motions in (n + 1)-dimensional space. These notes show a way to visualise a rigid motion of a cube in 4-dimensional space that flips the cube ‘as the page of a book’.

The two terms rigid motion and isometry are sometimes used as synonyms. Yet they do refer to different concepts. The first one has a purely kinematic connotation: the swing of a door or the movement of a piece of furniture pushed over the floor are described by rigid motions. On the other hand to ensure that two figures are isometric it is enough that there exists a correspondence between their points that maintains the relative distances.

In a previous note [1], Fermat's method of descente infinie was used to prove that the equations.

have no positive integer solutions. The geometrically based proof of [1] masked the underlying use of the difference of two squares. In the proofs of this article we shall make its use explicit, just as Fermat did [2, pp. 293-294].

We shall use the elementary idea of the difference of two squares to develop a powerful technique for solving equations of the form ax4 + bx2y2 + cy4 = z2. This will then be applied to three problems of historical interest.

Given two rational points on the graph of a cubic polynomial with rational coefficients, it is well known that one can construct another rational point by finding the third intersection point of the curve with the chord through the two points. For example, consider the two points A (2, 15) and B(0, 3) on y = 2x3 − x2 + 3.

The chord through A and B has equation y = 6x + 3 and this intersects the curve again at the point P, where

This cubic equation has three roots corresponding to A, B and the new point P,

P is the point

Step is a public examination administered by the admissions tests division of Cambridge Assessment (which is the parent body of the OCR examination board). It used as a basis for conditional offers by Cambridge and Warwick.

A grade one on a STEP paper is awarded to candidates who produce nearly complete solutions to four questions. The duration of the paper is three hours, so each question is designed to take a good candidate about 45 minutes to complete. Not surprisingly, then, the questions can be quite profound, often having depths far beyond what is required for the examination. One such question haunted me, on and off, for many years. It turns out that the underlying idea is elementary and far from new; but it is nevertheless very striking and not, I suspect, well known. A pleasing by-product is a relatively easy proof of the AM–GM (algebraic mean–geometric mean) inequality.