When I was offered the chance to become the President of the Association, I had no hesitation in accepting. How else could I react to the opportunity of joining such a distinguished list of predecessors? The list, which includes G. H. Hardy, Arthur Eddington and G. N. Watson contains almost all of the eminent UK mathematicians and educationalists of the twentieth century. So for the honour that you bestowed on me, thank you very much indeed. It wasn’t deserved, but it was hugely enjoyable and rewarding.

You should never judge a book by its title - but I nearly always do. A mathematical colleague recommended a book to me Concrete Mathematics [1]. Just recently, someone else also referred me to the same book. Well a wink’s as good as a nod to a blind man and I now realise what a mistake I’d made; concrete is a conflation of ‘continuous’ and ‘discrete’. Like all good textbooks in mathematics, it distinguishes itself not just by the quality of its contents and presentation, but also by the superb collection of problems that it includes. One such problem - question 58 in Chapter 6 - inspired this article. There was a hint towards a generalised result, but there were no details. It did give a reference, [2, 3], but despite some efforts I found this unobtainable. Hence this article and my attempt to cover the same ground. The question posed in the book is concerned with the linear recurrence relations satisfied by the successive powers of Fibonacci numbers - so we start with the Fibonacci sequence itself.

During this lecture I am working from the premise that all the world is a mathematical classroom. By the time children begin formal education they have spent up to five years in this classroom [1] and during their years of schooling the fraction of time spent in a formal classroom is minimal compared to the time spent out of it. I wish to argue that this broader definition of a classroom impacts in many strong ways on our conceptions of mathematics and helps to construct the misconceptions about the nature of the subject that impinge on the learning of mathematics and the value which we place on it. It is within the formal classroom that many of the misconceptions about mathematical knowledge and skills develop. Although a great deal of writing exists about these, and while teachers use this knowledge to inform their teaching I wish to place more emphasis on the less familiar conceptions and misconceptions of mathematics. I will be challenging you to examine your own conceptions by showing you examples from television, radio, advertising, newspapers, film and literature. I will ask you to reflect on how a consideration of these aspects of pupils' experiences of mathematics might affect our teaching. I am also going to ask you to think about whether influencing young people's conceptions of mathematics might affect their decisions as to stop or continue studying the subject.

An elementary algebraic problem attributed to Leonardo of Pisa is analysed and some illogical elements in its formulation and solution are exposed. The natural context in which the problem was formulated is then proposed, and some consequences are discussed.

How many times have you heard that somebody is a wizard? And how many times did you take it literally? Most likely, the answer to the second question is ‘never’. And yet, there are reasons to believe that some people among us are real wizards, of the kind described with so much charm in the recently published series of books on Harry Potter [ 1, 2, 3, 4 ]. These books have provided us with a wealth of details on wizards and their society.

Those who enjoy number patterns will no doubt have had many hours of pleasure exploring the Fibonacci sequence to various moduli, and especially in recognising the regularity with which various prime factors occur in thesequence (see [1]). Gill Hatch’s question is whether the occurrence of prime factors in generalised Fibonacci sequences is similarly predictable. Generalised Fibonacci sequences (Gn), abbreviated to GF sequences, are sequences of positive integers derived from the recurrence relation tn + 2 = tn + 1 + tn. In the case of the Fibonacci sequence (Fn), the first two terms are 1 and 1.

We wish to discuss some aspects of repetends, the repeating sequence of digits in the expansion of a fraction (for illuminating introductions to the subject see [1, 2]). For the most part we restrict consideration here to fractions with a prime denominator. But we do consider the general condition for the length of repetends and examine some special cases when the base of the number system is varied. An illustration of the use of other bases than 10 is given. Then we consider the multiplication of repetends and show a connection with group theory, giving an old result by a new twist.

C. A. B. Smith (c.1918-2002) holder of the Weldon Chair of Biometry at the Galton Laboratory was a member of the Mathematical Association for over sixty years. Amongst his scientific work, he developed the standard statistical methods now used to map genes onto chromosomes and he wrote a text book Biomathematics. He recently contributed a poem written by the mysterious ‘Blanche Descartes’ to the Gazette and it was published posthumously.

Recently a poem, written by Blanche Descartes in the 1930s, was republished in the Gazette, with the additional attribution ‘communicated via Cedric A. B. Smith’. Cedric had reported ‘it was written after Blanche heard that Hector Pétard was about to be married. She composed the verse as a wedding present. Hector is still active mathematically, though I suspect that he may be in his mid nineties’. We felt that readers might be interested to know something about the mysterious Blanche Descartes, and the origins of the poem. It makes quite a story. At the risk of boring you, we will start at the beginning.

A vivid memory of mine is of being shown that certain 2 × 2 matrices were ‘really’ complex numbers. The correspondence between matrices of the form

and complex numbers

seemed magical and the additional remark that

was an unexpected bonus. The correspondence has been examined in various articles in the Gazette [1, 2]. That it is an isomorphism is seen by comparing

with

in the case of addition, and, in the case of multiplication, by comparing

with

Lester’s theorem (1997) states that in any scalene triangle the two Fermat points F and F' (to be defined later), the nine-point centre N, and the circumcentre O, are concyclic, and that the pair of points O,F separates the pair N, F'. (In certain geometrical situations a line is regarded as a circle of infinite radius, so that the word ‘concyclic’ includes ‘collinear’ as a special case, but here ‘concyclic’ means ‘lying on a proper circle of finite radius’.) Previous proofs of Lester’s theorem have involved advanced techniques and/or computer algebra; to quote from Ron Shail’s recent article [1],

Morley’s theorem is well-known: if we trisect the interior angles of any triangle A0A1A2, then the common point of each pair of trisectors adjacent to a side is a vertex of the Morley triangle, M0M1M2, which is always equilateral (Figure 1). Less well-known is that the two triangles are in perspective, that is, the three lines AiMi, concur at a point M.

The pairs of trisectors further from the sides meet at W0, W1 and W2, the vertices of the anti-Morley (or perhaps the Worley) triangle. It, too, is in perspective with A0A1A2, the centre of perspective being W. Also, the Morley and anti-Morley triangles have a centre of perspective X, the points M, W and X being collinear.

Candidates are graded by their examination results in some such way as 1st B; 2nd equal A, D; 4th equal C, E; and so on. In how many ways gncan n candidates be graded? This problem was solved by Cayley [1, pp. 374-378] and later by others. It does not seem to be well known, and the following result may be of interest.

In a recent article [1], Ron Shail has given a Cartesian proof of an interesting theorem due to J. A. Lester. This states that, for any triangle, the circumcentre O, the nine-point centre O9 and the two Fermat points F and Fʹ, (which are the points of concurrence of the joins of its vertices to the vertices of equilateral triangles drawn outwards/inwards on the opposite sides), are concyclic. He refers to Lester's own treatment as needing complex coordinates with computer-assisted algebra; his own proof uses an unpromising method, and results in similar problems. Contemplation of the configuration would suggest that the location of the point of intersection of FFʹ with the Euler line OO9 might lead to a simple proof. The theorem is in fact a corollary from the properties of a remarkable configuration originating with Morley [2, p. 209], and shown in Figure 1. He did not deduce Lester’s result, nor label the crucial point J in the diagram, which was drawn without that particular intersection. Also involved in this figure is a rectangular hyperbola known by the name of its describer Kiepert [3]. It is helpful to discuss all three together. What follows is a journey through country nowadays rather unfamiliar, avoiding the computerised motorway and using older tracks via complex numbers, trilinear coordinates and Euclidean methods which reveal much more than is apparent from a Cartesian treatment.

Heron triangles are those with integer sides and integer area. It is well known how to construct them as the union or difference of a pair of integersided right-angled triangles with a common side. For example, the triangles with sides 8, 15, 17 and 20, 15, 25 may be united, with common side 15, to form an acute Heron triangle with altitude 15, base 28 and other two sides 17 and 25. Its area is 210. Alternatively, their difference may be formed to create an obtuse Heron triangle with altitude 15, base 12 and other two sides 17 and 25. Its area is 90. Triangles with rational sides and rational area may be enlarged to have integer sides and integer area, and so may be classed as Heron triangles also. There have been many articles about Heron triangles in recent times, both in the Gazette [1,2] and elsewhere [3,4, 5], to mention just a few. This is not surprising as the number theory involved has a direct and pleasing application.

Apart from false position and double false position, another numerical method for calculating roots of equations was known to the Ancient Chinese. Chia Hsien in the eleventh century is reputed to have given an algorithm for calculating roots as well as describing Pascal’s triangle. The algorithm was mentioned again by the twelfth century scholar Liu I. The Chinese used the method to solve quadratics and cubics as early as 100 BC, but it was not until 1247 that Ch’in Kiu-Shao from South China published its extension to higher order polynomials in his work, Mathematics in nine chapters. A year later in the book, Sea-mirror of circle measurements, Li Yeh, who was from north China, took root-finding for granted. The fact that these quite independent writers published similar work suggests that finding the zeros of polynomials was well known by the middle of the thirteenth century. It was left to one of China’s greatest mathematicians, Chu Shi-kie’ (ca. 1280–1303), to give this algorithm its name fan fa, which means the method of the Celestial Element or sometimes the Celestial Unknown [1,2]. Translations and spellings of these older Chinese words do not always give the same result. This algorithm eventually became known as the Ruffini-Horner Method or more simply Horner’s method.

Water waves are familiar to all of us and we encounter them in a variety of guises in many places, be it crashing to shore at the beach, rippling concentrically outward where a pebble lands in a pond or simply splashing at the sides of the bath. The study of waves can be simplified by idealising them as graphs, each graph being thought of as a cross-section of a physical wave at an instant in time. A sequence of such graphs can represent the progress of the wave as time passes.