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.

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.

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.

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.

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.

In a recent paper on Horner’s Method [1], which includes a compact method for dividing expressions, we mentioned that some Computer Algebra Systems (CASs) such as DERIVE could be used to make the calculations, but that such programs, even when obtained through educational establishments, are overly expensive especially when most PCs have spreadsheets on them that could equally well do the calculations. Here we describe the use of an Excel spreadsheet to divide one expression by another, first by the method of detached coefficients and second by Horner’s Method of Synthetic Division (or simply synthetic division). A third example uses Horner’s Method to replace x by (x + c) to form a new expression [2], useful in the determination of the roots of a polynomial.