Two prominent themes in the Presidential Address upon which this article is based were Enthusiasm and Challenge.

Many, if not most, practitioners of Mathematics can trace their interest in the subject to an enthusiastic teacher at some stage of their education. In this day and age, it is all too easy to get bogged down in a morass of assessment and quality assurance. Yet it is vital that teachers of Mathematics at all levels retain their enthusiasm and pass this on to their students.

Geometry is gradually coming back into the school syllabus, but so far only 2-dimensional geometry. I would like to make a case for including some 3-dimensional geometry as well, because the latter is vital for describing the world throughout science, engineering and architecture. Higher-dimensional geometry also comprises a major part of modern research within mathematics itself. Also 3-dimensional geometry fosters both our intuitive understanding and our geometric imagination. It teaches us to see things in the round. It also trains us to see all sides of an argument simultaneously, as opposed to algebra and computing which emphasise thinking sequentially.

The n th roots of unity play a key role in abstract algebra, providing a rich link between groups, vectors, regular n-gons, and algebraic factorizations. This richness permits extensive study. A historical example of this interest is the 1938 challenge levelled by the Soviet mathematician N. G. Chebotarëv (see [1] or [2]). His question was ‘Are the coefficients of the irreducible factors in Z[n] of xn − 1 always from the set {−1, 0, 1}?’ Massive tables of data were compiled, but attempts to prove the results for all n failed. Three years later, V. Ivanov [3] proved that all polynomials xn 1, where n < 105, had the property that when fully factored over the integers all coefficients were in the set {−1, 0, 1}. However, one of the factors of x105 − 1 contains two coefficients that are −2. Ivanov further proved for which n such factorisations would occur and which term in the factor would have the anomalous coefficients. A twist that makes this historical episode more intriguing is that Bloom credited Bang with making this discovery in 1895, predating the Chebotarëv challenge by more than four decades.

In [1], Tomislav Došlić studies a special case of the classic Ass and Mule Problem and determines when integral data give integral solutions. The question of when integral data gives integral solutions in the general case intrigued me some years ago; I even published the assertion that I could not see any simpler result than computing the answers and seeing if the results were integral [2]. However, I later found a satisfactory result for this and some variants of the problem, which are given in [3, 4]. Here I give this result, which describes an infinite set, and show it leads to a simpler derivation of Došlić's interesting result that his variation has only 16 positive integral solutions. A different method permits finding the nonpositive integral solutions. I will then state a similar but unsolved problem.

In [1] it is shown that every second-order linear difference equation with, in general, variable coefficients can be reduced to the form

It is also shown that (1) can be solved explicitly provided that a particular solution a (n) of the ‘auxiliary equation’

can be found, and many cases in which a solution of (2) is available are discussed. However, [1] is defective in that no method of finding a solution of (2) in the general case is given.

Given a mathematical operator, how should one reason about the outcome of its repeated invocation? This question is relevant in both mathematics and computer science, where iterative operator invocations are core, algorithmic elements.

An initial approach, which one may naturally follow, is to try the operator in diverse situations, observe the results, and suggest a general outcome. Such an approach embodies operational reasoning, where inference derives from ‘behaviours’ of invocation sequences. This may indeed reveal some behavioural characteristics, but is it sufficient for rigorous argumentation of the general outcome? Not quite.

The two oldest representations for the number π are infinite product expansions. The first,

is due to Vieta in 1592. The second is Wallis's product dating from 1655:

Consider the functional equation

Now let and These are elements of a group, G1 say, which is isomorphic to C2, the cyclic group of order 2, as can be seen from the group table:

Let p1, p2, p3, … be the sequence of primes. Take any integer n ⩾ 2 and partition the set {p1p2, … , pn} into two non-empty subsets A and B. Let PA be the product of primes in A and PB be the product of primes in B. No partition exists with PA = PB, because of the uniqueness of prime factorisation, but Fernando Castro’s neat result in [1] asserts that for each n there is a partition for which 1

The values of many functions, for example sinx or cosx, can be computed efficiently for arbitrary values of x from their known power series expansions. However, for the logarithmic function this is not so straightforward. There does not exist an expansion of ln x as a power series in x, the closest to it mathematically being an infinite power series in of the form

where (see [1]).

Just how many matrices have inverses? In elementary linear algebra courses, many of the matrices encountered are singular, but perhaps the reason is that such matrices provide rich and interesting examples. How many of them occur naturally? Many beginning students observe that a singular matrix can be made nonsingular by very minor tweaking – changing just one entry, for example, will make a matrix of rank n – 1 into a full rank (nonsingular) matrix. In fact, changing one entry just a tiny bit will do it. Looking at the question from that point of view, with a little experimentation students begin to discover that singular matrices are quite rare. Entire rows (or columns) have to be rigged exactly right in order to get one, while minor changes in individual entries undo all the work and give us another nonsingular matrix. If we were to reach into a hat full of all the numbers – each equally likely to be chosen – and draw enough to fill in a square matrix randomly, we would certainly expect to get a nonsingular one.

The importance of teaching our students at all levels how to use computers as a problem-solving tool is well-recognised. Using numerical approximations to discover unknown formulas can be a powerful example. It is usually pointed out to the students that computer exploration alone is not sufficient and should be followed up whenever possible with a proof or at least some additional analysis. The importance of this is illustrated in the example below, in which there is compelling numerical evidence for an incorrect conclusion.

In [1] the author investigated a method of measuring how good the spread of lessons is for a class in a school timetable. The method considered only the maximum number of days between two consecutive lessons. However, it was observed that not all configurations with the same maximum gap are equally good. For example, the configuration for a tenday cycle in Figure 1(a) would be preferable to that in Figure 1(b); the filled circles represent days with a lesson and the unfilled circles days without. Here we propose a model that takes into account the overall configuration when measuring how good the spread of lessons is.

A magic square is one in which all rows and columns, and the two main diagonals, sum to the same total. A knight’s tour is a tour of the board in which, using knight’s moves, all squares are visited exactly once. When the squares visited are numbered from 1 to 64, if the square is magic (but without including the two main diagonals), this is termed a magic knight’s tour. This paper describes two magic knight’s tours on an 8 by 8 board found in early 2003, the first new tours to be discovered since 1988, and the first irregular tours to be discovered since 1936.

In [1], Derek Ball made three conjectures about equiangular polygons in which the length of each side is an integer. He called these integer equiangular polygons. The first two conjectures were proved in an earlier note, and the third is proved here. It can be stated as follows.

The tiling conjecture: For each integer n ⩾ 3, there is a finite set Tn of tiles such that every integer equiangular n-gon can be tiled by sufficiently many congruent copies of tiles in Tn.