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.

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.

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:

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.

I wanted to call this article, Counting permutations that contain a specified number of cycles of a given length but the Editor pointed out the problems this might cause the Production Editor - it simply wouldn't fit the running head. Hence the chosen title and the subject matter. There is a very well known result, nonetheless startling, about the relative frequency of derangements amongst the permutations of [r] = {1,2,3,… , r }.

In many of the basic courses in Number Theory, Finite Mathematics and Cryptography we come across the so-called arithmetic functions such as ϕn), σ(n), τ(n), μ(n), etc, whose domain is the set of natural numbers. These functions are well known and evaluated through the prime factor decomposition of n. It is less well known that these functions possess inverses (with respect to Dirichlet multiplication) which have interesting properties and applications.

There are infinitely many regular polygons, but we find, on extending the idea of polygons to three dimensions, that there are only five regular polyhedra, the Platonic solids. What happens then if we try to extend this idea beyond three dimensions? It turns out that, of the five Platonic solids, just the regular tetrahedron, cube and regular octahedron have analogues in all higher dimensions, the so-called regular polytopes. Brief descriptions of these mathematical objects are to be found in [1], for example.

The real projective plane is a classical mathematical object which plays an important role in several areas of mathematics, including topology. This article gives methods for making three-dimensional models of the real projective plane out of wool. By describing these models, and how to make them, the article aims to help the reader to visualise the real projective plane more easily.

The aberrancy of a plane curve is a property of the curve that is invariant under both translation and rotation. It provides a numerical measure for the non-circularity of the curve at each point of the curve. (Recall that curvature gives an invariant numerical measure of nonlinearity.) The concept of aberrancy has been around for two centuries, but it has received very little attention. We hope to stir a little interest in the concept by presenting four different derivations of the formula for aberrancy. This may appear to be redundant, but there are some good reasons for doing so. First of all, the only derivation for aberrancy in the literature is a bit confusing and makes a few unjustified assumptions. Secondly, the derivations we present are all significantly different from each other and involve some interesting ideas in elementary real analysis. Finally, although the basic idea behind each derivation is simple, the details can become extremely messy unless a proper path is chosen. We encourage the reader to find one or two approaches of particular interest and use the concept of aberrancy as an extended problem set in an undergraduate analysis course or even a further mathematics class in an English school.

A tsunami usually starts on deep ocean water as a result of a submarine earthquake. A tsunami wave is very long, even as much as tens of kilometres, but of only very small amplitude, typically less than half a metre (Bascom [1]). In mid-ocean, the passage of a tsunami is imperceptible, but on reaching a shore it can achieve great heights and can deliver massive surges of water. Before the arrival of the first surge, and between subsequent surges, the water at a shore line usually retracts for a long distance, leaving bare large areas of ocean floor that are normally under water. This paper analyses the behaviour of a tsunami, and explains how its mid-ocean character is transformed to produce such massive surges of water at a shore line.