Euclidean geometry and fractions not only make unlikely bedfellows, but would seem to be a sure recipe for boredom. However, in this paper, with the help of a little history, we hope to prove the opposite.

Given a (rectangular) lattice, then a lattice polygon is a polygon whose vertices are lattice points. Pick's theorem [1] - the area of a simple lattice polygon is given by ½b + i – 1 where b is the number of lattice points on the boundary of the polygon, and i is the number in its interior – is well known and has been proved many times and in many ways.

A fairly well-known problem is that of counting the number of different colourings of the set of faces of a polyhedron, using n colours. Of course it depends on what is meant by ‘different’. As perhaps the simplest non-trivial example, let n = 2 (red and blue say) and consider the regular tetrahedron. If we regard the tetrahedron as static, then perhaps the answer is 24 = 16, since each of the 4 faces can be red or blue. However we are in the habit of picking up objects and rotating them around, so maybe a better (and more interesting) interpretation of ‘different’ is to regard two paintings as distinct if one cannot be rotated into the other. We are left with the following 5 distinct possibilities :

The subject of centrifugal force comes up at all of our mechanics INSET meetings and it seems that it continues to be a problem when we teach circular motion. Consider the problem of the conical pendulum. The forces acting on the bob are its weight and the tension as shown in Figure 1:

Frequently students put a force outwards in addition to these forces and label it the centrifugal force (or mrω2). It is undeniably true that the lay person (and therefore perhaps all of us at an intuitive level?) feels that this force is there and this is presumably why it appears on force diagrams; after all, it seems quite natural, ‘how else does the bob remain out at an angle if not held by an outwards force?’

First year mathematics undergraduates entering British universities are often very unprepared for the kind of proof activities that occur throughout undergraduate degree programmes. Their previous experiences of the idea of proof are frequently limited to situations where the ‘proof’ is simply an extended chain of calculations or algebraic manipulations. In fact, this is well-illustrated by their approach to proof by induction, where they can carry out the mechanical details of the inductive step, but have no understanding of why proof by induction works, nor what status the initialisation step and the inductive hypothesis have in the proof. There is no real grasp of what it means to prove that some assertion is true. Much time, in consequence, is spent in the first years of mathematics degree courses up and down the country attempting to bring about this understanding.

Thomas Kuhn made his reputation by pointing out that science is not just a steady series of rational developments, but is subject to paradigm shifts – revolutions in which the entire point of view changes. A less flattering word for the process is ‘fashion’. But what is fashion other than a paradigm shift in a rather trivial area of human activity? It is often thought that mathematics is immune to paradigm shifts. After all, mathematics consists of perfect truths, proven by impeccable logic. Doesn't it? Well, no, not really – but even if it did, that wouldn't prevent paradigm shifts. Paradigm shifts are not about the internal logic; they are about the external viewpoint. They are about seeing the same facts in a new light, incorporating them into a new framework.

We all have experience of waiting in a queue. We have waited in the dentist's surgery for treatment, we have queued at the supermarket check-out with our trolley or stood in line at the Post Office to buy stamps. No doubt we have had frustrating experiences in such situations, but before we rush to criticise the discipline of a queueing system we should think about the mad stampede for the next bus that can occur in some parts of the world.

The reader should first try the following exercise, which motivates the later work.

Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.

This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?

A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.