That Spring morning, the bent old man sat down at his large oak desk with his pen momentarily poised over the familiar blue foolscap sheet. He furtively glanced around the drawing room with its old china and flowers and books in abundance and wrote in one easy movement: The principles of book-keeping by double entry. He looked up once more but this time allowed a minute to gaze out through the window to the Cam gently meandering through the meadows towards the Cambridge colleges.

The early composers of change ringing music for English church bells were not mathematicians, yet they developed intricate algebraic ideas more than a century before mathematicians independently discovered them. We introduce the mathematical concepts of permutation group and symmetry group by means of elementary change ringing compositions of the 17th and 18th centuries.

The subject of polygon loops was introduced by Gerry Price, a County General Inspector in Hampshire for Mathematics, and Roger May, Teacher Adviser for Mathematics, Southampton, during an evening meeting for teachers of Mathematics in 1991. The question posed was, “Can you always make a loop with a set of congruent regular polygons?” Since then various articles have appeared in which polygon loops of various kinds have been discussed. Ann MacNamara and Tom Roper [4] described how Year 9 students worked with octagon loops in a trial Standard Assessment Test, and Jackie Davies [2] used a particular family of square loops with primary children. Grunbaum and Shephard [3, pp. 104-106] show plane tilings in which one can spot polygon loops of equilateral triangles, squares, and regular hexagons, octagons and dodecagons.

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.

This article considers the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle. The techniques are applied to produce new results in the geometry of the triangle. This is an extended version of part of the contribution: How do computers change the way we do mathematics? given at the Association’s 1994 Easter Conference.

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 :

In the geometry of the triangle, the Feuerbach quadrangle comprises the four points of contact between the nine point circle and the four tritangent circles. By omitting each of the four points in turn, we create four Feuerbach triangles. When the basic triangle is isosceles we find that two of the four points of contact coincide and the resultant single Feuerbach triangle is also isosceles. In the spirit of Steiner-Lehmus we address the converse question – if one of the Feuerbach triangles is isosceles, is the basic triangle necessarily isosceles? The answer is negative.

Most of us who teach “complex numbers” (especially to classes where the emphasis is primarily mathematical) take the opportunity at some stage to place the theory firmly on the concept of ordered pair (of real numbers), assuring the students that they have long been familiar with this notion (e.g. plane cartesian coordinates) and its manipulation (e.g. rational numbers as ordered pairs of integers and their arithmetic) without explicitly recognising it. Later we might ask (or even be asked!): How about ordered pairs of complex numbers? This article, consisting of chips from several quite old blocks, offers some forwardlooking ideas for investigation and development. An excellent general reference is [0].

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 mrω 2). 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?’

A standard application of calculus is to model some quantity which changes and to find the circumstances under which it takes some optimal value. There are a number of stock examples which are often used to show this in the teaching of calculus. This article considers how the extension of one of these can lead into a rich area of exploration.

Consider the statement (whose truth is obvious):

This is known as a pigeon-hole principle (or Dirichlet’s). It is one of the strategies which is often used for problem-solving. Although it is one of the most powerful tools of combinatorics, the principle can be used successfully in other branches of mathematics. Below you can find a collection of problems which can be solved easily by applying this principle to geometrical objects.

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.

Over the years 1830-1832, there appeared three volumes of a mathematical textbook in Welsh Elfenau Rhifyddiaeth (the elements of arithmetic) by John William Thomas. The book has long been forgotten and apparently only one specimen now remains, in the Salesbury Library in the Cardiff College of the University of Wales. The treatment and subject matter of the work are in places surprising for that era, and it is worth bringing the contents of the book to light.

I have always been intrigued by the Pythagorean triple (20, 21, 29). It interests me for two reasons. Firstly it looks really improbable and I am fascinated by the things that remain counter-intuitive in mathematics despite my repeated verification of them. Secondly apart from the basic (3, 4 ,5) triangle, it is the only small triple in which the first two numbers are consecutive. For ease of reference I shall call these special triples. I had never even been sure how many more of these there were, although I had tracked some down by computer at one point. In the middle of preparing some follow-up work for a lecture, I decided to look at these again to see whether there was anything suitable for the students to do. I uncovered a connection that certainly I had never known before.