Most school mathematics courses nowadays include a study of symmetry in the euclidean sense—isometries in 2- or 3-dimensional euclidean space which map a figure to itself. This article explores the theme of symmetry in a more general, topological sense. It could form the basis of a short self-contained unit in topology with a worthwhile end-point, which teachers might care to try out with students at sixth form level or perhaps in a college of education course.

My title seems self contradictory, for surely random happenings are essentially unpredictable—so how can they be regular? And yet this apparent contradiction is the key which justifies the mathematical model of probability theory. For the regularity is the basic reason for our model providing a new mathematical tool with powerful applications to analysis, potential theory, number theory, quantum theory, statistical mechanics....

Let us start by asking, “What is probability?” What do I mean if I say “When you toss a coin there is a probability ½ that it turns up a head” or “When you pick a card from a well shuffled pack there is probability that it will be an ace”?

A few years ago I tried making a model of an object from its photograph. Realising that, of course, one could not take direct measurements from the photograph, I had to try to guess the relative dimensions of the object. I ended up, unfortunately, with a model which did not seem to be in the correct proportions. Would it have been possible, I wondered, to find the precise ratios of the different lengths?

There is a popular view among sixth form mathematicians, and therefore presumably among some who teach them, that evaluation of an integral by the approximate Trapezium Rule is made more accurate by increasing the value of n.

This article is addressed to anyone who has ever doodled idly with a fixed compass on a blank sheet of paper. Perhaps he has drawn a circle centre O, picked a point P1 on it, drawn a circle centre P1 intersecting the original circle at P2 and P3, drawn a circle centre P2 producing two new points of intersection, slowly building up an interlacing hexagonal pattern II of points each a distance at least R from each other. Then he has picked some point P* not of II, and drawn a circle centre P*, and almost immediately the pretty pattern has been replaced by a jumble of intersections apparently covering the whole of the paper.

My family enjoys doing large jigsaw puzzles. We find a large table, lay out all the pieces face upwards, with the edge pieces separate, and build up the outside first. Then we usually try to follow obvious boundaries—between sky and trees, or shore and water, for example, though sometimes a single brightly coloured object catches our eyes and gets built up in isolation.

“The total work done by the external forces acting on a rigid body is equal to the change in the kinetic energy of the body.” O.K. ? Well, it all depends on definitions.

The work done by a constant force is usually (in nearly all textbooks ?) defined as the product of the magnitude of the force and the displacement of its point of application, this displacement being measured in the direction of the force. Consider then a sheet of paper on which a student has drawn in pencil a straight line which he is now about to erase. He causes the eraser as it moves along to exert a constant force on it, and he exerts another constant force with his other hand to keep the paper still. The desk (supposed smooth) exerts constant normal forces. Of all these forces on the paper the first force does some work, according to the definition, while the others do none. The total work done by them is positive, yet there is no change in kinetic energy of the paper.

The content of this article seems not only to have appeal for, but also to be well within the problem-solving capabilities of, many students of elementary geometry. No new results are cited (see [1]) but the concepts are presented differently, providing a viable and concrete means of explaining elementary partitions of space.

Group theory has been used in the study of various properties of musical scales. The cyclic group C12 provides a mathematical model for the tempered intervals (see for example Budden [1] Chapter 23, where the diminished seventh chords are shown to correspond to an appropriate subgroup of C12 and its cosets).

When I was younger, and had nothing better to do with my spare time, I sometimes used to play my own game of ‘patience’. I would take a pack of 52 playing cards and arrange them in order, ace to king of each suit and then the four suits one on top of the other. I would begin my game by taking the card at the top of the pack and putting it at the bottom; then taking the second card and placing it face upwards on the table; then the third card was transferred to the bottom of the pack; then the fourth card was placed on top of the second on the table; and so on, until all the cards had been placed in a new pile on the table. I would then turn the pack on the table face downwards and perform the whole operation again with the cards arranged in this new order. My aim was to see if, by repeating the whole operation enough times, I could get the cards back into their original order. If I had known that I would need to perform the operation 840 times, I probably would not have bothered to continue with my game.

A famous problem in euclidean geometry, associated with the names of J. Steiner (1796-1867) and D. C. Lehmus (1780-1863), is to show that a triangle is isosceles when two internal bisectors are equal. Although the converse is obvious, a direct proof of the statement is far from simple. A large number of demonstrations have been devised, some valid and some spurious, and a considerable literature exists on the subject.

The June 1975 Gazette (59, No. 408, 98-106) carried an article by Colin Tripp about the so-called “adventitious angles” problem: in the isosceles triangle in Fig. 1, for certain given values of the angles marked a, b, c, to find θ. Correspondence to which this gave rise was summarised in a “progress report” in March 1977 (61, No. 415, 55-58). This cleared up certain points raised by Tripp, using trigonometric methods; and its publication also had the desired effect of stimulating some further response, which has settled most of the outstanding questions. It seems now to be an appropriate time for the editor to produce a “final summary” for those readers who have taken an interest in the problem.

Recently I came across L. A. Graham’s delightful book of mathematical puzzles [1] for the first time and found there the problem I shall give in a moment. It occurred to me to try to solve it in a more general form.

I was able to find an unexpectedly neat and satisfying answer, using a famous result of E. Helly. The solution provides an excellent example of the type of use to which Helly’s theorem can be put, and a striking example of its power.

It has often been noted that modular arithmetic provides a rich source of supply of groups. Indeed, a remarkable theorem asserts that any finite commutative group can be found by a sufficiently diligent search through the multiplicative groups and subgroups of modular arithmetic! During the last few years two articles in this area have appeared in Mathematics Teaching. In the first [1], Tim Brand drew attention to the fact that such a multiplicative group can have an identity element other than 1. (For example, 8 is the identity of the group {2,4,8} under multiplication mod 14.) More recently [2] Geoff Saltmarsh described an ingenious way of finding the identity and put forward an interesting conjecture about these groups.

James R. Newman observed that “the most painful thing about mathematics is how far away you are from being able to use it after you have learned it” ([1], Vol. 3, p. 1978). Perhaps there is some sort of natural justice, then, in the fact that one of the most pleasant things about mathematics is that you don’t need to know very much about the subject in order to enjoy the puzzles, pastimes and toys that the study of mathematics has produced.

This article gives instructions for a new way to construct the mathematician’s toy which is called a “rotating ring”. I publish it in hopes that it may give you some pleasure and also provide a means by which you can entice your students to study the more serious aspects of solid geometry.

A quadrangle has four vertices, of which no three are collinear, and six sides joining the vertices in pairs. If the angle between each pair of the six sides is an integral multiple of π/n radians, n being an integer, the quadrangle is said to be n-adventitious [1]. A quadrangle is adventitious if it is n- adventitious for some n. For example, the quadrangle BCDE in Fig. 1 (the original adventitious quadrangle from which all the discussion started in [1]) is 18-adventitious. Various problems are posed in [1]; in a suitably generalised form these problems can be summarised as: find all adventitious quadrangles and prove their existence by elementary geometry.

Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by

The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.