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”?

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.

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).

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.

The simplest way of picturing a three-dimensional object in a plane is by orthogonal projection; that is, as it were, by taking a photograph from a distance. (Perspective, showing its appearance when viewed from nearby, is more accurate and lifelike but much more troublesome.) For technical drawing, it is customary to use a plan and elevation (and sometimes also side elevation). Quite often there are three perpendicular directions naturally associated with the object, such as those of the sides of a rectangular building. We call these the “left-right”, “front-back” and “down-up” axes, and the plan and elevations associated with these directions are usually easy to draw. Unfortunately, technical drawings of plans and elevations often appear rather artificial, and require training and practice to interpret properly.

I am most grateful to the Mathematical Gazette for publishing, in the March 1977 issue, my letter about the problem of finding five positive integers such that the sum of any pair is a perfect square. The response far exceeded my expectations and showed that the problem aroused great interest.

Most elementary texts on group theory define factor groups. Most beginners find factor groups difficult. The present exposition is based on the notion that it may be helpful to have a look at factor groups in a context slightly wider than that of “factoring out normal subgroups”.

Among the various logically equivalent sets of axioms for boolean algebra ([1] discusses four of them), that referred to in the title of this article is popular because of its symmetry (self-duality) and convenience in application. The textbook presentation is essentially as follows; the qualifiers ‘left’, ‘right’ are often omitted, although the symbolic version indicates that they are intended.