During the spring of 1973 we were able to visit the Soviet Union for 14 days to study mathematical education. Our visit was arranged by the British Council under the Anglo/Soviet cultural agreement. One of the most interesting features of the Soviet system which we hoped to study during our short tour was the specialised boarding school. Accounts of some of these have appeared already (for example, Bryan Thwaites’ article in the Gazette for December 1968) and it was intersting to compare our impressions with those which had been recorded by others.

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.

One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:

“For all x, y, z in ℝ:

1. 1. Exactly one of x = y, x < y, y < x holds.

2. 2. If x < y and 0 < z, then xz < yz.

3. 3. If x < y, then x + z < y + z.”

Boys who have learned to find π using jam jars and string are often mystified to learn that the number has been calculated to thousands of decimal places: the infinite series and products usually quoted cannot be justified without analysis.

This article is about a method whose derivation involves only elementary algebra and trigonometry (including square roots and Pythagoras), and whose use is a feasible proposition even when only four-figure tables are available for the square roots.

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved:

In the (n + l)th row of Pascal’s triangle, there are

Let G denote a finite non-commutative group. What can we say about the number of ordered pairs (x,y), where x,y ϵ G, such that xy = yx?

In order to prepare ourselves to answer this question, let us examine the Cayley table of S3, the symmetric group on three symbols. We take as generators the reflection (12) and the rotation (123). It is clear that we are examining the Cayley table for symmetry about its main diagonal.

Because of ‘modern mathematics’ most people are aware of different number scales. Ordinarily there is the decimal scale, while computers use the binary system and sometimes, I believe, the octal system based on 8s. Other scales may appear to be freakish, or mere excuses for arithmetical manipulations; yet at times they can be useful.

If on the sides of a triangle ABC equilateral triangles are constructed, inwardly and outwardly, and their circumcircles drawn, the 6 circles will in general have 11 points of intersection: A, B, C; 2 points on each of the 3 sides; a point F common to the circumcircles of the outer triangles, and a point N common to those of the inner triangles (Fig. 1).

This article uses complex numbers to investigate the figure, and has as its main object the establishment of an extremal property of N.

The word “lesson” came back to Pooh as one he had heard before somewhere. “There’s a thing called Twy-stymes”, he said. “Christopher Robin tried to teach it to me once, but it didn’t.” A. A. Milne.

In assessing the value of the mathematics taught in schools, considerable emphasis is normally given to the use which a minority will later make of it in their jobs. It is the purpose of this article to point out that even those who never use mathematics professionally could better understand important social issues if they had a good basic training in mathematics. This is illustrated by a set of mathematical models related to one such problem— the population explosion.

B. J. R. Bailey’s article in Gazette LVII, 169–174 (October 1973) worried me a little, because it seemed to suggest support for a widespread misapprehension, that mathematical statistics differs from other branches of applied mathematics in requiring a good deal of subjective argument. Every application of mathematics involves judgement, of course, but statistics does not really go beyond other applications in this respect.

“And though he made many elegant discoveries, he is said to have besought his friends and kinsmen to place on his grave after his death a cylinder enclosing a sphere, with an inscription giving the proportion by which the including solid exceeds the included.” Plutarch, on Archimedes, in The lives of famous men, the Life of Marcellus.

The paper begins with a description of a diagrammatical model which I used when lecturing on elementary calculus to a class of engineers. The second part suggests how the use of the model might be extended to provide a framework on which one could build up the theory of differentiation of functions of one or more real variables. For the purposes of this article it is assumed that the reader is familiar with this theory—the paper, therefore, merely gives the outline of a possible approach. The keynote of the lecture course to the engineers was ‘plausibility’ rather than ‘rigour’ and my use of ‘function boxes’ was sparked off by the fact that the class were using flow diagrams and ‘black boxes’ in a number of other courses—most noticeably in that given by the person who lectured immediately before me and who, in his progress up the technological scale to the electronic computer, had clearly not encountered that mundane educational aid, the blackboard duster. The grand finale of the one-term course was to be an introduction to partial differentiation and, in particular, the chain rule—a piece of mathematics which students often find difficult and which I hoped to make easier by means of an approach using a simple type of ‘flow’ diagram.

Newton was fascinated by the fact “which is very remarkable” that the two “principal cases of attractions” (i.e. the inverse square and direct distance laws) are distinguished from others in two independent ways:

1. (i) the attraction of a spherically symmetric body at a point outside “preserves the same law of increase or decrease” as that of a point mass (call this the spherical property);

2. (ii) a particle can move in an elliptical orbit, though with different centres of force in the two cases (call this the elliptical property).

1. You all remember that in that well-known play, Waiting for Godot, the eponymous hero never actually appears. In the same way my talk should perhaps have been entitled Waiting for transcendental numbers—for while transcendental numbers are at the back of everything that follows, they will not become, for reasons of technical difficulty, proper part of the action.

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.

One of the oldest and best known of chessboard problems is to place the largest possible number of similar pieces on the board such that no two of these pieces are attacking each other. In his book Amusements in mathematics [1], Dudeney considered this problem on a generalised square chessboard containing n 2 cells, and proved that for rooks, queens and bishops this maximum number is equal to n, n and 2n – 2 respectively.

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.