The growing importance of the elements of group theory in sixth form teaching has been recognised by a number of recent publications, but a very simple presentation of the theory of groups by means of free variable axioms which was introduced by Lorenzen more than twenty-five years ago, and which is particularly well suited for elementary teaching, does not appear to be as well known as it deserves to be.

Lorenzen defines a group in terms of division x/y, instead of multiplication, a device which obviates the need to postulate the existence of a neutral element and an inverse.

The exponential law of increase has been described by Professor D. J. de Solla Price as “the fundamental law of any analysis of science”, and considered to hold over long periods of time [1]. The same writer suggested that low-grade work doubled in about ten years, but that the period taken was about twice as long for very high quality activity. It is useful to remember that a doubling every fifteen years corresponds to a multiplication by about a million in three centuries, or an annual increase of nearly five per cent.

Mr. Parrack's paper on cliques [in the Gazette of February 1966, pp. 43-46] links up with some results which I obtained in 1962 on “ double-identities ”. [Mimeographed copies of the paper are available for any who wish them.]

Briefly, the simplest kind of double-identity is an equation which, like:

is an identity in any additively-written abelian group when “ o ” is replaced by “ + ” and, also, when “ o ” is replaced by “ - ”. More precisely, an (e, o)-double identity is an equation between terms constructed from variables, “ o ”, and a constant, “ e ”, which is an identity when “ e ” is replaced by “ 0 ” and “ o ” is replaced by “ + ” or by “ - ”.

In Spring 1963 the first National Mathematical Contest here was organized by Mr. F. R. Watson (now of the Dept. of Education, Keele University). The papers were those of the Mathematical Society of America and Society of Actuaries (past papers are published in the two “Contest Problem Books” (Random House)). About 1000 people, aged 14-18, took part. There were no prizes, only the interest, for teacher and pupil, of submitting to an American test in which even those of modest talent can put up a satisfactory performance.

[This is part of a lecture delivered at the Annual Conference of the Association on 31st March 1967. It contains an elementary teaching approach to matrices.]

Let us look at this diagram and consider what it might depict. It might be a map of airline routes between airports in three countries; with two airports, x 1 and x 2, in the first country, three airports y 1, y 2 and y 3, in the second country, and two, z 1 and z 2, in the third. The numbers beside the lines in the diagram might indicate the number of alternative flights (perhaps planes departing at different times) along the routes.

In the first war, not only did computers not exist, but the labour of calculating trajectories by a step by step process was regarded as quite prohibitive, and approximate formulae were the only resource. In the end these proved surprisingly accurate. The methods inevitably worked on approximations valid in the first instance for small values of a parameter, ø, say. But it was found possible to find formulae correct to the fourth or fifth order of ø, and as a result there emerged the principle that “In Ballistics 25° can be a ‘small’ angle”; results for 25° could be correct to 0·1%.

Some of the problems which come under the heading of “ inferential puzzles ” are useful for illustrating a varied selection of topics in modern mathematics. Consider the following problem.

Tom, Dick and Harry are married to Jane, Mary and Susan, but not necessarily in that order. Each couple has one child; the children's names are Alan, Brenda, and Charlie.

• (i) Mary's child and Harry's child are the two stars of the school football team.

• (ii) Tom's son is not Alan.

• (iii) Dick's wife is not Susan.

• Find who are the parents of each child.

• This problem, unlike some of its type, is easily solved by a straightforward commonsense argument.

As an introduction to elementary programming for digital computers, it has been found advantageous to make a start with a desk machine. A very simple code was devised for this purpose. Interested sixth-formers tackled problems prominent in their ordinary mathematical work by this method and were soon able to make programmes and get results themselves. They were later able to visit a digital computer installation and to try their hands at using an established “autocode” (Sirius). They found that their previous experience with the desk machine was most valuable in initiating, very quickly, the kind of thinking required for the more sophisticated procedures. The total experience, although very brief, appears to have been well worth while, as a means of showing them something of the potentialities of modern computers in relation to problems within their own competence.

F. Bowman has found the following pretty result.

The minimum value of the definite integral

for real a is given by I = 1/(n + 1).

For convenience we shall repeat here the definition by Hawkins, Hill, Reeve and Tyrrell (1, hereafter referred to as HHRT) of a k-map:

When teaching elementary group theory, the ideas of coset, normal subgroup and quotient group are difficult to introduce.

A possible approach is to make use of the patterns which appear on the multiplication of a group.

It is easy to pick out from the multiplication table of a group the elements which form a subgroup. All that is needed is to find a set of elements which is closed under the operation.

In the group of Table 1, clearly {I, C} form a subgroup.

The study of convex polyhedra from a mathematical point of view began in ancient Greece well over two thousand years ago. It is believed that Euclid’s “Elements” was written to inspire a wider appreciation of the properties of the five regular polyhedra or “Platonic solids”, as well as to place their study on a secure mathematical foundation. The discovery of convex polytopes (the analogous figures in four or more dimensions) and their investigation by the Swiss mathematician Ludwig Schläfli just over a century ago, led to a renewed interest in polyhedra, and since that time hundreds of papers on this subject have been published. However, there are still many unsolved problems, and the purpose of this article is to state a few of these. These problems have widespread appeal because they can be readily understood with very little previous mathematical knowledge, and although they may be quite difficult, there is always the possibility that someone with a very minimum of mathematical background will devise a method of approach that leads to a solution.

To continue the debate (Mansfield and Bruckheimer [1], Broomhead [2]) :

1. Scalar Quantities are things like length, area and mass. Vector Quantities are things like force, velocity and acceleration (in 2 or 3D). But let us not, for the present anyway, call vector quantities “vectors.”

Let P be a given convex polyhedron and L any plane. Then if L∩P is not empty it is called a plane section of P. Every plane section of P is either a convex polygon, an edge or a vertex of P.

VI. If a convex polyhedron P has v vertices (v ≥4), does there always exist a plane section of P which is a convex n-gon with n ≥ √ v ?

A more general statement of this problem is to ask whether there exists a constant d (0 < d < 1) such that every convex polyhedron P with v vertices possesses a plane section which is an n-gon with n ≥ vd. Problem VI is then equivalent to asking whether d ≤ ½