The National Council of Teachers of Mathematics in the USA took the occasion of the 1966 International Congress of Mathematicians as the opportunity for arranging with the Russian authorities a tour of Soviet mathematical centres by a small party of selected American mathematicians with special experience and interests in various aspects of education. The tour covered Kiev, Moscow and Leningrad and it was hoped that, in each city, a full range of mathematical education would be seen, from the department of mathematics in the University to the famous special boarding schools for pupils of outstanding mathematical ability and finally to the elementary level.

I was honoured to be invited to join the party as the only non-American and am very grateful to the Schools Council for enabling me to go by offering to meet the costs of travel and subsistence.

There is grave difficulty in showing elementary school students how mathematics is applied in practical situations (at least in Canadian schools). Since current emphasis is on pure mathematics, applied mathematics (and applying mathematics)—procedures of model building, approximation, analytical manipulation and interpretation—may remain obscure or unknown to students leaving school despite essential simplicity of principles.

Thus it is important to keep a look-out for problems which have an obvious practical interest or economic moment, but which can be tackled with the most elementary mathematics. Emphasis on the strategies of model building in the schools seems all the more sensible (and feasible) now that automatic numeration is permitting easy command of even complicated analyses.

I feel that many of the descriptions of linear programming are either too elementary or else expressed in terms of n variables with m constraints using matrices and innumerable Σ's. This is an attempt to arrive at a description somewhere between the two and in it I take n = 7, m = 3.

Suppose that the function to be minimized is given by

and that in addition to xq ≥ 0 for all q, the equations of constraint are

Solving equations (2) for any three of the variables, say x1, x2, x3, gives equations like

Suppose that solving for the variables x1, x2, x3 gives non-negative values of β1, β2, β3; if it does not, three other variables must be chosen until positive numbers are obtained on the right-hand side.

As the name suggests, the Multinomial Theorem is an extension of the Binomial theorem, and it was when I first met the latter that I began to consider the trinomial and the possibility of a corresponding Pascal's triangle.

My mathematics master suggested that I construct the triangle myself. It became apparent that such a triangle may not be drawn, as the result is a pyramid or a tetrahedron where the apexes of the tetrahedron are occupied by the coefficients of xn, yn, zn in the expansion of (x + y + z)n.

This may be shown by forming expansion coefficient triangles for a trinomial of power 0, 1, 2, 3.

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.

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 “ - ”.

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.

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.

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

In the Gazette (176) for May, 1925, Miss J. M. Brown wrote a delightful article on “ Mathematical Clubs ” in Schools. It is interesting today to recall some of the ideas mentioned.

“ The obvious function of a Mathematical Club is to stimulate the interest which may have been aroused in the classroom, or if possible to awaken an interest hitherto dormant.”

The History of Mathematics was considered a starting point but was not found to be a very suitable topic for arousing enthusiasm. Mention is made of graph work (there was no Cundy and Rollett, Mathematical Models, in those days)—and of interest shown in Flatland (Dr. Abbott).