Many communist countries hold internal olympiads in order to identify and thereafter foster special talents. Since 1959 they have held an annual International Mathematical Olympiad (IMO) for all European communist countries (except Albania) and subsequently Mongolia, Cuba and Vietnam. Non-communist countries were first invited in 1967 and France, Sweden, Italy and Great Britain accepted the invitation. Italy has dropped out and Austria, Netherlands and USA now regularly participate. Belgium, Finland and, last year, Greece have also taken part. Each country sends a maximum of eight competitors and two adults.

[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.

A vivid memory of mine is of being shown that certain 2 × 2 matrices were ‘really’ complex numbers. The correspondence between matrices of the form

and complex numbers

seemed magical and the additional remark that

was an unexpected bonus. The correspondence has been examined in various articles in the Gazette [1, 2]. That it is an isomorphism is seen by comparing

with

in the case of addition, and, in the case of multiplication, by comparing

with

Eperson's conjecture is that if n ⩾ 3 is odd then 3n2 can be written in at least two ways as a sum of three squares. We shall give a fairly elementary proof of this, from scratch.

Let q be a real or complex number with

Define for n ⩾ 1.

(In reference to A. C. Dixon’s note on lm + 2m + 3m +...+nm, on pp. 283-4 of this volume, the following further notes may be of interest. They are taken, with some alterations, from a paper communicated to the London Mathematical Society in February, 1910.)

Space does not allow much room for “reporting” in The Mathematical Gazette, though that deficiency is greatly offset by the Newsletter. As it happens, however, two very contrasting items arrived while this Number was in the proof stage and so they are included, as typical of much else, to give some information about mathematical activities in general.