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.

In the classical Monty Hall problem you are a contestant on a game show confronted with three identical doors. One of them conceals a car while the other two conceal goats. You choose a door, but do not open it. The host, Monty Hall, now opens one of the other two doors, careful always to choose one he knows to conceal a goat. You are then given the options either of sticking with your original door, or switching to the other unopened door. What should you do to maximise your chances of winning the car?

In a previous note the present author has given an account of the conjugations known as isogonal conjugation and isotomic conjugation with respect to a given triangle and has shown how to generalise the idea of conjugation involving a pair of points so that P(l, m, n) is q-conjugate, by means of a construction involving a transversal q, to the point (p2/l, q2/m, r2/n) and he has also demonstrated how to perform the geometrical constructions involved. When there is a self-conjugate point at the incentre I one has isogonal conjugation, and when there is a self-conjugate point at the centroid G one has isotomic conjugation.

This Association came into existence nearly one hundred years ago with the prime object of improving the teaching of geometry and although its field of interest has widened considerably the teaching of geometry remains a very important issue. Of course, the question “Whither Geometry?” is rather meaningless unless one specifies whether one is talking about geometry as taught to primary, junior or secondary school levels, to Colleges of Education and the Polytechnics, to undergraduates and postgraduates at University, or geometry as understood by professional research mathematicians. Moreover the motivation for studying geometry may well be quite different at these different levels. I shall try to deal with the question as it affects all these various levels of mathematics.

For a long time prime numbers have attracted the attention of mathematicians, especially those primes that possess some sort of a symmetry. The mysterious repunits An = 111… 1(n), whose decimal representations contain only units, form an important class of them. For a repunit to be prime the number n of its digits must be also prime. But this condition is far from being sufficient: for instance, A3 = 111 = 3.37 and A5 = 11111 = 41.271. Some of the repunits are nonetheless prime: A2, A19, A23, A317 and A1031, are the only known examples. The question of primeness of the repunits was discussed by M. Gardner and later in [2–4]. It is not clear whether the number of prime repunits is finite or infinite.