It is well known that G. Tarry [1] was the first to publish a proof that the famous thirty-six officers problem posed by L. Euler [2] in 1779 has no solution but it appears to be less well known that he was the first to devise a systematic method of constructing bimagic and trimagic squares, that is, magic squares which remain magic when each entry is replaced by its square and, in the case of trimagic squares, also when each entry is replaced by its cube. Tarry's method was outlined in [3] and was explained and slightly improved upon in a book by E. Cazalas [4] published in Paris in 1934. Recently, there has been renewed interest in this topic.

This article does not advocate dispensing with the calculus in the least; it suggests ways in which some of the traditional problems of calculus may be studied by other methods much earlier in the school. This makes a range of very interesting problems available to a much wider audience, and also shows how some of the problems of the sixth form science course can be treated with pupils who have the misfortune to be studying science without the calculus to help them.

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.

The object of this paper is to discuss some properties of the Isosceles Tetrahedron, in which each edge is equal to its opposite. The last three theorems are possibly new, the remainder are included for their own interest or for completeness in listing elementary results; in the latter case, proofs are not given.

In a recent paper (1) P. Holgate obtains the following equations for pk(t) [k = 1,2,…, n], the probability that, at time t, the n animals in a given area are grouped into k herds.

The Simson line property is normally associated with points on the circumcircle of a triangle. It is embodied by the following theorem.

Given any triangle ABC and a point P in the plane of the triangle, if perpendiculars from P on to the sides BC, CA, AB meet those sides at L, M, N respectively then L, M, N are collinear if and only if P lies on the circumcircle of triangle ABC. The line LMN is then known as the Simson line of P.

The number π (as all—or very nearly all—the world doth know) signifies the ratio of the circumference of a circle to its diameter and is also equal to the ratio of the area to the square of the radius. This suggests two generalisations of π which apply to plane figures of any shape: