Most tennis players have two types of serve in their repertory. One is a safe serve, which is relatively accurate, but also relatively easy for the opponent to return effectively. The other is a risky serve, which is more likely to be a fault, but also more difficult to return if it is good.

In a recent article [1], Ron Shail has given a Cartesian proof of an interesting theorem due to J. A. Lester. This states that, for any triangle, the circumcentre O, the nine-point centre O9 and the two Fermat points F and Fʹ, (which are the points of concurrence of the joins of its vertices to the vertices of equilateral triangles drawn outwards/inwards on the opposite sides), are concyclic. He refers to Lester's own treatment as needing complex coordinates with computer-assisted algebra; his own proof uses an unpromising method, and results in similar problems. Contemplation of the configuration would suggest that the location of the point of intersection of FFʹ with the Euler line OO9 might lead to a simple proof. The theorem is in fact a corollary from the properties of a remarkable configuration originating with Morley [2, p. 209], and shown in Figure 1. He did not deduce Lester’s result, nor label the crucial point J in the diagram, which was drawn without that particular intersection. Also involved in this figure is a rectangular hyperbola known by the name of its describer Kiepert [3]. It is helpful to discuss all three together. What follows is a journey through country nowadays rather unfamiliar, avoiding the computerised motorway and using older tracks via complex numbers, trilinear coordinates and Euclidean methods which reveal much more than is apparent from a Cartesian treatment.

A teacher can make a great deal out of some of the questions which pupils and friends put to him from time to time. In an article in Mathematics in School a few years ago [1] I listed a number of such questions and discussed one, which involved a geographical application of matrices, in detail. The present article continues this theme, but its main aim is to stimulate discussion of the place of “mathematical modelling” in the curriculum. For it is felt by some that, in spite of the opportunities offered by (say) linear programming, mechanics and probability, insufficient emphasis is placed on modelling in the secondary school. It has been said too that we have not been notably successful in finding good applications of topics new to our syllabuses and incorporating them into our teaching [2].

The purpose of this short article is to give a simple method for constructing a magic square of any given odd size. The method consists basically of two cyclic permutations perpendicular to each other.

The 5 × 5 square’s construction is illustrated, a proof outlined, and a 17 × 17 square quoted.

The sequence 0,1, a, aa, a(aa), . . . has certain unexpected features which may easily be demonstrated with the aid of a programmable pocket calculator. The investigation of the convergence of this sequence provides an exercise which is both interesting and informative.

When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [1, p. 171f] (see also [2, p. 59], [3, p. 222f] and [4, p. 42ff]).