Professor Forder’s elegantly simple method in Note 3068 (Math. Gazette 47 (1963), 237-8) can be extended to prove a more general result. Let k be any positive integer, ξ any irrational number, ar the (r + l)-th quotient in the simple continued fraction for ξ and pr/qr the (r+ l)-th convergent, so that

Beginning a new study involves considerations both of strategy and of tactics. The strategic objective is to impart to the study a sense of purpose, to arouse interest in it, to encourage the pupil to accept the challenge which is implicit in the new situation which confronts him. The tactical question involves the choice of method.

So far as strategy is concerned, the older student beginning algebra has no difficulty in recognising its practical importance in science and technology, where the power of its language to exhibit and clarify relational patterns is manifested. The appeal by way of vocational and cultural interests has been made successfully in various popular books for adults and “second-tryers”; these interests simplify the teaching problem, and the teacher can proceed at an early stage to give training in analysis and expression.

Imagine six drawing pins stuck in the wall. If we stretch threads, either red or blue, between every pair of pins, we inevitably create at least one triangle all of one colour. This we shall call Theorem 1: the proof follows.

Consider any pin. From this emanate five threads (since there are five other pins), of which three must be of one colour. We may without loss of generality assume that they are three red threads.

Mathematica is a computer software system for symbolic manipulation and numerical computation; it has excellent graphics, a high-level, mathematically-oriented programming language and, very importantly, it takes seriously the need for work to be interactive, properly presented and communicated. This article is in fact prepared as a Mathematica “notebook”. An extensive library of supporting software “packages” has been developed, and several books about using it are now available.