For many sixth-formers and undergraduates, the method of proof-by-induction seems to resemble nothing so much as a confidence trick. They appear to have little or no appreciation of the logical foundation of the method; frequently, they have been given a mysterious recipe which might just as easily be “eye of newt and toe of frog”. They slavishly follow the rules given and so it is little wonder that they are confused when, having been cajoled incessantly not to beg the question, they are now seemingly allowed to assume the answer. However, it is not my intention in this article to present yet another justification for the principle of mathematical induction, important as that might be; in any case, perfectly good articles on the subject have already appeared in the Gazette [1,2].

One card up the sleeve of many a teacher of mathematics involves the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,…, in which each number is the sum of the preceding two. These numbers and the closely related golden ratio (√5 − 1):2 have intriguing geometric and algebraic properties and appear mysteriously in nature ([1], 160-172). The main purpose of this article, which stems from a recent talk to prizewinners in the Scottish schools’ problem competition, Mathematical Challenge, is to advertise some less widely known occurrences. A reader interested in further references to any part of this article is invited to contact the author, who himself is grateful to several colleagues.

John Branfield’s request [1] for the mathematics of bowls was too tempting to resist. What follows comprises a simple model for the curve upon which the bowls move, an explanation of how and why they move as they do and an attempt to refine the model of the curve. The search for a model proved to be addictive and effectively displaced many other activities. One of the reasons for this was the way in which the investigation involved so many people with interesting contributions to make. In order to give a flavour of the investigation, I have taken the liberty of giving a somewhat personal rather than strictly mathematical account, the mathematics being no less important.

Since there is far more mathematics even quite elementary mathematics than anyone can carry in their working memory, it must follow that results which are unused will be forgotten, and are then often rediscovered after many years. A tour of some old haunts which are now being opened up again would therefore seem not out of place in a centenary number of the Gazette. Tourists, however, will need some preliminary briefing if they are to enjoy the tour fully.

A frieze pattern is an array of numbers arranged in staggered rows, like the bricks in a wall. In the case of a multiplicative frieze pattern the top and bottom rows consist entirely of ones, and for each ‘diamond’ of four elements

the unimodular rule

ad = bc + 1 (2)

holds. These frieze patterns have many curious properties, and for further details the reader is referred to the article Triangulated polygons and frieze patterns by J. H. Conway and H. S. M. Coxeter which appeared in Gazette 57, 87–94 and 175–183 (1973).

In the first part of this article we studied the relation between mathematics and the child and tried to suggest that a greater place could be given to mathematics in the schools by taking into account how the child learns. We tried to show that, to give the child a knowledge of mathematics, various approaches can be used, each of which can serve as a basis for the following one if we follow the development of logic, using the set of tools mastered at each stage in order to progress from one level to the next.

Venn diagrams are usually considered to be a method for visualising set-theoretic identities and suggesting true theorems. It is frowned upon to prove a set-theoretic identity using them, such proofs being considered non-rigorous. One of the better kept secrets of the trade is that they can be made rigorous by proving the right kind of ‘metatheorem’. A proof is quite instructive, and might be of interest to set-theoretically minded sixth formers.

Project work has become a major constituent of Applied Mathematics courses at most British polytechnics and universities. There is a general consensus among academics as to what constitutes a project, how to discover and vet suitable projects, what their value is as a teaching aid, and how a student’s project work should be assessed.

Some time ago a Hungarian post-graduate, Dr. P. Erdös, studying under Prof. L. J Mordell at Manchester University, enunciated an interesting theorem connected with a triangle, for which he asked that a proof should be given.