Thomas Kuhn made his reputation by pointing out that science is not just a steady series of rational developments, but is subject to paradigm shifts – revolutions in which the entire point of view changes. A less flattering word for the process is ‘fashion’. But what is fashion other than a paradigm shift in a rather trivial area of human activity? It is often thought that mathematics is immune to paradigm shifts. After all, mathematics consists of perfect truths, proven by impeccable logic. Doesn't it? Well, no, not really – but even if it did, that wouldn't prevent paradigm shifts. Paradigm shifts are not about the internal logic; they are about the external viewpoint. They are about seeing the same facts in a new light, incorporating them into a new framework.

The following is a well known pursuit problem: n bugs are at rest at the vertices of a regular polygon. At a certain instant each bug starts crawling towards its neighbour on the right with a constant speed, always altering its course so as to be headed directly towards its neighbour. How long does it take for the bugs to meet at the center of the polygon and what paths do they trace out in getting there? The earliest reference I could find to this problem was in a collection of mathematical puzzles edited by L.A.Graham. A more recent reference to it occurs in the amusing and informative collection of geometrical curiosities by David Wells. The “n-bug” problem was also discussed briefly in two of Martin Gardner’s columns in Scientific American. The cover of the July 65 issue of that magazine shows the equiangular spirals traced out by four bugs as they “square dance” their way towards their tryst at the centre.

In certain computer applications to engineering problems such as the automatic control of machine tools the computing equipment derives its input in digital form, from, for example, an angle-indicating device on a rotating shaft. The indicating device may take the form of a multiple commutator or a coded disc read by photoelectric means. Since the shaft is rotating at mechanical speeds and the reading device is operating at electronic speeds a possibility of false readings exists when for example a change from 1999 to 2000 is taking place. In fact any of the readings 1999, 1990, 1900, 1000 or 2000 may occur in the same sort of way that a mileage recorder could be misread under similar circumstances.

Mr. A. M. Cohen in his article (The Mathematical Gazette, LIII, Feb. 1969, p. 41) makes some suggestions for school work in numerical methods and number theory. These prompt me to describe some work actually done in which, for reasons which will appear, the emphasis is rather different from his.

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.