The Mathematical Association is very kind to its Presidents. They are relative strangers, they are placed for a short-time in the most intimate contact with all the Association’s affairs, and they are given up to two years in which to agonise over the subject of their Presidential Address. In my own case I chose the above title about 18 months ago, and was then filled with worry and embarrassment when last year Frank Budden unveiled his title “Accuracy is a virtue”. Would my lecture be viewed as an unworthy attack on a very distinguished predecessor? More important, was I completely mistaken in my view? My conclusion was that I should go ahead with the talk as planned asking you to view Frank Budden’s remarks and my own as two elements in a necessary dialectic, another example of the tensions which are inherent and necessary to good mathematical education.

Areal measurement of a two-dimensional plane figure is a problem of ancient origin and one which continues to hold both mathematical and practical interest. There are various approaches possible, ranging from counting squares to the more sophisticated techniques based on integration.

A particle is free to move on a smooth horizontal circular table having a circular insurmountable barrier Γ. The particle is projected from a point P of the table and it subsequently suffers reflections at the points P1, P2, P3, … on Γ due to elastic impact. Investigations are made of the possibility of its passing again through P at some stage.

This article starts with a problem in probability and leads on to some natural connections with Fibonacci numbers. Details of some of the proofs are left for interested readers (or their students!).

The roots of any event are often hard to trace, but at some point some years ago now, some member of the Open University maths faculty was approached by a maths advisor and asked if we could arrange the programme for a one day mathematics outing for sixth form students. Since we have run all sorts of mathematical events for our own students, it seemed like a good opportunity to try out ideas on a different sort of student. This was the first of a burgeoning number of events we have planned and run, and the purpose of this note is to indicate as clearly as is possible in print what we do, with the thought that others might like to try something similar.

In a sequence of Bernoulli trials suppose the outcomes are either successes (S) or failures (F). A list of these outcomes will consist of runs of S’s and F’s and it is the purpose of this article to investigate such runs. Initially the investigation will be pursued theoretically. Then the theoretical model will be applied to data provided by the Test Matches which have been played between England and Australia for many years, to give a comparison between the expected results and those that have actually been achieved.