In a recent article in the Gazette (March 1983) Chris Haines described the problems which arose when he was arranging a tenth birthday party for his son. There were seven boys in addition to his son at the party, making eight all together. In order to prevent the party from getting too boisterous, Mr Haines decided to split the party into pairs who would play various boxed games for two players: for example, halma, draughts, snakes and ladders. The plan was to have a sort of tournament in which:

1. (1) each child would play every other child once and once only;

2. (2) in a given session, different games would be played by each pair as only one set of each game was available;

3. (3) each child would play as many of the available games as possible.

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 one dimension, a set of rods of equal length may be used to cover the line. In two dimensions, an area may be covered by joining up sets of identical symmetrical squares, equilateral triangles or regular hexagons. In three dimensions, there are at least four different symmetrical solids which can be packed together to fill space. Of these, the cube is most obvious, two others are moderately well known (see [1] for example), while I have been unable to find any record of the fourth solid.

In the summer at Avignon a little Disneyland train conveys tourists from the Palais des Papes to the summit of a little hill behind the cathedral called the Rocher des Doms. In the garden on this hill there is a most remarkable sundial; the visitor himself provides the gnomon, and a linear scale on the flagstone pavement shows him where to stand at different dates. His shadow then falls towards one or other of a ring of fixed pyramidal markers which indicate the hours of the day.

The entertaining book, There’s one born every minute, by Harry Blackstone, presents many situations where most people would intuitively be prepared to wager even money on the outcome, whereas a proper study of the probabilities concerned shows that the bet is heavily biased. One particular problem is interesting because the author claims to have sought evaluation of the probability concerned, without success. It is as follows:

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.

Most direct proofs of Morley’s theorem are very involved. Dick Hogg’s neat geometrical proof (Note 67.3) circumvents the problem with its slightly indirect approach, but is very pleasing, none the less. It may however not be generally known that there is a direct straightforward proof using complex numbers which is simple to those familiar with the technique involved; indeed the proof is almost inevitable, once the right start has been made.