There are probably few readers of the Gazette who also read the weekly periodical Commercial Motor, but it was a table appearing in an article in the latter which caught the author’s eye and provided the inspiration for the present work.

In recent years legislation has been introduced which requires that lorries carrying dangerous substances must also bear panels on which appear, inter alia, codes identifying the substances and giving an indication to the fire brigade of what action should be taken in the event that the vehicle is involved in a collision or other emergency. Examples of the codes are shown in Figs. 1 (a) and (b). The former is probably the most commonly seen code as it is that displayed on petrol tankers; the latter is a code seen on a vehicle carrying liquid nitrogen.

The best mathematicians, from the days of Euclid and before, have defined the terms they use. Now a definition may be a complete fraud, and Euclid was a case in that which hath no part; although to be fair to our greatest text book writer, I am sure he knew a fraud when he perpetrated one. On the other hand, a definition may be a perfect hidden gem. That mighty man Eudoxus was able to define what was meant by the equality of r1 and r2, without knowing what r1 and r2 were. A lesser man would have thought that impossible, but Eudoxus was not a lesser man, by definition. Euclid and Eudoxus, those great definers, would be proud of much of what we now do in the way of definition. There are still frauds, such as definition by example, so beloved of biologists and new mathematicians. (By the latter we mean those persons who consider their business to be new maths.) And there is that modern phenomenon, the railway definition, where properties are hitched together in series just like the carriages of a train, to create one monster definition, which, on the face of it, and sometimes not only superficially, has no justification and no relevance.

The introduction of group theory in the sixth form often raises the question of its relationship with other areas of mathematics. The purpose of this note is to develop a group theoretical solution of a simply described counting problem; a problem which may, in fact, be solved in several other ways. The development of the solution yields useful examples of several algebraic notions; in particular the notion of equivalence relation is exploited in a variety of ways. The particular group theoretical/combinatorial result that finally yields the answer to our problem is known as Burnside’s Lemma. This result is not normally presented until a second or third year university course. I hope to convince the reader that the manner of presentation given here is perfectly accessible at a more elementary level.

After the publication of the Mathematical Association’s booklet on sixth-form work in pure mathematics in 1967, P. R. Buckland published the results of a short multiple-choice test, given to graduates training to teach mathematics, in the Gazette in 1969. He discovered what he called “alarming gaps” in students’ mathematical backgrounds. Is the situation any different 15 years later? In an attempt to obtain an answer, Buckland’s test (with a few changes and a new ‘historical’ section) was given to a group of graduates and undergraduates.

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.