Matches of a certain popular brand are contained in a box designed to hold 100 matches. This box has two compartments so that if you open the box casually and without thought as I do, then you will select a match at random from one compartment or the other with equal probability. At some stage you will empty one of the compartments; and having more statistical insight, perhaps you will be less surprised than me by the large number of matches often left unused in the other compartment.

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.

A-level courses and examinations are with us for some time to come. Even if Mevel is introduced, for most pupils who spend two years in the I sixth form A-levels will remain the basis of their studies. It seems to me that we need a major debate about A-level mathematics; we need to review it as it is and try to decide what developments are necessary. The discussion I paper ‘Mathematics at A-level’ has already raised a number of points, some of which are touched on here. This article is not a statement of official SMP policy; any opinions expressed in it are my own. The aim is to provoke discussion. I shall consider a number of issues which seem to me to be important, hoping that others will be stimulated to contribute comments and suggest other topics which need attention.

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 rationale for the introduction of the system of three points for a win in the Football League in 1981-82 was, presumably, that more reward for winning games would encourage more positive attitudes from teams and that the consequent more attractive and attacking football would bring in bigger crowds. Certainly the philosophy of many away teams, or at least their managers, that they began a game at 0-0 with one point to hang on for had become commonplace in recent seasons.

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 way this problem is stated tempts one to think of it as an isolated puzzle. But there are a number of reasons why the problem may be worth investigating more closely. Perhaps you found it so mind-boggling that you couldn’t do it at all. Or you may have succeeded in finding one such list, but could not find a convincing proof that your list was the only possible solution. In either case it would have been worth having a go at a simpler version of the same problem first. But even if you solved the puzzle completely, the mathematician in you should be mildly inquisitive as to why it asks for a list of eight whole numbers, rather than say seven, or nine, or even six hundred and thirty four.

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.

The dramatic increase in the speed of numerical calculations made possible by electronic wizardry has opened up new vistas in school mathematics in the last few years. One application is the solution of equations by iterative procedures. Interest so far has been only on the speed of convergence of the iteration and not on how a given iteration will succeed or fail—an equally interesting area. At school level this is a relatively uncharted area, and one where a sixth-former with a calculator could produce results of interest.

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.

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.

A recent unfortunate incident in the medical world leads to some interesting mathematics [1]. The incident centred around the half-life of a drug used in treating arthritic people; elderly patients having longer drug half-lives than young people, e.g. due to impaired liver or kidney function. Doses were being repeated before the half-life period of the patient, with a dangerous build-up of the drug in the system, leading to death in some cases. A mathematical model can be developed to describe the normal drug concentration variation with time, and the cumulative effect of repeated doses.

One day, as I was struggling to construct examination questions on vectors in geometry, the following thought (1) occurred to me.

• (1) If the position vectors of points A, B, C are denoted by a, b, and c respectively, then the position vector of the centroid G of ΔABC is ⅓(a + b + c), but similar formulae for the circumcentre, orthocentre and incentre are not commonly known.

After the thought had drifted about in my mind, it prompted a second observation (2).

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.

An interesting and important problem in probability theory is known as ‘Bertrand’s paradox’. Consider a chord chosen ‘at random’ inside a circle: what is the probability that the length of the chord exceeds the length of side of the inscribed equilateral triangle? Three different answers have been given to this problem p = ½, ⅓ or ¼.

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 traditional method of teaching group theory, based on the axioms closure, associativity, identity, and inversion, leads to a large measure of abstraction in the learning procedure. The reason for this is the nature of group theory itself, with its abstract sets and binary operations defined within them. This traditional method of learning group theory could be compared to the learning of geometry without drawing triangles, something quite unreasonable, at least at the high school level. A further difficulty in studying group theory is the large number of concepts which are required from the beginning.

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.