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.

An ordered pair whose second member (or value) is a number, while its first member (or object) may be anything, will be referred to as a quantity. By a consistent class of quantities—briefly, c.c.q.— we mean a class of quantities that does not contain two quantities with equal objects and unequal values. Reviving Newton’s term, we will refer to c.c.q.’s also as fluents. The class of all objects (of all values) of the quantities belonging to a fluent is called the domain (the range) of the fluent.

This story is an object lesson in the art of picking other people’s brains.

After one of my fifth form had pointed out for me the result in Classroom Note 248 (ii) (June 1971), it occurred to me that some attractive patterns might result from re-writing Pascal's triangle with the numbers reduced modulo n. I tried it at first with n = 2, using a red dot for odd numbers and a green dot for even. The pattern was, as I had hoped, pleasing. Since we had an Open Day in the near future, I persuaded another fifth-former, P. W. Swinyard, to try the same thing for other small n. Attractive patterns resulted for n = 3, 4, and 5, a rather less attractive pattern for n = 6, but nice ones again for n = 7, 8, and 9. (I recommend readers to try this for themselves.)

Since antiquity mathematicians (and not only them) have taken an interest in constructing magic squares. Probably the first magic square ever created is the one shown in Figure 1. Its origin is shrouded in the mystical legends of ancient China. It became to be known as Luo Shu (Luo river writing). There was no clear connection between this configuration and mathematical study until the time of Yang Hui, even though it was described in the sixth century. Another well-known magic square (Figure 2) is in the painting Melancholia made by the famous renaissance artist Albrecht Dürer in 1514 (the year is formed in the middle of the lowest row).

The following extends the result given by the author for the Nth power of a square matrix to the case of negative and fractional values of N. See The Mathematical Gazette, 1965, XLIX, p. 421.

This extremum problem is really a classical beauty. It has had a long and interesting history since it was formulated by Fermat in the 17th century. Given three points A, B and C, the task is to find a point P such that the sum of distances PA + PB + PC is minimal; (see Figure 1). After a few years Torricelli found the solution: P should be situated so that the angles between the half-lines PA, PB and PC are all 120° (except when one angle of the triangle ABC is greater than or equal to 120°). The solution has then been rediscovered many times in new and interesting ways.

This article is focussed on a letter that perhaps the greatest mathematician ever, Carl Frederich Gauss, sent to a female mathematician, Sophie Germain. The letter is interesting because it is the first that Gauss wrote to Germain in the knowledge that she was a woman, and it contains a piece of mathematics that reveals better than the text of the letter what Gauss really thought of Germain. To put the letter in context I shall give some of Germain’s biography and also briefly describe female mathematicians that pre-date Germain.

The solution of linear differential equations under given boundary and initial conditions has been greatly expedited in recent years by the use of the method of integral transforms. It is the purpose of this note to show that an entirely parallel method may be developed for the solution of linear difference equations and that this leads to a new view of the generating function of a sequence as a transform of the sequence, analogous to the Laplace transform of a function.

In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]

The purpose of this paper is to consider the question: if p is the probability that a player P will win a given rally against a player Q, what is the probability that P will win a given game? The answer, of course, depends upon the method of scoring; and the methods considered here will be those used in squash and, for comparison table tennis.

In table tennis P earns a point each time that he wins a rally. But in squash he must serve the rally that he wins in order to gain a point; and he relinquishes the service when he loses a rally that he serves.