As an industrial mathematician, who has done some teaching and has had some connections with the educational establishment, I feel extremely honoured to have been asked to serve the Association, this year, as its President. I feel it is a culmination of the growing links and dialogue between education and employment. I am happy to have been a part of that process.

In his article on bingo in the June 1985 Gazette, M. G. Sykes makes the implicit assumption that the fifteen numbers on a bingo card may be chosen from the set {1, 2,…, 90} without restriction. This is not correct—there are severe constraints on the choices. He also assumes in his calculations that the cards are printed at random. Whether this is ever the case I do not know, but the bingo cards produced for various tabloid newspapers in recent years are produced in a highly systematic manner, some details of which will be given in this article.

Mathematics at all levels depends on infinite processes. This is a “fact of life”. But, like those other “facts of life”, this one has traditionally been hushed up—especially when teaching elementary mathematics to beginners. Of course, in our liberated age, adult mathematicians are no longer embarrassed by infinite processes, and even discuss them openly. But it is not easy to see the connection between the way they discuss infinite processes (as in, say, an undergraduate analysis course) and those parts of elementary mathematics which depend on infinite processes, and which should provide the soil in which the beginner’s seedling intuitions about infinite processes can take root. So even today most children, adolescents, and their teachers must either make do with the traditional smutty versions, or work things out for themselves.

It is generally accepted that the presentation of numerical methods will depend to a large extent on the type of student being taught. Mathematics students should be exposed to error analysis and possible pitfalls of the various techniques. However, other groups of students, such as engineers or physicists, usually require a different approach, and the emphasis tends to be more on showing how to use the method on a calculator or, preferably, a computer. Indeed, at Napier College, students are encouraged to write their own programs and/or use the microcomputer laboratory. This practice, though commendable, does have the danger that the students begin to believe every answer that the computer/calculator generates. They forget that rounding errors can become important, and in fact teachers often forget to mention this, since more than 8 figure accuracy can always be assumed.

A dance is to be arranged for n married couples in which all the women’s names are put into a hat enabling the men to select their partners at random. What we are going to consider is the number of married couples who end up dancing together.

I read the first page of John Hersee’s article in the June 1984 Gazette and sat down immediately, “stimulated to contribute comments” on his question. “What do we mean by ‘investigationary work’… in the context of A-level?”

This article outlines a simple locus problem which leads to some interesting conclusions. It enables three elementary loci to be seen as special cases of a more general locus.

The problem might present a sixth-former with a challenging, open-ended investigation enabling him or her to practise some newly-acquired techniques in seeking the answer to an apparently simple question.

Magic squares have long provided a fruitful source of classroom examples and stimulating exercises in recreational mathematics. Attention is often devoted to the so-called normal magic squares of a given order n in which the entries are the integers from 1 to n2. In these the size must be at least 4 for examples to be sufficiently plentiful: there is essentially only one normal magic square of order 3 (apart from rotations and reflections), while there are 880 of order 4 and several millions of order 5. However, if we drop the requirement of normality, the 3 × 3 square becomes a surprisingly rich subject for investigation, which can challenge pupils at all levels. In this note we discuss the following problems: (1) find a formula to generate all 3 × 3 magic squares with a given magic number m (the sum of entries in any row, column or diagonal), with no restriction on the entries—they can be any real numbers; (2) find conditions under which the square has all entries positive integers; (3) determine how many essentially distinct positive integer squares there are for a given m.

The problem of finding a geometric construction in a finite number of steps to trisect an arbitrary angle using straightedge and compasses alone is a very old one, originally proposed by the mathematicians of ancient Greece. It was only with modern algebraic techniques in the nineteenth century that it was shown conclusively that no such construction can exist. This fact has in no way deterred a legion of crank angle-trisectors from presenting their alleged solutions to the problem! However some angles (even non-constructible ones) can be trisected: we’ll see later, for example, that given an angle π/7 it can be trisected. We can also consider the more general problem:

The two most common measures of central tendency and dispersion in statistics are the mean and standard deviation on the one hand, and the median and absolute deviation on the other. For most purposes the former measure is preferred for two very good reasons; the first is that the squares of quantities are easier to handle analytically than their moduli; and secondly for all the common symmetrical distributions, such as the normal, uniform and binomial distributions, for which the mean and median coincide, if a sample is taken to estimate the central value, then the mean of that sample has a smaller variance than the median, and is therefore relatively more efficient as an estimator of the central value of the parent population.

I, in common with many others, dislike pretentious titles, particularly ones which disguise human problems in the jargon of computer hardware, so I sympathise if mine leaves you cold. The difficulties which I want to address are the emotional and academic ones so often encountered in the transition from secondary education to a degree course containing a major component of mathematics. They are difficulties of which I have been aware for many years, but on which my attention has been focused more sharply by two recent conferences: the first was the Mathematical Association annual conference in Manchester at which I was asked to lead splinter group discussions on this topic, and the second was the latest University of Nottingham Undergraduate Mathematics Teaching Conference at which I participated in a group discussing “Perennial difficulties” (and which will issue a report later this year). The very title used at the latter meeting expresses clearly the intractability of many of these problems.

In 1972 Kenneth Arrow was awarded the Nobel prize for economics. Much of his work was in the theory and philosophy of decision making, in particular the theory of voting, and in this field “Arrow’s theorem” is a truly remarkable result. It is a genuine application to the social sciences of some elementary mathematics and as such I feel that it is a pity that it is not in the repertoire of more teachers. Hence this article.

Expressions like

are known as continued radicals and they occur frequently in the mathematical literature, if only because of their orderly structure and their close links with continued fractions. But it’s worth bringing them out again here because they can make very interesting classroom exercises on sequences, and they lead to unusual ways of expressing π and the golden ratio, etc.

1. Introduction

Nearly forty years ago (and I barely knowing the three times table), my father taught me the twenty-one card trick, amid background murmuring from mother as to its mysterious mathematical nature. Now, with a rather more questioning six year old, I have been made to look at how it works, and see what variations a budding Paul Daniels might introduce.

The idea of a recursive function is an important one for students of mathematics and computer science so it is especially interesting to find an elementary though non-trivial example based on traditional mathematics. In problem 68. A (Gazette, March 1984, solutions October 1984) we are given the values of three positive integers bv b7, b3 with the remaining terms of a sequence (bn) obtained recursively by the determinantal condition