I don’t know how many readers of this journal would have been watching the draw for the opening matches of the 1990 football World Cup, made in Rome on Saturday, 9 December 1989, and televised worldwide. But I do hope that most of you that did—and who were paying attention!—at least thought “Hey, wait a minute, that can’t be right!” Sure enough, the draw was flawed: it was not as fair to all concerned as it might have been. This was a great pity given that the world football authority, FIFA, were at pains to stress how fair these proceedings would be, and the agonies of FIFA trying to be seen to live up to their “fair play” motto were indeed prolonged. The errors lay not in the process of drawing balls from urns itself, but in the algorithm used to define the drawing procedure, as I shall explain below. The problem was caused by FIFA’s decision to keep the four South American countries involved apart and, in particular, by their inability to implement this decision in the most fair manner. There is no allegation of anything being “fixed” here, just one of incompetence on behalf of football’s governing body. That there should be errors of this kind in such an overtly public affair is, however, yet another sad indictment of the general lack of understanding of matters numerical and probabilistic in the public at large.

In a first year university course on programming and numerical methods (in BBC BASIC), it was decided to give the students the quartic equation:

x4 + 2·88x3 − 19·23x2 − 36·11x + 91·56 = 0

to solve numerically. One root of this equation is an integer—to help the students get started—but a second root is close to the first to make it more interesting. Initially we did not realise quite how interesting this would prove to be.

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.

A recent excellent book by Paul J. Nahin on Oliver Heaviside, a self-taught genius of the Victorian age, contains several nice examples of problem solving by the master. Two of these, concerning the flow of electricity along telegraph cables, are suitable for sixth formers and are described here. In passing it may also be mentioned that these cables, used first for telegraphic signals and later for voice signals, were a triumph of both the practical and theoretical skills of the nineteenth century, and were in use for world-wide communications for over a hundred years up to the nineteen seventies.

One way to make a tetrahedron is to use four equilateral triangles. Let’s see why this works. When we fit two equilateral triangles of the same size together along an edge, the edges match up.

A standard introductory calculus problem is to calculate the size of small squares that have to be removed from (say) a unit square of metal so that the resulting net folds into a tray of maximum volume. With notation as in the figure, this requires that be maximised which occurs when x = 2/3 giving a maximum volume of 2/27. (Surprisingly, this conclusion may also be reached without using calculus via the inequality of arithmetic and geometric means:

with equality if and only if or x = 2/3.)