Suppose someone gave you a large collection of identical regular tetrahedra. Would it be possible to glue them together face to face in such a way that you formed a nontrivial ring, or doughnut, in the sense that, during the glueing process, you could glue a tetrahedron both to your last tetrahedron and your first? Tetrahedra would be allowed to cut into each other if necessary.

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).

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.)

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.

It seems that, ever since first meeting the ham sandwich theorem fourteen years ago, I have had a lurking interest in halving shapes. Somehow a few months ago I turned my attention more specifically to the problem of halving triangles and tetrahedra—a problem with which I am still wrestling. This problem was going to be the subject of an article for the Gazette, but it did not really work out well. In a letter about this abortive article Douglas Quadling drew my attention to the significance of hyperbolae for halving triangles—something which I had previously not noticed.

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.

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’.]

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.