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.)

On the reverse side of the Moon, just south of the Sea of Moscow, is a crater named Tsu Ch’ung-Chih. Who is Tsu Ch’ung-Chih? What did he do to be thus admitted to the honoured company of Maxwell and Hertz, of Mendele’ev and the Curies, of Lomonosov and Tsiolkovsky?

Tsu Ch'ung-Chih was a Chinese mathematician-astronomer who nourished in the fifth century A.D. He made notable contributions to the calendar calculation and determined several constants with remarkable accuracy. For example, he gave a value of 27·21223 days for the length of the nodical month, the modern value being 27·21222 days. As another example, he found that the planet Jupiter completes seven and one-twelfth circuits of the heavens in every seven cycles of 12 years; this corresponds to a sidereal period of Jupiter of 11-859 years, which differs from the modern value by only 1 part in 4000.

There is an ever-increasing need for a collection of statistical experiments that can be used in the classroom. Ideally they should be linked to (and motivated by) attempts to model or simulate real situations. Mandel attempts to satisfy this need by studying some simple classroom experiments in physics from a statistical viewpoint. Unfortunately most of the experiments deal with artificial situations. Downton discusses the role of statistics in schools and lists a dozen suitable classroom experiments. This paper is strongly recommended and is full of interesting and controversial ideas.

It was at St. Barnabas’ School at Ravenshoe in North Queensland that I met the boy with the stockwhip. I forget his name but it does not matter. His whip was some ten or twelve feet long, and he demonstrated for me a little trick that he used to do. An assistant, the volunteer from the audience, would stand facing him with arms in front of face in case of accidents, the boy would swing the stockwhip towards his assistant with what in lawn tennis is called a forehand drive, the lash at the end would crack behind the assistant’s back, and then curl painlessly round his waist. Presumably the cracking destroyed a lot of kinetic energy.

From its formation to the present time, one of the major concerns of the Committee has been the identification of a common core of mathematical knowledge that pupils with one or two A-levels in mathematics could, and should, be expected to possess. The ‘Core Syllabus in Pure Mathematics’ (published by the Association in 1968 with the support of the Schools Council) was one outcome of that concern. The first report is currently being revised, along with a companion report on Applied Mathematics (and Mathematics Applied).

It is part of the folklore of symbolic logic, and more specifically of the statement calculus, that there is an arithmetic method for determining the truth values of statements of that calculus. A brief survey of texts in logic reveals that the matter seems to be dealt with rarely if at all, and usually only as an exercise ([1], p. 76). It is the purpose of this article to draw the attention of the teachers of mathematics to this rather amusing sidelight of symbolic logic.

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.

Consider the hyperbola y = 1/x. (It will be convenient to restrict attention to that part for which x > 1.) Everyone will agree that the x-axis is an “asymptote” to this curve; however, different people are likely to advance different reasons for this fact.

Statisticians are commonly confronted with the problem of expressing the relationships among three variables and often wish to illustrate these relationships. Various ways have been tried with more or less success. For some types of data isometric block diagrams or graphs have been found satisfactory. But unless there is a fairly steady surface trend this solution is not practicable, and the whole picture can be seen only in a solid model. Where the data are a number of points the information can be represented as small balls or other symbols standing on stalks . The co-ordinates of each point are given by the height of its stalk and the position of the base of the stalk on the horizontal plane. But the problem becomes acute with scatter diagrams where the purpose of the exercise is to see the relationships among the points and to identify clusters, particularly when the variables themselves are principal components and of little interest or meaning. Gyllenberg and Rauramaahave attempted a solution by plotting the points from a component analysis with respect to two perpendicular axes and representing the distance in the third dimension, i.e. height above the plane of the paper, by symbols. Although the height of each individual point can be roughly inferred, it is impossible to see the overall distribution of the points. Other similar techniques are commonly employed, but the impressions they give are usually rather confused.

Let f(x) (x≥0) be a positive continuous function. Associated with f(x), we can define a sequence of numbers {an} by setting an = f(n). Conversely, given any sequence of positive numbers {an}, we can find a positive continuous function f(x) such that f(n) = an. Of course, in this latter case, there is more than one such function f(x), but this is not relevant.

In the theory of special relativity one encounters the notion of time dilatation. A commonly cited example is that “a clock moving relative to an observer is found to run more slowly than one at rest relative to him” [1]. This concept, although readily acceptable on a purely mathematical basis, often appears somewhat mystical to a student, when compared with his everyday experiences in the physical world. In order to bridge the gap between the mathematics and the physical world, a simple physical analogy (which could easily be constructed as a classroom demonstration) illustrates in a nonrelativistic manner this rather unintuitive notion of relativistic time dilatation.

The Reuleaux triangle of Figure 1(a) is obtained by drawing three circular arcs of radius d with centres at the vertices of an equilateral triangle of side d. It can be rolled between a pair of parallel lines at distance d apart and is said to be of constant breadth d.

Applying an analogous construction in three dimensions to a regular tetrahedron of side d, we obtain a solid figure bounded by four spherical triangles intersecting in six circular arcs, the centre of each spherical portion being at the opposite vertex of the tetrahedron and its radius of curvature being d. This figure has two modes of rolling between parallel planes. In the first mode a vertex touches one of the planes and the opposite spherical surface rolls on the other; in the second mode, opposite circular edges touch the planes. The distance separating the planes in the first mode is d; in the second mode, the separation is d when the points of contact with the tetrahedron are at the opposite ends of the circular arcs, but this distance increases to d(√3 − ½√2), or l·025d, when the centres of the circular edges touch the planes. By slightly rounding one edge in each of the three opposed pairs, a solid of constant breadth may be produced. The geometry of this process is illustrated at Figure 1(b).

The transfer from pen and paper methods to hand calculator methods which has been made by many teachers still uses visible denary digits. We are now beginning the transfer to a computer and I feel that the Mathematical Association’s admirable new report does not place enough emphasis on the possibilities of computer system errors which are invisible to the user. The teacher looking at numerical error must not put untested faith in its abilities, and must not expect it always to behave in a regular denary fashion, as will be shown in the tests below. He must also be warned that the computer reference manual rarely has the reliability of a text-book.