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.)

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.

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.

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.

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).

In a common development of the special theory of relativity one considers two events P and Q, P being the emission of a flash of light and Q the reception of the light. We suppose Q to be infinitesimally close to P and, by using the principle of the invariance of c, the velocity of light, argue that the infinitesimal coordinate differences between P and Q must satisfy

Iand

II

where dx; etc. are the coordinate differences in one inertial frame whilst dx' etc. are those in a second inertial frame (e.g. [1], p. 16).

Readers familiar with the nine-point circle property of a triangle (Figure 1) may have wondered whether there is a similar result for a tetrahedron.

In general the answer is “No”: there is no equivalent of the orthocentre for a tetrahedron, since the altitudes from the vertices onto the opposite faces do not intersect. There is, however, a special kind of tetrahedron for which these four lines do have a point H in common.

In this article, we are going to be looking at certain finite sequences composed of zeros and ones.

Given n dimensions, the obvious way of representing n classes is by a hypercube of side 2. Such a hypercube contains 2n unit hypercubes, one for each combination of the n classes. With only one dimension available, it is still possible to represent each of the 2n combinations. Take a line OX with any point P1 marked on it. The interval OP1will represent all combinations excluding the first class A1 and the interval P1X all combinations containing it.

The article by Mr. D. C. D. Potter in Gazette LVI, No. 396 (May 1972), pp. 109-10 attracted a number of interesting letters. We cannot print all of these in full, but the main points are included in the following extracts.

First, Dr. K. Szymiczek of the Silesian University, Katowice, Poland (writing from Cambridge) offered some information on the history of the problem. “According to L. E. Dickson, L. Euler was the first to show that there are infinitely many square triangular numbers and S. Roberts proved in 1879 that Euler's formula gives all square triangular numbers ([1], pp. 13, 27).”