Some time last year I clipped from the Guardian a report on the bungled draw for the quarter-finals of the Benson and Hedges Cup. The relevant section is

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.

I understand that this problem appeared many years ago in a mock examination paper in the Granta. I heard of it from Dr. A. C. D. Crommelin in 1919, when A, B, G and D (or more strictly C, C’, D and E) were about to start on the eclipse expeditions which verified Einstein’s prediction of the deflection of light. Crommelin in an after-dinner speech hinted that this problem might arise. I made use of it in my book New Pathways in Science, and rashly gave my solution-—the result being a considerable increase in my correspondence. I hope the following explanation will be sufficiently clear to avert a similar sequel to its appearance in the Mathematical Gazette.

The problem how to express very large numbers is discussed in ‘The Sand-Reckoner’ of Archimedes. Grains of sand being proverbially ‘innumerable’, Archimedes develops a scheme, the equivalent of a 10 n notation, in which the ‘Universe’, a sphere reaching to the Sun and calculated to have a diameter less than 1010 stadia, would contain, if filled with sand, fewer grains than ‘1000 units of the seventh order of numbers’, which is 1051. [A myriad-myriad is 108; this is taken as the base of what we should call exponents, and Archimedes contemplates 108 ‘periods’, each containing 108 ‘orders’ of numbers ; tlle final number in the scheme is 108. 1015.] The problem of expression is bound up with the invention of a suitable notation; Archimedes does not have our ab , with its potential extension to aa a . We return to this question at the end ; the subject is not exhausted.

The famous birthday problem is well known to regular Gazette readers. For those readers not familiar with it, the problem is to find the size of the smallest group of people for which there is a probability greater than one-half of two members of the group sharing a birthday. The answer (23 people) is a result which the eminent probability specialist William Feller described as “astounding”.

In a previous issue of the Gazette J. G. Colman gives a formula for the nth derivative of the composite function f[g(x)], states that this is proved by induction and asks for a more direct and illuminating proof.

During the course of Civil Service committee meetings it is socially acceptable to draw doodles instead of smoking. The two drawings exhibited here are elaborations of a committee-meeting triangular doodle which consists of one sixth of the first drawing. This triangular doodle differs from one independently discovered by Robison (1954)* in that the present one has smaller angles and is taken right to the centre of the triangle. In fact Robison had such large angles that there was no illusion of any enveloped curves in his diagram. I am told by Mr. R. A. Fairthorne that these curves have for long been called ‘pursuit curves.’ By fitting congruent right-handed and left-handed triangular doodles together at random we get an agglomeration of ‘spearheads’ and ‘hyperboloids,’ as in the first drawing. It consists entirely of straight lines and can be drawn without using any skill. The method could be used for producing a dazzling form of wall-paper, possibly suitable for a scientific exhibition.

When a point source of light is placed low over one side of a gramophone record and the eye is diametrically opposite it at about the same height, then the bright reflections on the record take the form of a rectangular cross, symmetrically placed. The direct image of the source is coincident with the centre of the record.

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved:

In the (n + l)th row of Pascal’s triangle, there are

A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the nonnegative integers and with determinant 1. To our great surprise we discovered that:

THEOREM. In M the matrices

are the only primes, and any member of M (except I) can be uniquely factorised into a product of those two.