The method of writing , … as recurring decimal fractions by cyclic permutation of the six figures 142857 is well known, but it is perhaps not so well known that this is but a tiny piece of a general pattern of behaviour of all fractions with denominators which are prime to the radix.

Shiu has shown [1] that there are infinitely many triangular numbers that are also square. He remarked that while his theorem did give a list of such numbers, it did not give all of them. It seemed that there might be a reiterative method, working in integers, that gave more of these triangular square numbers, and which included all the known smaller numbers.

The following paper is an attempt to explain the action of a class of three-dimensional mechanisms, first from a theoretical standpoint and afterwards to suggest some practical applications. One or two rather more general mechanisms are investigated than have previously been studied and well-known mechanisms are deduced as special cases of these. There is no modern textbook or recent publication dealing with these mechanisms and their properties are not generally known.

As a subject of study statistics differs from other subjects in being non-existent by itself. A “statistic” is a measure—e.g. mean, variance, standard deviation, median—of a sample, and therefore of a sample of “something”, and that original “something” pertains to some other “subject”—e.g. biology, agriculture, medicine, industry, economics. Or, if we investigate or research in the theory on which our techniques are based, we are at once plunged into mathematics. This somewhat nebulous quality of the subject, together with the peculiarly subtle mixture of down-to-earth algebra and logic on the one hand and subjective intuition and philosophical approach on the other, all wrapped together in a sort of cotton-wool of probability, is often for the beginner a real difficulty and thorn-in-the-flesh, though largely contributing later, it is hoped, to the fascination of the subject. I have found in over ten years of teaching statistics at Sheffield University that the student can be much helped over this initial period by simple experiments in which he collects his own data, in the course of which he builds up an idea for himself of the underlying probability situations and in some of which he learns something of the relevance of design and the practical difficulties associated with the carrying out of such simple-sounding instructions as “randomise.” When it comes to analysing his data he experiences a certain “vested interest” in the results just because they are his own observations.

In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s.

Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given. In [2] Wetzel asked when a given triangle fits into a given square. In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s. For the sake of convenience, let α, β, γ denote the angles opposite sides a, b, c respectively, and we may assume without loss of generality that a ≥ b ≥ c, which implies that α ≥ β ≥ γ.

When I was offered the chance to become the President of the Association, I had no hesitation in accepting. How else could I react to the opportunity of joining such a distinguished list of predecessors? The list, which includes G. H. Hardy, Arthur Eddington and G. N. Watson contains almost all of the eminent UK mathematicians and educationalists of the twentieth century. So for the honour that you bestowed on me, thank you very much indeed. It wasn’t deserved, but it was hugely enjoyable and rewarding.

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.

Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:

where n ≥ 3. They can also be defined by Binet-like formulas:

1. L. J. Mordell has shown very simply (Math. Gaz. Lii (1968), p. 135) that the minimum value of the definite integral

Nowadays, when so much computing is done with the use of the binary scale, it is arguable that some knowledge of binary arithmetic should be part of the mathematical equipment of the normal grammar-school pupil. The basic electric circuits for binary addition are comparatively simple, and a satisfactory machine for demonstrating the principles can be constructed with standard multiple switches and a flash-lamp battery and bulbs. In the U.S.A. the progressive firm of Berkeley Enterprises, Inc., of Newtonville Mass. has popularized “Geniac” kits from which various simple computing devices can be manufactured from interchangeable components. I am indebted to the Geniac Manual for some of the basic circuits I have used, though I have modified them considerably