The genesis of this problem, which was communicated to the first author by Robert Vertes, is in a proposed ‘ice-breaking’ activity at a party attended by an even number of guests. Each guest is assigned a unique number from the set {1, 2, …, n}, where n is even, and their task is to form themselves into ½n pairs so that the sum of the numbers in each pair is a perfect square. It is not difficult to verify that this cannot be accomplished with fewer than eight people, while eight people can be split into the four pairs {1,8}, {2,7}, {3,6} and {4,5}, each summing to the square 9. Sixteen people can also be split into eight pairs with the required property, viz. the four pairs listed above together with {9, 16}, {10,15}, {11, 14} and {12, 13}. There is no requirement that every pair must sum to the same square.

As I write (Spring 2012) I am aware that I have been a member of the Mathematical Association for fifty years. I joined during the annual conference in 1962 when Dr Combridge was the main figure in the society. The highlight was a session at the conference (at King's College, London) by Geoffrey Matthews showing how matrices might be introduced into the school curriculum. Some treated this as hilarious, others as a signpost to the future. My friend Frank Budden and I regarded it as a premature ploy, which, we thought, might not tum out to be such a good idea. We subsequently wrote a successful book Mathematics through Geometry (1964) arguing in detail why the modem mathematics revolution—which was then gathering pace—might end in tears. We identified spatial imagination as the heartland of mathematics, and contrasted it with a denial of spatial imagination in the abstract calculi which were all the vogue at the time.

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

I Admire the ingenuity of the method by which Prof. Nanson establishes the expansions of sin x and cos x as power series in x. But I must confess that his attempt to ‘simplify’ the ‘accepted accurate’ proofs of these expansions seems to me a fundamentally mistaken one. Nor can I admit that the result is at all satisfactory ‘from the elementary didactic point of view.’ His proof is to my mind essentially an artificial verification, and altogether unnatural, in the sense that it has no place in any natural and logical way of developing the ideas which lead to the result. I speak with diffidence as one who has had far less experience of teaching than Prof. Nanson. But I am convinced that my criticism is a fair one: and I think that many of the proofs usually given, in English text-books, of many of the theorems of ‘Higher Trigonometry,’ are open to fair criticism on similar grounds, when (as is too seldom the case) they are not to be condemned for ‘hopeless complexity’ or utter lack of force.

The following paper was read at the meeting of the London Branch, which took place on the 8th of March last at the London Day Training College. It contains in outline the substance of the first few chapters of A School Course in Geometry, to be published shortly by Messrs. Longmans, Green & Co. Many details, necessarily omitted in a short preliminary paper, will be found fully treated in the forthcoming volume.

Here is a published example of what I mean by the sinusoidal fallacy: