It has now been possible to study the many comments received since the publication of the “interim statement” in Gazette No. 407 (March 1975). The recommendations given below are based on the original suggestions, taking these comments into account. Sidelining indicates a change or modification of the original suggestion.

It is proposed also to publish these recommendations as a pamphlet, so that they may be widely available to authors and others.

A few years ago, a well known laundry detergent manufacturer ran a sales-promotion competition in Australia called “Jack-in-the-box”, with the following rules:

1. 1. Draw a continuous line through 13 of the numbered squares (see Fig. 1). The numbers in the squares through which the line has been drawn are then added to give a total score. Entries achieving the highest total scores will qualify for one of the prizes.

2. 2. The line must not go through any square more than once. It also cannot be drawn diagonally through the corner of any squares_only up and down or across (see the example in Fig. 1, for which the score is 723).

3. 3. The line must begin at the square J (for Jack-in-the-box) and may finish in any square. The J square does not count as one of the 13 squares.

A good deal of interesting mathematics at various levels arises from the situation where a loan is repaid by regular equal instalments, with interest being charged on the balance outstanding at each stage. The loan may be a mortgage, in which case the instalments are reckoned as annual (even though they are actually paid monthly), or a hire purchase debt or a personal loan of some sort with interest reckoned monthly. The following topics are in roughly increasing order of difficulty.

Apart from the conic sections, many well known curves have physical and geometrical applications which are readily understood by sixth formers and undergraduates. The catenary, cycloid, spirals, involutes, evolutes and many others appear in courses in civil engineering and control mechanisms, and frequently in A level calculus questions.

Dr T. J. Fletcher’s article in last December’s Gazette [1] was of great interest mathematically in illustrating how an eminently practical problem (here, essentially a combinatorial problem) could be solved using some surprising and unexpectedly theoretical mathematical ideas. (These included the field GF(22), the group (C2)4 and finite geometries, as well as excursions into Pappus’ and Desargues’ theorems.) But to myself, at least, the interest was augmented because the article provided the background for three or four end-of-term lessons with a third year grammar school form. The purpose of this article is to describe how some of the material was adapted.

“Groups are important because they arise from structures as systems of automorphisms of those structures . . . Group tables and group diagrams are devices to make groups explicit or to visualise them, but they are utterly inefficient tools to introduce groups or to prove that some system is a group ... Are there valid arguments to teach a school group theory, different from the genuine one? I would doubt it.” H. FREUDENTHAL: Address to the ICME congress at Exeter, 1972, on What groups mean in mathematics, and what they should mean in mathematical education.

An interesting example of the usefulness of group theory occurs in the solution of first order differential equations. If such an equation retains its form under each member of a group of transformations on the plane then, under certain circumstances, we can determine from the structure of the group new variables in which the equation is separable and hence solvable by integration. In the first section we show how to construct the new variables and in the second we prove separability.

The death of Mr John Lloyd on 4th April moved the members of the Association gathered at Liverpool for the Annual Conference a week or so later. Conference members had become accustomed to his annual report as the Association’s Treasurer from April 1972 to the Autumn of 1975 and to his characteristically direct recommendations as he planned to keep the Association’s financial position secure despite the difficult national economic circumstances.

Some years ago an interesting article appeared in the Gazette [1] about a young student J. E. Cubbon who had found a theorem on the number of odd coefficients appearing in a row of Pascal’s triangle. Whilst trying to prove Cubbon’s conjecture in my own way, I found an unexpected relation between Pascal’s triangle and the construction of regular polygons.

Last summer a colleague of mine suggested a counting problem which I eventually solved by a curious method involving the counting of subgraphs. The problem was to find the number of different ways of arranging twelve Dienes logic blocks which differ only in shape and colour in a closed chain, in such a way that there is one difference in attribute between each pair of neighbouring pieces in the chain.

In a Gazette article some years ago [1] Derek G. Ball proved the theorem that the only regular polygon which can be constructed on a square pinboard is a square. This note gives two more proofs of this result.

It appears not to be widely known that there is an analogy of the theorem of Pythagoras for a ‘right-angled tetrahedron’, i.e. a tetrahedron which can be formed by slicing off a vertex from a cuboid. The result is: the square of the area of the face opposite the vertex containing the three 90° dihedral angles is equal to the sum of the squares of the areas of the other three faces.

Many results in A level statistics textbooks are quoted with the observation that the proof is “outside the scope of this book”. The result discussed here, which many students find surprising when they meet it for the first time, is obtained by a method which requires only some knowledge of moments and of the use of a probability generating function in the calculation of the mean and variance. It may be found by the students themselves with a little prompting.

Most tennis players have two types of serve in their repertory. One is a safe serve, which is relatively accurate, but also relatively easy for the opponent to return effectively. The other is a risky serve, which is more likely to be a fault, but also more difficult to return if it is good.

In their recent article [1] B. Grünbaum and G. C. Shephard asked whether, for every pair of natural numbers k, r, there exist sets of k tiles that admit precisely r distinct tilings of the plane. (Two tilings are said to be distinct if they cannot be brought into coincidence by a rigid motion, and it is to be understood that in the tilings each distinct tile must occur at least once.) They gave examples for k = 1, r = 1, 2 and for k = 2, r = 1, and referred in a footnote to a later discovery of a solution for k = 1, r = 3.