There are many ideas, fundamental to the whole of mathematics, that are difficult to teach explicitly and are rarely examined. They include the concept of mathematical proof, the idea of a converse, necessary and sufficient conditions, and those elements of elegance and rigour without which mathematics can lose its beauty and even its claim to be called mathematics. It is these ideas I shall refer to as mathematical background; it must be emphasised that this background is common to both traditional and “modern” mathematics, and that its cultivation in their pupils would seem a desirable objective for all teachers, both conventional and progressive. Most of these ideas were clearly expressed in the Mathematical Association’s booklet, “Suggestions for Sixth-Form Work in Pure Mathematics”, which seems admirable in everything but its title, since most of the work should be begun much earlier than the sixth-form. (If it is left until then, there is justification for one reviewer’s remark that it is “an attempt to paper over the cracks of sixth-form mathematics”.)

I consider here another 3 problems, whose solutions are “not easy to see, but easy when seen”. Each of them involves in some way the notion of the convex hull of n points in a plane.

In many branches of mathematical physics it is of some importance to know how a given potential field is disturbed when a finite sphere is introduced into it. The new potential is usually obtained as the superposition of the former one and a suitable “perturbation potential” which becomes vanishingly small at large distances from the sphere. Such problems arise in electrostatics, hydrodynamics and steady heat flow. The nature of the perturbation potential depends on the boundary conditions on the surface of the sphere. In this note a method is given for finding this perturbation potential for several different types of boundary condition. The field need not be axially symmetric. The method is useful in several cases for determining image systems directly and without using special geometric properties of inverse points in a sphere.

Everybody is familiar with the game of “Snakes and Ladders”, but not so many realise that the game constitutes an interesting example of a Markov chain. We feel that a good way of introducing Markov chain theory is by solving the problem, How long does a game of Snakes and Ladders last? Doubtless the reader has found that the last game before sending a child to bed seems to drag on interminably, and we used a computer to determine the actual average number of moves when 2 people play on a typical board.

The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?

We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties of convex bisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.

Whereas the works on Trigonometry, Geometry and Determinants were published over the name of Charles L. Dodgson, Student and Mathematical Lecturer of Christ Church, Oxford, the two small works on Logic—The Game of Logic and Symbolic Logic, Part I — and the two interesting logical notes in Mind appeared over the signature of the creator of Alice. Thus on the theory of double personality which the Rev C. L. Dodgson tried so hard to establish (and so unsuccessfully, since his was a completely integrated though singular personality), the formal works on logic are to be classed with those informal ones which are the delight of the civilised world.

The Farey series of order n, Fn, consists of all the fractions between 0 and 1 (inclusive) with denominators less than or equal to n, arranged in order of magnitude and expressed in their lowest terms, so that for example The most obvious way of computing one is to assign a long series of stores and then insert the terms with denominator 1, with denominator 2, … , shifting the terms already there to make room for the new ones in their correct positions. This may require a large amount of storage and is fairly slow because of the large amount of shifting involved: for instance F1025 has 319765 terms, and occupies 400 printed pages (see Neville [1]). Neville also gives a table related to F100, which has 3045 terms. His procedures, a heroic application of pencil-and-paper methods, are described at the end of this paper.

It is well known that G. Tarry [1] was the first to publish a proof that the famous thirty-six officers problem posed by L. Euler [2] in 1779 has no solution but it appears to be less well known that he was the first to devise a systematic method of constructing bimagic and trimagic squares, that is, magic squares which remain magic when each entry is replaced by its square and, in the case of trimagic squares, also when each entry is replaced by its cube. Tarry's method was outlined in [3] and was explained and slightly improved upon in a book by E. Cazalas [4] published in Paris in 1934. Recently, there has been renewed interest in this topic.

One of the consequences of Descartes’ new approach to geometry (1637) was an increased interest in instruments for drawing conic sections. Conic section drawers, which is the name I shall give to instruments for drawing conic sections in a continuous movement, had been described earlier, but the central role of conic sections in the geometry of Descartes made them by about 1640 into a new topic of research. This article sketches the background to this growing interest and then describes some of the instruments designed by the Dutch mathematician Van Schooten (1615/6 – 1660). It concludes with an application of this material in mathematics teaching.

It has often been noted that modular arithmetic provides a rich source of supply of groups. Indeed, a remarkable theorem asserts that any finite commutative group can be found by a sufficiently diligent search through the multiplicative groups and subgroups of modular arithmetic! During the last few years two articles in this area have appeared in Mathematics Teaching. In the first [1], Tim Brand drew attention to the fact that such a multiplicative group can have an identity element other than 1. (For example, 8 is the identity of the group {2,4,8} under multiplication mod 14.) More recently [2] Geoff Saltmarsh described an ingenious way of finding the identity and put forward an interesting conjecture about these groups.

The June 1975 Gazette (59, No. 408, 98-106) carried an article by Colin Tripp about the so-called “adventitious angles” problem: in the isosceles triangle in Fig. 1, for certain given values of the angles marked a, b, c, to find θ. Correspondence to which this gave rise was summarised in a “progress report” in March 1977 (61, No. 415, 55-58). This cleared up certain points raised by Tripp, using trigonometric methods; and its publication also had the desired effect of stimulating some further response, which has settled most of the outstanding questions. It seems now to be an appropriate time for the editor to produce a “final summary” for those readers who have taken an interest in the problem.

The importance of teaching our students at all levels how to use computers as a problem-solving tool is well-recognised. Using numerical approximations to discover unknown formulas can be a powerful example. It is usually pointed out to the students that computer exploration alone is not sufficient and should be followed up whenever possible with a proof or at least some additional analysis. The importance of this is illustrated in the example below, in which there is compelling numerical evidence for an incorrect conclusion.

There is a widely held view that the jump from school to university mathematics is a big one. Recently James and Fiala have dealt respectively with the interdependence of school and university courses and the teaching of mathematics at university level. It seemed pertinent to obtain the views and opinions of students at the start of their mathematical studies at university.