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.

Have you heard of the game of Amida? It seems to be unknown in Europe, but in Japan is very commonly used as a method for drawing lots. It also provides an engaging example of a one-one correspondence which may be useful to teachers in several contexts.

The procedure is as follows. The adjudicator draws on a sheet of paper a sort of compound ladder consisting of a number of uprights, one for each contestant, joined by horizontal rungs, in such a way that each rung joins a pair of adjacent uprights, and no two rungs have a common endpoint on an upright (Fig. 1). Subject to these rules, the adjudicator may place the rungs as he pleases, and as many or as few as he likes, even none at all.

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.

An artist friend, Mr Graham Day of Corsham, Wiltshire, recently sent me the following, asking whether he had accidentally discovered a ruler-and-compass construction for a regular pentagon. His own main interest in the configuration stemmed from a similarity to one in Indian mystical art.

The usual basic topological concepts, as seen by students for the first time in a course such as analysis, often seem unintuitive. The following procedure may be more digestible.

Virtually everyone can recognise a boundary point of a set A in a space X. It is merely a point of X all of whose neighbourhoods contain both a point belonging to A and a point not belonging to A. The boundary bA of A consists of the boundary points of A.

During a study of the probabilities involved in an experiment with dice I needed to determine the frequencies of appearance of certain categories of display when six dice are thrown simultaneously. The theory of this led to the polynomial which is the subject of this note.

A computing colleague asked me recently if I could devise an algorithm for obtaining the ith permutation in an alphabetical list.

Suppose that the n! permutations of n distinct letters A, B, C,..., N are listed in alphabetical order. The problem is: Given i (where i = 1,2,...,n!), find the ith permutation.

A favourite exercise on Simpson’s rule is to evaluate π as the area of a quadrant of a circle of radius 2, i.e. as the area under y = √(4 – x2) from x = 0 to x = 2. It is somewhat discouraging, and a poor advertisement for the technique, to arrive at the answer 3·127 008 with intervals of width 0·2.

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.

Let π be the plane spanned by the vectors b and c. The product vector b × c is normal to this plane and hence a × (b × c), being perpendicular to this vector, will be in π.

In the article Problems, solved and unsolved in the Gazette for March 1976, and in the postscript to its sequel in December 1976 (60, 1–11 and 284–291), proofs are given that

is integral when a and b are positive integers. The search for a combinatorial proof was unsuccessful.

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.

In Note 60.3 (March 1976) Mr. Gubby revived the old problem of the two ladders. If (Fig. 1) the lengths of the ladders AD, OB are m and n, and they meet at a height p above the ground, then the problem is to find OD, the distance apart of the walls, in terms of m, n and p.

In a recent Note [1] a nomogram was described for calculating the times of sunrise, sunset and twilight in terms of the latitude of the observer and the time of year.

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.