Matrix models are used extensively in demography and population biology. The example discussed here shows how a simple 2 × 2 matrix may prove of use in the study of bird populations. Teachers may find it an interesting illustration of the application of matrices. The knowledge of matrices that is required to appreciate the model and investigate it numerically may be summarised as (1) the expression of two linear equations as a matrix equation, (2) the multiplication of matrices. A deeper understanding of the behaviour of the model comes from the use of eigenvalues.

In his interesting article on Redesigning the dartboard, published in Gazette No. 413 (October 1976, pp. 171–178), Keith Selkirk refers to a dartboard he has designed by attempting to minimise the magnitude of the vector resultant obtained when vectors of magnitude equal to the score and in the appropriate direction are added.

“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.

The idea of a probability generating function seems to be a difficult one for sixth formers, and this may well be because the formalism is introduced too abruptly. The t or z or whatever letter is used in a p.g.f. has to be explained as having “no physical significance”, to quote from a recent M.A. booklet on sixth form probability and statistics. The following gentler approach I have found to lead to better understanding on the part of ‘lower fliers’, and it has the advantage of drawing attention to an underlying structure.

In its simplest form the theory of Markov chains states that, if the (i,j) element of matrix M is the probability pij of a one-step transition from state i to state j, then the (ij) element of the matrix Mk gives the probability of transition from state i to state j in k steps. Since the probabilities of transition to the different states add to 1, the sum of the elements in any row of M is necessarily equal to 1. (Readers are warned that in some elementary treatments the alternative convention is adopted of writing pij in the (j,i) position, so that the column sums are 1, in order that the probability vectors should be column matrices, which are more familiar in school mathematics.)

The sequence 0,1, a, aa, a(aa), . . . has certain unexpected features which may easily be demonstrated with the aid of a programmable pocket calculator. The investigation of the convergence of this sequence provides an exercise which is both interesting and informative.

The discussion of symmetry in modern mathematics syllabuses can be taken a stage further in the sixth form to show how symmetry can often be used to simplify the solution of certain types of problems. This article gives just one example, that of determining the eigenvalues and eigenvectors of an n × n matrix. The theory involves two steps. In the first we show how the eigenvalues of a matrix M can under certain circumstances be found from those of a commuting matrix P. In the second step we show how to construct, from the symmetry properties of matrix M, commuting matrices P whose eigenvalues are readily found.

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.

In the Gazette for June 1975 (59, No. 408, 98–106) Colin Tripp showed how the familiar problem illustrated in Fig. 1 (in the isosceles triangle, for certain given values of the angles marked a, b, c, to find θ) could be tackled for most of the possible triples (a, b,c) revealed by a computer investigation.

The idea of this article is to find a ‘simple’ expression for the number of circuits of length 7 in the directed graph illustrated; that is, a path along

the arcs of the graph which passes through each vertex exactly once and returns to the vertex from which it started.

Several writers have drawn attention to the fact that there is a connection between determinants and sets of loops in directed graphs. This fact is sometimes useful in evaluating the determinant of a matrix M, as was explained by Greenman in his recent article [1]. Towards the end of his article he introduces Δij-subgraphs and relates them to the cofactors Cij of the matrix M. Now if M is invertible, then M−1 = adj M/det M, where the (j, i)-entry of the adjoint is just the cofactor Cij. Hence one would expect to be able to give a graphical explanation for the fact that M(adj M/det M) = I, and this will be done in the present article.

It is well known to ‘curve-stitchers’ that, if equally spaced points around a circle are numbered 1, 2, 3,..., n (and then repeated cyclically), then the chords joining 1 to 2, 2 to 4, 3 to 6, ..., n to 2n envelop a cardioid. (See [1], p. 41, §16, where the cardioid is described as “the caustic of a circle with respect to a point on its circumference”.) Correspondence with E. H. Lockwood has led to a rather unexpected result from this envelope.

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.

The Witch of Agnesi (or versiera) is a curve named for Marie Agnesi who first studied it. It may be defined geometrically as follows: In a circle with centre at and diameter a (hence tangent to the x-axis at the origin), a secant through the origin is drawn, intersecting the circle in A and the line y = a in B. The intersection of lines through A parallel to the x-axis and through B parallel to the y-axis determines a point P. The Witch of Agnesi is the locus of the point P as the angle the secant makes with the x-axis assumes all values from 0° to 180°.

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.

1. It is well known that if p is a prime number not equal to 2, then exactly half the non-zero residues of p are quadratic residues (that is, squares in the arithmetic mod p). What happens if the modulus m is not prime? We can look at this problem in two ways: we can insist that our residues are relatively prime to m, or we can consider all the residues.

The method described in this note is, like other standard approaches, based on the “Euclidean algorithm”, but a matrix format is used (assuming the coefficients have no common factor) to reduce the equation ax + by = c to a form such as

or a similar equation with matrix (k l),(k −l) or (−1 k). The following lemma is easily verified.

There are various ways of introducing a discussion of the completeness of the system of real numbers. In this article we take the unusual approach suggested by the provocative question of the title.

In any elementary treatment of the convergence of sequences and series, a discussion of series of positive and negative terms will lead to the theorem that an absolutely convergent series is convergent.

The following ‘paradox’ was produced by one of my fourth year (14-year-old) pupils in an attempt to solve a quadratic equation. (The fourth years have no knowledge of techniques for solving quadratics, nor of course of complex numbers.)

The property of “φ sequences” which D. A. Barwick describes in Note 60.31 was proved by reference to the theory of Farey sequences. When the author first showed me the sequences I set about trying to obtain a proof from first principles. This note is the result of that investigation.