In 1972 Kenneth Arrow was awarded the Nobel prize for economics. Much of his work was in the theory and philosophy of decision making, in particular the theory of voting, and in this field “Arrow’s theorem” is a truly remarkable result. It is a genuine application to the social sciences of some elementary mathematics and as such I feel that it is a pity that it is not in the repertoire of more teachers. Hence this article.

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

The idea of a recursive function is an important one for students of mathematics and computer science so it is especially interesting to find an elementary though non-trivial example based on traditional mathematics. In problem 68. A (Gazette, March 1984, solutions October 1984) we are given the values of three positive integers bv b7, b3 with the remaining terms of a sequence (bn) obtained recursively by the determinantal condition

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relation

generates a cycle of period 5 for almost all values of u1 and u2 [1].

The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is then

Lyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.

No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].

The harmonic series seems to be a topic of enduring mathematical interest. Readers will be aware of the many neat methods which have appeared in the Gazette for showing its divergence but let us here investigate one of its discrete properties.

In “The music of the spheres”, Jacob Bronowski takes us on a mathematical tour of the Alhambra in Grenada. This includes a visit to the Sala de las Camas, that most private inner sanctum, in order to study the tessellations on the walls. We have used this film with teachers in training at Leeds University for some years now. It provides a stimulating introduction to the history of mathematics and enables consideration of the place of historical references in the teaching of mathematics in schools. We have also compiled a worksheet, based on the film, which includes the task of constructing one of the tessellations, illustrated overleaf.

1. The preceding paper considers the most general theorem on change of variable in a Riemann integral: If g(t) is integrable over [a, b] and f (x) is integrable over G ([a, b]), then (G(t))g(t) dt exists and equals (x)dx.

The article Problems, solved and unsolved in the Gazette for last March (60 (411), 1–11) drew reactions from a number of readers. There is not sufficient space to show in full all the proofs sent in, but I will try to indicate the methods used. For statements of the problems themselves I must ask readers to refer to the earlier article.

When the Mathematical Gazette was founded, statisticians were mainly concerned with the collection and presentation of data. The mathematical side consisted mainly of developments from the work of Gauss and Laplace in the early part of the 19th century, though some surprisingly modern work was done by Edgeworth towards the close of the century. 1889 saw the publication of Galton’s Natural Inheritance, introducing correlation and regression. These two workers influenced Karl Pearson who, in 1900, began the modern era of mathematical statistics with the concept of the significance test usually known by the symbol he used, chi-squared.

In a recent article (this Gazette, 65 (1981), 87-92) Dr Anderson raised the question of the derivation of the well-known formula for the sum of integer squares, in contradistinction to its mere verification by induction. Some time ago, in response to a request from a colleague, I derived each of the well-known formulas

by arguments based upon the decomposition of rectangles. As a complement to Dr Anderson’s article, these may be of interest; especially, in view of the derivation, to those requiring the formulas for elementary integration from first principles.

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.

This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.