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.

In the June 1981 Gazette, Ian Anderson discussed several ways of finding the sum of the first n squares:

His methods are mostly algebraic and I wondered if there were more geometric ways of obtaining the formula. After some time, I have found several methods which I present here. These methods are based on counting balls in piles and are suitable for exploratory activity at school level.

In making a study recently of “whiplash injury” in vehicle impacts, I came upon a fact which surprised me, and further investigation led to some pleasing results involving only elementary mathematics, together with the design of a model which might have a future as an “executive toy” related to Newton’s cradle.

In this article we describe how a group of third year mathematics students of the University of Rome who were following a mathematics education option explored some of the properties of hyperbolic plane geometry before preparing a unit on non-Euclidean geometry to be used with sixth-formers at the Liceo Virgilio, Rome. We recall here that the Mathematical Association considered the possibility of such work in its 1923 report on geometry [1]. Nowadays the report is associated with the three ‘stages’ of geometry teaching, A, B and C, but in fact it proposed two further stages. The fifth stage included non-Euclidean geometry both ‘for a few gifted specialists’ and also because it might interest ‘able boys who are not mathematical specialists’.

We will consider two related maximisation problems relative to an arbitrarily given triangle. Their solutions will illustrate the power of cyclic quadrilateral constructions and the utility of Ptolemy’s theorem for obtaining symmetric formulae.

The great mathematician Henri Poincare [1] described the method of proof by mathematical induction as mathematical reasoning par excellence. Induction is, undoubtedly, a method of great significance for mathematics. Unfortunately, induction is also a rather complicated form of proof. Furthermore, it is taught in schools at sixth form level. In fact, it is probably the only method of proof explicitly taught in British schools. Perhaps it is not surprising that some students find the method difficult to apply. In view of the importance and difficulty of the method, mathematical induction is worthy of further attention. In this article, the history and equivalents of mathematical induction are discussed.

The widespread availability of electronic computers has widened the domain of possible investigations at all levels in mathematics, in the classroom no less than in advanced research. For instance, provided convergence is not exceptionally slow, series may be summed quite accurately and very quickly using a pocket calculator. This enables us to check theoretical work like the evaluation of certain integrals which can be expressed as series.

The cracking or tearing of thin sheets of material is of considerable interest in modern-day engineering. A recent investigation into the behaviour of concrete indicated that it would be of interest to consider a probabilistic model in which a crack is regarded as some form of non-self-intersecting random walk in the plane. This paper considers some simple models and in the Appendix are included some results on paper tearing.