To live to a ripe old age, untroubled by health problems, physical or mental, is an almost universal aspiration. But most people are not so lucky and will likely be in care homes for their final years, with varying levels of disease, disability and dementia. Kinley et al [1] maintain that over a fifth of the population of developed countries die in care homes. Moreover, the financial cost of this end of life care, which is the focus of this paper, can be daunting and require much planning [2]. It was reported in 2017 that, in the UK, care home costs are rising up to twice as fast as inflation [3]. Consequently the question arises about the long term affordability of such care to those having to fund it, a question that ever more people both nationally and globally are having to confront.

Eperson's conjecture is that if n ⩾ 3 is odd then 3n2 can be written in at least two ways as a sum of three squares. We shall give a fairly elementary proof of this, from scratch.

Let q be a real or complex number with

Define for n ⩾ 1.

As a student in the Department of Textile Industries of the University of Leeds—then the Yorkshire College—I was taught to emulate the successes of my forefathers in the spinning of yarns and in the weaving of fabrics,—and also to perpetuate some of their errors. Empirical work, without any discrimination or “imaginative insight,” was the fundamental basis of practically all instruction.

Some students on first meeting the axioms of a vector space are puzzled by the inclusion of the scalar unit law: 1v = v for all vectors v. It is the purpose of this note to discuss the consequences of omitting this law from the list of axioms.

For several years, the notion of ‘mathematical proof’ has been at the centre of much discussion, writing and educational research. To a very large extent the discussions have focused on the nature of the acceptability of the reasons for a mathematical conclusion. Admittedly there have been contributions concerning the role of proof in mathematics and also the mechanics of pupils‘ proof behaviour, including works which discuss both affective and pedagogical aspects, but the dominant context has usually been one of validation. That is to say, discussions have largely been in the domain of what ‘counts as a proof’ and for whom. However, I claim that there is another dimension to proof that is of central importance in the intellectual development of students at school and university. This relates to the phenomenon in mathematical thinking in which a section of mathematical structure is mentally compressed into a single unit, small enough to fit into the conscious focus of attention at a given time, and possessing an interiority which is able to both guide manipulation of the unit and also be subsequently expanded without loss of detail. This article discusses the role of this phenomenon in the manipulation of mathematical statements involved in proof. An analysis in these terms of the dynamics of certain types of mathematical proof suggests both an explanation for students’ difficulties and a possible means of help in some cases.