Suppose that Pn is some proposition about the integer n, which we want to prove for all n≥n0 (usually n0 = 0 or 1). The form of inductive argument most commonly taught in schools is the following:

A. Simple induction. If Pn0 is true, and Pn ⇒ Pn+i for each n > n0, then P" is true for all n > n0

Some time ago, Geoffrey Matthews and others introduced the idea of ‘shopping’ matrices as a method of introducing matrix multiplication in schools, and it recently occurred to me that these shopping matrices might also provide a palatable realisation of something that undergraduates traditionally find pretty indigestible when they first meet it, namely the idea of dual space, and particularly the transpose or adjoint mapping associated with a given linear mapping between two vector spaces.

After constructing Platonic solids, Archimedean solids and some of the Kepler-Poinsot polyhedra, I became interested in zonohedra, like the rhombic dodecahedron and triacontahedron shown in Fig. 1. These

models were new to me and most of what I knew about them at that time is summed up by H. S. M. Coxeter in his book Regular polytopes as follows:

It has been the experience of one of the authors of this article that students in senior secondary and early tertiary courses tend to regard functions defined on split domains and concepts such as differentiability as somewhat pathological and of interest principally as traps for students in pure mathematics.

Many readers will be familiar with the puzzle illustrated in Fig. 1, in which the triangle ABC is isosceles and angles a, b, c are given. If a, b, c are integers when expressed in degrees then it might appear, at first sight, that so must angle θ be. But this is not the case. The reader will quickly find that θ is not accessible by simple ‘angle chasing’. By this I mean marking in the

angles obtained by making the angles in any triangle sum to 180° and by making the exterior angle of any triangle equal the sum of the interior opposites. The following question arises: Given a, b, c can angle θ be obtained by pure geometry? My conjecture is that the answer is yes in those cases when angles a, b, c and θ are all multiples of the same angle. However, I am far from proving this, and have obtained geometrical derivations only for a limited number of cases, as will be seen.

In 1951 received a letter from Prof. H. Davenport, F.R.S., with some information about the factors of 12n ± 1. The results he gave were interesting and not obvious; whether they were novel I do not know. It occurs to me (whether strictly novel or not) they might well be worth publishing, as of general interest and as a tribute to a great and kindly man. However, since Prof. Davenport’s letter was not intended for publication as written, I paraphrase it here with explanation of the background.