It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not alwaysdefine, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.

Oxbridge Tripos examination questions originating from the turn of the century have had a marked effect on the teaching of mechanics both at School and at University. These questions, reproduced in the standard textbooks of the day, were the stable diet of the undergraduate population over so many years and set the pattern for a style of question in which the student applies mechanical principles by rote without ever questioning the validity or relevance of those principles. An unfortunate consequence of this state of affairs is that mechanics has been largely overlooked as a vehicle for the teaching of mathematical modelling. There have been moves recently to make questions more “relevant” and it is not unknown to see illustrations of “real” boats, shotputters etc. However, the student is still expected to answer the question “mechanically”, if the reader will excuse the pun, without any discussion of the modelling process. How refreshing it is, even with first year undergraduates, to look at the problem of a tin can being knocked off a wall by a bullet and, instead of asking the students to apply two conservation laws in order to determine the speed with which the can hits the ground, to ask under what assumptions the two conservation laws would appropriately model the physical situation? It is quite staggering how the lecture theatre erupts with ideas, suggestions and interest.

The problem I am going to discuss is the generation of the class of all mappings of the space S×S×.. × S, with any number of factors, into S itself, from a single mapping of the class, where S is a finite set. The problem has its origin in the study of truth tables and it is in this setting that the problem may be most simply presented.

Geometry in a Mathematical Syllabus seems to me to occupy much the same position that Greek does in the Classical. The two have similar merits and similar defects, and I am doubtful whether either deserves its place except in specialist classes.

In a paper with the same title, Carlitz [1] proves two inequalities about a triangle ABC and an internal point O:

1. Introduction

In [1] the following three questions have been asked, and answered: For which values of f is there a convex polyhedron whose faces consist of

• (i) f triangles? (Ans f = 4, 6, 8, 10, 12, 14, …)

• (ii) f quadrilaterals? (Ans f = 6, 8, 9, 10, 11, …)

• (iii) f pentagons? (Ans f = 12, 16, 18, 20, 22, …)