In recent years we have been teaching mathematical analysis using the computer to give experiences to help students understand the concepts [1]. It has proved to be a valuable aid in many ways, but the method depends on giving a suitably rich environment enabling students to experience the full range of possibilities. A great weakness of the use of regular computer languages is that they hold numbers in memory as a finite expansion in base two, so, effectively, all numbers in such a system are rational. This meant that we were unable to demonstrate any ideas that depend on the difference between rationals and irrationals; for instance, we could not plot graphs of the kind

It is known that if a solid is turned right-handedly through an angle θ about a line with direction cosines ni, a particle with co-ordinates xi moves to yi, where

The solid which is the three-dimensional analogue of the plane quadrilateral appears to have been neglected by geometers, possibly on account of the difficulty of visualising it adequately from a plane figure. A model, however, makes an interesting object and is readily constructed from thin stiff wire and thread. The extraordinary number of collinearities, which are its principal feature, then become at once obvious. For the benefit of any one who wishes to construct such a model I would suggest the desirability of commencing with the external diagonal triangle PQR and the line P′Q′R′ (see Figs. 1 and 2), as otherwise it is difficult to get a compact model.

Many people have had the experience of borrowing a sum of money which is to be repaid with interest by making equal monthly payments over a year or a number of years. Two problems suggest themselves:

1. If the annual rate of interest is known what is the value of each monthly payment?

2. If the monthly payment is known, what is the annual rate of interest?

An order-n Latin square contains numbers, each of which is one of a set of n real numbers, , arranged in the form of an n × n matrix, in such a way that each row and each column of the matrix contains all n numbers. Euler (1707-1783) was the first to study the properties of Latin squares and they have been the focus of continued attention since. Studies of Latin squares naturally lead one to elements of group theory and of matrix theory. As will be shown in this note, both of these features may offer interesting investigative opportunities for classroom discussions of the permutation group on three symbols and of the algebra of the associated permutation matrices.