In a game where there are two ways of adding to your score, but one way gives you more points than the other, some scores may be impossible to obtain. For example, you could never achieve 19 points in a game where each score was either 4 points or 9 points.

The result obtained in this short paper was discovered by chance in the course of constructing some examples for the first-year students of numerical analysis. It may therefore be of particular interest to others who have similar duties to perform. The relevant example from those which I had set is:

The process of cutting squares from rectangles described by D. G. Ball in the June 1974 Gazette, and its connection with continued fractions given by Professor Goodstein in Mathematical note 3359 (December 1974), has an interesting application to electrical circuits.

The number π (as all—or very nearly all—the world doth know) signifies the ratio of the circumference of a circle to its diameter and is also equal to the ratio of the area to the square of the radius. This suggests two generalisations of π which apply to plane figures of any shape:

In Mathematical note 3356 (Gazette No. 405, October 1974) George Berzsenyi produces new groups from old by redefining multiplication in an appropriate fashion. Though the group axioms are easily checked relative to the new multiplication, the structures of the new groups are not immediately clear. This is a common situation when groups are presented by their multiplication tables which, in the present author’s view, are of little use. They do not give a clear picture of how the groups ‘work’, as anyone who has tried to verify the associative law from a table knows to his cost. Indeed, a multiplication table of a group is usually constructed only after its structure has been worked out by some other means. Take for example the non-commutative group of order 6. Much more informative than a multiplication table are its abstract presentation in terms of generators and defining relations, and its concrete realisations as the group of all permutations on three symbols and the group of isometries of the plane that map an equilateral triangle onto itself.