It is generally accepted that the presentation of numerical methods will depend to a large extent on the type of student being taught. Mathematics students should be exposed to error analysis and possible pitfalls of the various techniques. However, other groups of students, such as engineers or physicists, usually require a different approach, and the emphasis tends to be more on showing how to use the method on a calculator or, preferably, a computer. Indeed, at Napier College, students are encouraged to write their own programs and/or use the microcomputer laboratory. This practice, though commendable, does have the danger that the students begin to believe every answer that the computer/calculator generates. They forget that rounding errors can become important, and in fact teachers often forget to mention this, since more than 8 figure accuracy can always be assumed.

Magic squares have long provided a fruitful source of classroom examples and stimulating exercises in recreational mathematics. Attention is often devoted to the so-called normal magic squares of a given order n in which the entries are the integers from 1 to n2. In these the size must be at least 4 for examples to be sufficiently plentiful: there is essentially only one normal magic square of order 3 (apart from rotations and reflections), while there are 880 of order 4 and several millions of order 5. However, if we drop the requirement of normality, the 3 × 3 square becomes a surprisingly rich subject for investigation, which can challenge pupils at all levels. In this note we discuss the following problems: (1) find a formula to generate all 3 × 3 magic squares with a given magic number m (the sum of entries in any row, column or diagonal), with no restriction on the entries—they can be any real numbers; (2) find conditions under which the square has all entries positive integers; (3) determine how many essentially distinct positive integer squares there are for a given m.

Expressions like

are known as continued radicals and they occur frequently in the mathematical literature, if only because of their orderly structure and their close links with continued fractions. But it’s worth bringing them out again here because they can make very interesting classroom exercises on sequences, and they lead to unusual ways of expressing π and the golden ratio, etc.

The idea of a recursive function is an important one for students of mathematics and computer science so it is especially interesting to find an elementary though non-trivial example based on traditional mathematics. In problem 68. A (Gazette, March 1984, solutions October 1984) we are given the values of three positive integers bv b7, b3 with the remaining terms of a sequence (bn) obtained recursively by the determinantal condition

First it may be necessary to remind some readers of what is meant by the representation of a group by a Cayley graph. Each element of the group is represented by a point, the points being connected by lines (or edges) each of which represents a generating operation of the group. Suppose, for example, that x and y are two elements of a group, while another element;? is taken as a generator, and that these are connected: xp = y. (Throughout this article we shall take the identity element of a group to be 1, inverses, to be indicated x-1, while xp means “first x, thenp”.) This would be graphed: