One of the questions that I set my students in a recent A level revision lesson was to express (√2 - 1)6 in the form a + b√2, where a and b are integers. While they were calculating the answer, it occurred to me that, as a consequence of the fact that (√2 - 1)n → 0 as n → ∞ (since 0 < √2 - 1 < 1 ), a and b could be used to give us a ‘good’ rational approximation to √2. Within a minute or so everyone agreed that (√2 - 1)6 = 99 - 70√2, allowing us to write √2 ≈ 99/70.

The development of algorithms has been one of the principal concerns of mathematics throughout its history, but now that machines can perform algorithms more efficiently than people our attitude to algorithms has to adjust. It is hardly possible for a school to implement the National Curriculum in mathematics without having the computer language LOGO among its resources, but the curriculum's prescriptions are minimal, and it is quite possible to introduce the opening steps of the language and then do no more. What further use can be made of the language? In this article I show how LOGO can provide a systematic approach to some traditional problems of combinatorics.

In Note 1568 (V for Victory) very brief reference was made to Generalised Chess, that is, an expansion of the ordinary game as played on the 8 × 8 board. In that note attention was called to the existence of the 5-leaper, whose moves are from (0, 0) to (5, 0); (0, 5); (3, 4); (4, 3). Here it is proposed to give some account of chess in three and four dimensions. In anticipation it must be pointed out that fractions do not exist in chess geometry—a special kind of lattice geometry; but that lengths such as √(a2 + b2) and √(a2 + b2 + c2), which are not integral, but where a, b, c are, do exist, since they can be constructed from integral lattices.

A topic of current interest is that of finding defining sets (preferably smallest defining sets) for various combinatorial structures such as latin squares and block designs, so it seemed interesting (though of no practical use) to try to solve the same problem for magic squares of small size. (The germ of this idea was suggested in the Gazette Note [1] of 1988 under the sub-heading ‘Squares from clues’.)

The advent of the modern calculating machine has reduced the labour of computation in an extraordinary degree; continued products, long division, and so on, offer no terrors to anyone who has access to any of the latest excellent machines; even the extraction of a square root can be performed to ten digits in a minute or two. Under these conditions the numerical solution of algebraic equations, a process stigmatised in the seventeenth century as intolerably laborious, can even be in danger of becoming a vice : the present note owes its origin to a mild indulgence.