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.

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.

In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the exception of the final section where it is advantageous to employ Cartesians.

The topics are:

(1) examples of collinearities where the distance ratios for important points vary with triangle shape;

(2) a pencil of lines which includes the Fermat, Napoleon and Vecten axes;

(3) two more circles related to the Lester circle;

(4) the family of rectangular hyperbolae which circumscribe the triangle of reference ABC and whose principal member is the Kiepert hyperbola [2];

(5) Cartesian coordinates leading to a solution of the ‘Euler disc problem’ [3].

I was interested to see in a recent issue of the Gazette a note from Hoare and Wright [1] about the distribution of leading significant digits, l.s.d., and “Benford’s Law” that this distribution is logarithmic. This distribution has fascinated many people since Benford’s original observation [2] in 1938 of the dirty pages at the beginning of a book of log tables.

It was at the Fifth Southeast Asian Conference on Mathematical Education (June 1990, in Brunei Darussalam) that Brian Bolt posed the question: Can the numbers 1 to 12 be arranged within the hexagram below to give equal sums in the six indicated directions? This is the Magic Hexagram Problem.

Just how many matrices have inverses? In elementary linear algebra courses, many of the matrices encountered are singular, but perhaps the reason is that such matrices provide rich and interesting examples. How many of them occur naturally? Many beginning students observe that a singular matrix can be made nonsingular by very minor tweaking – changing just one entry, for example, will make a matrix of rank n – 1 into a full rank (nonsingular) matrix. In fact, changing one entry just a tiny bit will do it. Looking at the question from that point of view, with a little experimentation students begin to discover that singular matrices are quite rare. Entire rows (or columns) have to be rigged exactly right in order to get one, while minor changes in individual entries undo all the work and give us another nonsingular matrix. If we were to reach into a hat full of all the numbers – each equally likely to be chosen – and draw enough to fill in a square matrix randomly, we would certainly expect to get a nonsingular one.

The eminent physicist, Peter Guthrie Tait, having constructed mathematical models of the flight of a golf-ball and having carried out an extensive series of experiments to evaluate their parameters, specified the maximum distance a golf ball could be made to carry by the most accomplished golfer. He was then upstaged by his son who hit a monumental drive which pitched even further. The golfing press of the day - what there was of it - was happy to run the story and scientists themselves seemed to enjoy the joke at Tait’s expense. Macfarlane, for example, related the anecdote in the following terms in a lecture delivered just eight months after Tait’s death: