The harmonic series seems to be a topic of enduring mathematical interest. Readers will be aware of the many neat methods which have appeared in the Gazette for showing its divergence but let us here investigate one of its discrete properties.

Suppose that an even number 2n of teams take part in a football league, and that each team is to play each other team once. Suppose also that the games are to be scheduled for 2n 1 days, each team playing one game on each day. How can a league schedule be produced? There are many ways of doing this, but one particularly simple method will be described.

In “The music of the spheres”, Jacob Bronowski takes us on a mathematical tour of the Alhambra in Grenada. This includes a visit to the Sala de las Camas, that most private inner sanctum, in order to study the tessellations on the walls. We have used this film with teachers in training at Leeds University for some years now. It provides a stimulating introduction to the history of mathematics and enables consideration of the place of historical references in the teaching of mathematics in schools. We have also compiled a worksheet, based on the film, which includes the task of constructing one of the tessellations, illustrated overleaf.

It is a well-known fact that the number of hours of daylight varies with the time of year, except at points on the equator. It is sometimes thought that at latitudes between the Arctic and Antarctic circles the variation takes the form of a sine curve. However there is no theoretical basis for this idea, although it is true that a sine curve may give quite a good approximation.

The word “game” includes a variety of human activities, some like football or cricket, involve physical strength and skill, others like bridge, involve luck and skill, and some involve only mental skill, the best example perhaps being chess. This is the type of game which we are going to look at in this article, although to analyse a game as complex as chess by the methods below, (although theoretically possible) would be completely impractical—even with computer help.

The Steiner point S of a triangle ABC is such that the total distance SA + SB + SC is as small as possible. The nice fact about S is that the angles ASB, BSC and CSA are all 120°. I mentioned this casually in a lesson with 5Ma recently, and they decided we would try to prove it. It became clear quite quickly that we would have to invent a bit of partial differentiation, but nothing too advanced.