Although a number of earlier researchers had used the geometric mean as a convenient statistic to summarise observational data, Gallon is usually credited with being the first to consider its sampling distribution. At Gallon’s request, in 1879 McAlister undertook a pioneering mathematical study, which eventually led to the modern large-sample theory. However, some sixty years elapsed before much attention was paid to small samples from particular parent distributions. Since about 1960, new techniques have made it possible to derive exact sampling distributions for a much wider class of parent distributions. Some work has been done on producing approximate general relationships between the moments of the parent distribution and those of the sample geometric mean but they are of very limited value for small samples and even now it is difficult to find any general description of how the distribution from which a sample is drawn will affect the distribution of its geometric mean and how this will vary with sample size.

A ‘magic’ hexagon has rows of numbers in three directions that add to the same total. It is possible to construct such a hexagon from a honeycomb array of hexagons and from an array of triangles (Figure 1). There is known to be only one magic hexagon formed from hexagons. However, numerous magic arrangements are possible for the hexagon of triangles and these arrangements have many additional interesting properties. In this article we give the reasons why such configurations are possible, we look at the number of arrangements possible when there are 2 triangles on each side of the hexagon and we show that in general the arrangements are balanced in many ways – including that they are physically balanced, a property we call ‘magic moments’.

The authors of the recent Note [1] exhibit an odd preference. They derive recurrence relations for the odd terms, un = F2n+1, n ≥ 0, in the sequence of Fibonacci numbers, Fn, defined by

Many problems in topology are easy to state, but some are extremely difficult to solve. And even if they are easy, they may have surprising solutions. For example, take a tyre inner tube, that is, a toroidal surface with a hole (for the valve!). Can one turn it inside out? The answer is ‘yes’ but if one actually carries out this procedure the result is entirely unexpected.

You never know from where your next mathematical problem will come. I was surprised to find it, literally, in my own back yard. Recently, I was trying to determine the optimal way to water the lawn. I say ‘optimal’ because there are a couple of constraints that I try to minimise. First, I try to minimise the area of the wasted water. Water is wasted if it lands on the concrete that borders the lawn or if it lands on a region of the grass that has already been watered. Second, I try to minimise the number of times that I have to move the sprinkler. More often than not, I forget to move it anyway. So, it might as well be as a minimum.

This paper presents a number of theorems about hexagons whose three pairs of opposite sides are parallel. The first of these is a well-known result that the vertices of such a hexagon lie on a conic. Theorems 3 and 4 show how such conies are related to the Cevians of a triangle, and which Cevians lead to such conies being circles. When they are circles they are called Tucker circles. None of the results is at all obvious, yet it seems that some of the results presented here were known in the late nineteenth century or the early twentieth century. They seem to be of more than passing interest which should not get lost by the passage of time.

Facetting, or faceting, is the process of removing parts of a polyhedron, without creating any new vertices. It has received far less attention than the process of stellation, which is its exact reciprocal. Facetting is perhaps less easy to visualise clearly, and the results may not have the same immediate appeal. But once broached, it has a fascination all its own. Also, to understand the rules of stellation properly, it can be illuminating to study the reciprocal rules of facetting. Now that suitable software is available (see Further investigations), the study of facetting is gaining momentum.