The change from essentially two party contests to three-, or even four-cornered fights, has provided the impetus for a fresh look at both diagrammatic representations of an election outcome, and the measure of swing from one election to the next. The representation of the votes in a three-cornered contest has been considered, amongst others, by Ibbetson [1], by Hollingdale [2] and by Miller in a series of papers and books (see for example [3] and [4]). Miller puts forward three ways of describing the outcome of such a contest, the ‘box-corner’ method (three-dimensional coordinate axes), homogeneous coordinates, and the ‘rectangular coordinate derivation’, but, as he notes, these three are fundamentally the same. Miller states that ‘mathematicians will instinctively go for the homogeneous coordinate representation’, and Hollingdale kindly obliges, although Miller himself plumps for the third representation, [3].

Have you ever looked at a scenic photograph and wondered where it was taken from? The problem of calculating the location of the camera turns out to have a rather elegant solution. In the first place we want to make as few assumptions as possible about the optical systems used to produce the photograph. In particular, we will not, in general, know the focal length of the camera lens (or that of the enlarger). Neither will we know the precise direction in which the camera was pointing: the photograph we are looking at may be an enlargement of just a portion of the original photograph, so the centre of the photograph may not correspond to the centre of the optical system of the camera. However, we have to assume that there is no distortion, as is often evident on photographs of interiors or buildings taken with a wide angle lens. Fortunately, scenic photography usually involves bright light conditions with the camera stopped down to a small aperture so that it is a good approximation to an ideal “pin-hole” camera.

5, 11, 17, 23, 29 is an arithmetic progression of prime numbers. It is conjectured that there are such progressions with arbitrarily many terms (there cannot be infinite sequences because a, a + d, a + 2d,… sooner or later contains a + ad). The longest such known to the authors has first term 4943 and common difference 60 060 with 13 prime terms, but longer ones will have been found by now. In general, additive problems about the multiplicatively defined prime numbers are inevitably difficult (e.g. the Goldbach conjecture and twin primes) and may even be candidates for undecidability. We have been trying to prove a much more general conjecture of Erdos, spurred on by his offer of $3000 to the first proof or disproof of it.

The famous birthday problem is well known to regular Gazette readers. For those readers not familiar with it, the problem is to find the size of the smallest group of people for which there is a probability greater than one-half of two members of the group sharing a birthday. The answer (23 people) is a result which the eminent probability specialist William Feller described as “astounding”.

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.