In the Name Game, letters of the alphabet are drawn at random, and a player marks off all occurrences of the letter in his name as it is called. The winner is the player whose name is deleted first; but a tie can occur when players' names have letters in common. For the two-player game, the probability of a player winning depends not only on the length of his own name but on how many letters occur only in the other player's name. (See [1] for probabilities involving more players.) For example, if Stephanie plays against Georges, then there are 11 letters in the union, 2 in the intersection, 3 that are in Georges but not Stephanie, and 6 that are in Stephanie but not Georges. In this case, the probability of a tie is 2/11, the probability of Stephanie winning is 3/11, and the probability of Georges winning is 6/11. These probabilities are easily derived by considering the 11! permutations of the letters in the union.

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

This article came about as a response to the Supplement to the Gazette Number 514 [1] which consists of an extended version of Sir Christopher Zeeman’s Presidential Address to The Mathematical Association at York in 2004. His intention was to stimulate the teaching of 3-dimensionaI geometry in schools, and his publication will certainly satisfy some long-felt needs. It contains a short section on conies, which seemed relevant to a topic I am already investigating: the level of knowledge of conic geometry potentially available to the Greeks before the time of Apollonius. Professor Zeeman has encouraged me to publish the first stage of this research in the Gazette to provide a background to his discussion.

In an article in the Gazette for November 2005 [1] Martin Griffiths described some ideas which he had explored with a Further Mathematics class based on extensions of the Platonic solids in more than 3 dimensions. This reminded me of an investigation I had myself undertaken with similar groups of students. Whilst giving a glimpse of ways in which mathematics can develop through generalisation, it also led to some unusual applications of the binomial theorem.

It has been known for some time that when the sun is reflected in a circular cylinder, the illuminated part of the cylinder's base is bounded by part of a nephroid (the curve obtained by rolling a circle around another circle with twice the radius).

A few years ago, Shail [1] described earlier computer-assisted work on the Lester circle. Shail then showed how a Cartesian proof was ‘just feasible by hand’ but that it was more convenient to use computer algebra systems.

However, not relying on computer assistance does have the advantage that it forces us to search for improvements in our methods. In this instance, using vectors referenced to a special set of three coplanar axes enables us to avoid involved computations.

In this article we discuss two main theorems relating to two different decompositions of the fraction r/n as a sum of three fractions of numerator one. Each decomposition is associated with three positive integers a, b, m such that m is a divisor of an + b. Through this discussion we devise a procedure to find values of a, b and m for a given n.

Certain numeric puzzles, known as ‘magic letters’, each have a finite permutation group associated with them in a natural manner. We describe how the isomorphism type of these permutation groups relates to the structure of the magic letters.

It never ceases to amaze me how often an apparently innocent-looking problem can lead on to something altogether more deep and interesting. As an example of this, I had recently set one of my Lower Sixth classes, as part of a statistics revision lesson, the task of devising some original questions on probability when the following problem emerged:

‘A person tosses a fair coin six times. Given that 4 heads and 2 tails result, what is the probability that the proportion of tails obtained at each point before the last toss is less than one third?’