Euler made many contributions to what is now called combinatorics. Some of these arose from recreational mathematics, such as magic squares and knight's tours on a chessboard; others from his study of lotteries; and, perhaps his most important work, from the study of partitions of an integer. In what follows I shall attempt to show the breadth of his work. Euler's papers will be referred to by their Eneström numbers, such as E338; this cataloguing was carried out in the early twentieth century by the Swedish Mathematician Gustav Eneström, who was to Euler what Köchel has been to Mozart. Euler's papers can all be studied in detail on the Euler Archive, details of which are given at the end of this article.

There is a very easy way to produce the Euler line, using transformational arguments. Given a triangle ABC, let AʹBʹ'C be the medial triangle, whose vertices are the midpoints of the sides. These two triangles are homothetic: they are similar and corresponding sides are parallel, and the centroid, G, is their centre of similitude. Alternatively, we say that AʹBʹC can be mapped to ABC by means of an enlargement, centre G, with scale factor –2.

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].

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.

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?’