In [1] and [2] we demonstrated how Pascal-like triangles arose from the probabilities associated with the various outcomes of a particular game (see Definition 1 below). It was also shown that they could be considered as generalisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate the existence of simple relationships between these Fibonacci-like sequences and the Fibonacci sequence itself. In addition we will investigate a generalisation of the binomial coefficients that appears when considering an extended version of the game. We start by describing this game.

Dr T. J. Fletcher’s article in last December’s Gazette [1] was of great interest mathematically in illustrating how an eminently practical problem (here, essentially a combinatorial problem) could be solved using some surprising and unexpectedly theoretical mathematical ideas. (These included the field GF(22), the group (C2)4 and finite geometries, as well as excursions into Pappus’ and Desargues’ theorems.) But to myself, at least, the interest was augmented because the article provided the background for three or four end-of-term lessons with a third year grammar school form. The purpose of this article is to describe how some of the material was adapted.

Some time ago a friend, who was organising a chess league, rang me up and asked “How many different ways are there of arranging a roundrobin league tournament?” Thinking of his phone bill I told him I’d ring him back. Which was just as well, because the problem was much more difficult dian I had realised. Indeed it is unsolved in general. In the language of graph theory it is the problem of finding the number of onefactorisations of a complete graph. These and related objects have been the focus of considerable study both in combinatorics and recreational mathematics and, although the enumeration question remains unanswered in general, it is, even for small numbers of players, remarkably full of interest.