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.

“If pairs of common tangents be drawn to three circles, and if one triad of common tangents be concurrent, the other triad will also be concurrent.”

This is Casey’s enunciation * of a familiar theorem; Salmon † has the same unqualified assertion, and appends a footnote: “This principle is employed by Steiner in his solution of Malfatti’s problem…. [His] construction is: Inscribe circles in the triangles formed by each side of the triangle and the two adjacent bisectors of angles; these circles having three common tangents meeting in a point will have three other common tangents meeting in a point, and these are common tangents to the circles required.”

The urgent necessity for examining thoroughly the content of the Mathematical Curriculum in all types of schools at the present time is my reason for introducing a subject which is becoming a well-worn one, and I hope that in the course of the afternoon I may be successful in showing you what I believe to be a fresh and more fundamental approach to the problems involved, and that by being provocative I may invite a discussion on a very important question.

The Fifteen puzzle is well known. In its original form it consists of a shallow square box that can just hold sixteen small squares. Squares, numbered 1 to 15, are placed in any order in the box, leaving the sixteenth position empty (e.g. Fig. 1). The problem is to slide the squares, without lifting them out, until the numbers are in their natural order (Fig. 2).