I am continually surprised by the mathematical structure which surfaces in unexpected situations, and even more so when arranging a tenth birthday party for my elder son. The invasion of the home by some eight rather energetic boys on a cold February afternoon was viewed with some trepidation. The instinct of self preservation was uppermost in the search for activities which would allow the afternoon to pass with the minimum of discomfort and the maximum enjoyment. A family friend is very good at organising team games at parties, but with a space problem (our house does not have a large living room as so many modern houses have these days) it was important to split the children up for their activities, but yet devise a programme which was coherent.

A popular way of introducing the binomial coefficients is by means of the number of routes in a grid or lattice of lines. In Fig. 1 the grid in the first quadrant of the x−y plane consists of those points for which at least one of x and y is an integer.

One of the standard results of the theory of Newtonian gravitational attraction is that the potential outside of a spherical shell can be obtained by regarding all the mass as concentrated at the centre of the shell.

Frank Budden’s article in the March 1982 Gazette concerned various problems in probability about the formation of triangles from, for example, a glass rod broken into three. That article, and an Oxford Entrance Examination question, set me thinking about the following connected problems. The first is from the 1972 examination.

1. Introduction

Since Poincaré introduced algebraic or geometric topology in a series of papers during the years 1895-1901, the subject has grown rapidly to become one of the major branches of mathematics with ramifications in many other branches. However, apart from some recreational level ‘rubber-sheet’ geometry, very few students, even at first degree level, meet algebraic topology.

School mathematics is customarily taught in three stages which can be described as practical, pure and applied respectively. They differ from each other in the way that they make use of scientific data; [1]. The first stage involves teaching mathematics through its applications; pure mathematics makes no use of applications and at the third stage known pure mathematics is applied to other scientific problems.

We shall try to show that, in some well-defined sense, every right-angled triangle is ‘approximately similar’ to a Pythagorean triangle: and then we’ll ask how large the Pythagorean triangle must be.

“What is the probability that a randomly drawn triangle is acute-angled?” This question has been discussed in recent Gazettes (for example by Trevor Easingwood in [2], followed up by Stephen Ainley in [1]). As Easingwood pointed out, the answer to this question depends on what properties of triangles are assumed to be random and on what one means by random. And as Ainley pointed out some of these assumptions can lead to a contradiction. A statistical model is necessary to define the sense in which the chosen properties (lengths of sides, angles, co-ordinates of vertices,…) are assumed random. This model may (and preferably will) relate to a possible physical ‘random’ mechanism for generating the random triangle.