The nth triangular number is defined to be the sum of the first n positive integers:

Thus

In a letter to Mersenne in 1638 [1, p. 61], Fermat claimed that every positive integer can be written as a sum of at most three triangular numbers. This remarkable result was eventually proved by Gauss in 1796, at the age of 19.

Just over sixty years ago Coxeter, Du Val, Flather and Petrie wrote The fifty-nine icosahedra. A new edition, with redrawn illustrations and useful additional material, has recently been published, making this a convenient moment to see if there is anything new to say. Coxeter once described the work as reflecting a certain ‘youthful exuberance’, so do not be too surprised at what we find.

Anyone who has taken first-semester calculus has spent a considerable amount of time studying polynomial functions and their graphs. Calculus I students typically learn how to locate certain special points with respect to the graphs of polynomials, such as their x- and y-intercepts, their local maxima and minima, and their points of inflection (if any). They then learn that these points, together with information provided by the first and second derivatives of the functions, can be used to sketch the graphs of polynomials.

A rational number is said to be expressed in Egyptian form (or Egyptian presentation) if it is written as a sum of fractions where each fraction has the numerator 1 and all of the denominators are distinct. The individual terms of die sum are often referred to as unit fractions. The problem of finding such expressions, and many related questions, has a long history, apparently beginning with Ahmes in the Rhind Papyrus. It is for this reason that we talk of ‘Egyptian’ form.

Two recent papers published elsewhere [1, 2] set out the original development of logarithms by John Napier: in [2] spreadsheets are used, and the computation of these logarithms is well within the grasp of a student looking for an unusual PC based project. It is an interesting exercise to continue the computer aided investigation of the subsequent development of tables of logarithms in the early 17th century by that other major participant, Henry Briggs (1560–1631): only now can previously unknown flaws in Briggs’ extensive numerical work be found, without the trauma of spending years of arithmetical drudgery. This work is also suitable for a student project using a PC maths package, making use of variable place arithmetic. The story that unfolds is remarkable: some parts, though well-documented, are possibly not so well-known as they deserve - e.g. D. T. Whiteside [3] has shown that Briggs discovered the binomial series expansion for (1 + α)1/2 for small α, while devising a finite difference algorithm for extracting square roots. Florian Cajori [4, pp. 5-14] pieced together the analytical aspects of the development of logarithms in the early years of the 20th century, but largely ignored the numerical aspects of the work, especially that of Briggs (and Mercator). The aspects of Briggs’ work investigated here is guided by his usually good intuition, while of course there are pitfalls into which he occasionally stumbles, as one might expect from someone engaging in numerical work ab initio: with the analytical methods of the calculus still being largely over the mathematical horizon, Briggs presents his methods without formal proof, but uses numerical examples instead.

Games arouse enthusiasm and present challenge. Mathematical games add a form of scientific discovery. The beauty of mathematical games lies in the challenge of discovering patterns hidden in a (sometimes very simple) set of playing rules. Once a pattern is identified, the triumph of discovery brings the satisfaction of mental achievement and the joy of defeating an opponent. When the games can be programmed and colourfuUy demonstrated on a computer, the sense of satisfaction can expand even further.

The aim of this paper is to give some elementary tools to solve problems of the following type: let x be a real number, n a positive integer and ε a positive real number; to find the best approximation of x by a fraction in the interval Iε = [x - ε, x + ε], with denominator at most n and possibly satisfying some additional constraints.

I recently came across a very appealing result about the product of the lengths of the chords of a regular polygon with n sides inscribed in a unit circle. If one vertex is selected and the product of the chord lengths from that vertex to each of the others is evaluated, the result is just n. There is a simple proof. Consider such a polygon and denote its vertices by the points A0, A1, … , An. Suppose that O is the centre of the circle and the origin of the complex plane. The roots of the equation

are just where ω is a primitive nth root of unity. Each of these roots represents one of the vertices. Suppose for convenience that the real number 1 is represented by A0.

The year 2000 was the centenary of not only Hubert’s Problems but also Dehn’s solution of the Third Problem, which was the first to be solved. The Third Problem is concerned with the Euclidean theorem that two tetrahedra with equal base and height have equal volume [5, Book XII, Proposition 5].

Today one proves this theorem by integration, showing that the volume of a tetrahedron is a third base times height. This 3-dimensional theorem is the analogue of the 2-dimensional theorem that the area of a triangle is a half base times height. But Euclid did not have access to integration, nor the real numbers, nor the product of real numbers, so he did not use the concept of an area being the product of two lengths or a volume being the product of a numerical area and a length. For instance when he wanted to state the triangle theorem he said that two copies of the triangle were ‘equal’ to a rectangle with the same base and height, where by ‘equal’ he meant here that the two triangles could be cut into pieces and reassembled into the rectangle. So Hilbert’s problem was equivalent to asking whether the same could be said about a tetrahedron. Can three copies of the tetrahedron be cut into pieces and reassembled into a rectangular prism with the same base and height? Dehn proved that it was impossible, and later Boltianskii simplified his proof. Dehn’s proof invented a real number invariant of a solid, that was invariant under cutting and reassembling, and he showed that two particular tetrahedra with the same base and height had different values of his invariant.

Imagine a horizontal rectangular tray with vertical sides, and a disc lying flat in the tray and rolling around the inside of the tray in such a way that its circumference is always in contact with the edge of the tray. We shall suppose that the disc has been rolling for ever in the past, and that it will continue to roll for ever in the future. What can we say about this physical system? Our aim is to illustrate the complexity and the discontinuous nature of the system under arbitrarily small changes in the size of the tray (a feature that is typical of many dynamical systems).