“A derivative is a linear transformation which approximates a function near to a point.” I heard this definition for the first time in 1964, in a lecture at Oxford on the calculus of several variables (the lecturer was D.A. Edwards). The words came as a revelation, and from then on I had a new conception of differential calculus. At school, I thought that I knew about calculus because I could write out proofs of differentiation ‘from first principles’. I enjoyed going through the hoops and proving that δy/δ tends to a limit, but somehow the idea of differentiation never became ‘part of me’.

I believe many people have been inspired by Grunbaum and Shephard's book on tilings and patterns. On reading the book, as a computer programmer, I thought it would be a challenge to display the tiling patterns in colour on the screen of my home computer. Not having a hard disc, I had found the storage for fractals too demanding, notwithstanding the time needed to compute the intricate patterns. Hence the idea of producing a variety of interesting patterns without such overhead was attractive.

An {r, s}-leaper is a generalized knight that can jump from (x, y) to (x ± r, y ± s) or (x ± s, y ± r) on a rectangular grid. The graph of an {r, s}-leaper on an m x n board is the set of mn vertices (x, y) for 0 ≤ x < m and 0 ≤ y < n, with an edge between vertices that are one {r, s} -leaper move apart. We call x the rank and y the file of board position (x,y). George P. Jelliss raised several interesting questions about these graphs, and established some of their fundamental properties. The purpose of this paper is to characterize when the graphs are connected, for arbitrary r and s, and to determine the smallest boards with Hamiltonian circuits when s = r + 1. or r = 1. (A Hamiltonian circuit is a closed path that visits every cell exactly once.)

We are so used to using calendars that we do not often ask when they were first used and why they were used. One of the earliest places where calendars are known to have been in use is ancient Egypt. Egyptian agriculture was based on using the Nile flood water. The Nile floods at the same time every year but the cause of the flood is not connected with the weather in Egypt itself. It was therefore essential to be able to predict the time of the Nile flood in advance if advantage is to be made of the flood waters. However this is a special situation. For example when deciding when to prune my roses in the spring I could consult a calendar. However if I had no calendar I could easily find the right time of year for pruning the roses by observing when some other plant had reached a certain stage of development. In fact given the difference between one season and another this method is probably better than using a calendar anyway.

Here is a published example of what I mean by the sinusoidal fallacy:

In three earlier articles we drew attention to an astonishing error which appears in several texts in the United States, even some of otherwise very high quality. The error relates to a fallacious procedure for the calculation of the Euler characteristic of a polyhedron, producing, in particular, the extraordinary statement that the Euler characteristic of a torus is 2.

Take any positive integer written in the usual base 10 notation, reverse the digits and add to the original number. If the result is a palindrome, stop. If not, continue. Will a palindrome always occur at some point? Let us do a fe examples to see the sort of thing that happens.

In MacKinnon‘s article he makes mention of Gregory of St Vincent‘s method of squaring the circle, giving the impression that this method involved ruler and compass, an impression that is reinforced by the title and extravagant frontispiece of the book. In this article I intend to correct this impression by describing St Vincent‘s method, which used a kind of integration method, to calculate, among other things, the area of the circle.

Can we produce a ratio ψ to rival the golden ratio ϕ? In this article I shall construct a variant of the classical golden rectangle by choosing a different geometrical property. The rectangle described has more advanced mathematical properties than its golden relation, though it is not extravagant. In this sense it could be described as a supergolden rectangle.

Starting with a finite set of numbers {xl, x2, x3, …, xn} we can generate a sequence of numbers {yi} by the rule

so that yi is the average of the previous n terms of the sequence.

The National Lottery is upon us, and as you read this you may be clutching your numbers waiting for the live draw on November 19. This article will discuss the probabilities of winning, how to improve your chances, and expose a loophole that may be worth millions to a determined syndicate.