In this article, we are going to be looking at certain finite sequences composed of zeros and ones.

While reading a book by David Wells on curious and interesting geometry, I came across the following remarkable theorem named after Holditch.

In Figure 1 the point R divides a straight stick ST into lengths p and q, where p, q 0. We restrict the end points of the stick, S and T, to lie on a plane, simple, closed, convex contour, C1, and ST slides around C1. Assuming C1 is such that ST can pass around C1 once, the locus of R is another plane closed contour, C, inside C1.

Most students in their undergraduate careers ensure that they have many examples of groups at their finger tips. These include not only whole systems such as cyclic groups and symmetric groups but also specific examples, in particular groups of small order. These latter groups are useful for trying out new ideas as soon as they are encountered, such as the bogeys of cosets, normal subgroups and factor groups. If we consider only groups of order 7 or less, then we already are furnished with nine examples of cyclic, non-cyclic, abelian and non-abelian groups.

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.)

In the June 1981 Gazette, Ian Anderson discussed several ways of finding the sum of the first n squares:

His methods are mostly algebraic and I wondered if there were more geometric ways of obtaining the formula. After some time, I have found several methods which I present here. These methods are based on counting balls in piles and are suitable for exploratory activity at school level.

The flexagon was defined by Mr R. F. Wheeler (M. G. 1958, p.1) as a strip of equilateral triangles which can be folded in various ways into a hexagon to exhibit different colour combinations. Can this idea be generalised? Suppose a plane figure to consist of m regular n-gons, with a common vertex, which can be folded together in pairs.

George Boole gave (in his books, listed under [1] and [2] in the references at the end of this article) a method for solving any equation in his algebra. This part of his work is now largely ignored in the many modern books on boolean algebra which are making their appearance (I can think of several reasons why). Nevertheless, the method can be adapted to conform with our standards of rigour.

This essay-review of Jeffreys’ very welcome and valuable Tract with the above title has been written at the editor’s request. Many readers of the Gazette must have heard of Heaviside’s operational method of solving the equations of dynamics and mathematical physics. If they have tried to learn about them from Heaviside’s own works, they have attempted a difficult task. Nothing more obscure than his mathematical writings is known to me. A Cambridge Tract is now at their disposal. Prom it much may be learned; but the air of mystery still—at least in part—remains.