Eddington's ‘liar problem’ received its first truly public airing in his book New Pathways in Science [1], although it enjoyed a ‘prehistory’ (beginning with a mock examination paper) more fully detailed in his later discussion [7]. As posed, it reads:

For any three points X, Y, Z, let ⊙XYZ denote the circle through X, Y, Z (the circumcircle of ∆XYZ) or, if X, Y, Z happen to be collinear, the line XYZ. (We shall often regard lines as special circles, circles of infinite radius.) This paper is about the following theorem, and extensions of it:

Theorem 1: Given ∆ABC and a point P, reflect ⊙PBC, ⊙APC, ⊙ABP in the lines BC, AC, AB respectively. Then the three reflected circles have a common point, Q (see Figure 1).

I do not know if this theorem is new, but I have not come across it in the literature. The reader is invited to prove it by angle-chasing, using circle theorems: let two of the reflected circles meet at Q and then prove that this point lies on the third reflected circle. This method is rather diagram-dependent, and does not seem to lead to the extensions of Theorem 1 referred to above in any very obvious manner, so we shall adopt a different approach.

The fact that an abelian group can be associated with an irreducible cubic curve has been known for a long time, and many results of great beauty are known about such groups. Ever since Wiles' proof of Fermat's Last Theorem, some of these results have become well known even outside the mathematical world.

It may come as a surprise to the reader that there is a natural way to associate an abelian group with a much better known and simpler class of curves – the conic sections! In this article we explore some properties of these groups.

An order-n Latin square contains numbers, each of which is one of a set of n real numbers, , arranged in the form of an n × n matrix, in such a way that each row and each column of the matrix contains all n numbers. Euler (1707-1783) was the first to study the properties of Latin squares and they have been the focus of continued attention since. Studies of Latin squares naturally lead one to elements of group theory and of matrix theory. As will be shown in this note, both of these features may offer interesting investigative opportunities for classroom discussions of the permutation group on three symbols and of the algebra of the associated permutation matrices.

This paper is motivated by the work of V. Dubrovsky [1] who described an n-dimensional cube. It is our aim to build a fractal, based on such a cube as well as to derive a formula for a general volume of that fractal. Two special cases n = 2 and n = 3 are investigated in detail.

Recently, when solving a jigsaw puzzle, I noticed that many of the pieces were symmetric in some sense; they had either reflective or rotational symmetry. This made me wonder if it was possible for all the pieces in a puzzle to be different shapes, and yet have some prescribed symmetry.

The central role of the resolvent cubic in the solution of the quartic was first appreciated by Leonard Euler (1707-1783). Euler's quartic solution first appeared as a brief section (§ 5) in a paper on roots of equations [1, 2], and was later expanded into a chapter entitled ‘Of a new method of resolving equations of the fourth degree’ (§§ 773-783) in his Elements of algebra [3,4].

In this paper we present a simple formula that can often be used to assist in the approximation of the sum of an alternating series

The ideas presented here were motivated by translating Euler's papers [1] and [2]. Suppose we begin by summing a terms of this series exactly, then the approximation that we will obtain is given by