In [1] it is shown that every second-order linear difference equation with, in general, variable coefficients can be reduced to the form

It is also shown that (1) can be solved explicitly provided that a particular solution a (n) of the ‘auxiliary equation’

can be found, and many cases in which a solution of (2) is available are discussed. However, [1] is defective in that no method of finding a solution of (2) in the general case is given.

Given a mathematical operator, how should one reason about the outcome of its repeated invocation? This question is relevant in both mathematics and computer science, where iterative operator invocations are core, algorithmic elements.

An initial approach, which one may naturally follow, is to try the operator in diverse situations, observe the results, and suggest a general outcome. Such an approach embodies operational reasoning, where inference derives from ‘behaviours’ of invocation sequences. This may indeed reveal some behavioural characteristics, but is it sufficient for rigorous argumentation of the general outcome? Not quite.

Let p1, p2, p3, … be the sequence of primes. Take any integer n ⩾ 2 and partition the set {p1p2, … , pn} into two non-empty subsets A and B. Let PA be the product of primes in A and PB be the product of primes in B. No partition exists with PA = PB, because of the uniqueness of prime factorisation, but Fernando Castro’s neat result in [1] asserts that for each n there is a partition for which 1

A magic square is one in which all rows and columns, and the two main diagonals, sum to the same total. A knight’s tour is a tour of the board in which, using knight’s moves, all squares are visited exactly once. When the squares visited are numbered from 1 to 64, if the square is magic (but without including the two main diagonals), this is termed a magic knight’s tour. This paper describes two magic knight’s tours on an 8 by 8 board found in early 2003, the first new tours to be discovered since 1988, and the first irregular tours to be discovered since 1936.

In [1], Derek Ball made three conjectures about equiangular polygons in which the length of each side is an integer. He called these integer equiangular polygons. The first two conjectures were proved in an earlier note, and the third is proved here. It can be stated as follows.

The tiling conjecture: For each integer n ⩾ 3, there is a finite set Tn of tiles such that every integer equiangular n-gon can be tiled by sufficiently many congruent copies of tiles in Tn.