This article started life as an investigation into certain aspects of the Fibonacci numbers, ‘morphed’ seamlessly into the structure of some infinite matrices and finally resolved into a general set of results that link structural aspects of Fibonacci numbers with trigonometric and hyperbolic functions. It is a surprising fact, but while I can find evidence that the link between these areas has been noted in the past, I can find no evidence that the link has been systematically developed.

The well-known simple proof that the number e is irrational consists of writing e as its exponential series, multiplying by a suitable factorial, r!, and deriving the contradiction that, if e were rational, there would be an integer lying strictly between 0 and 1. This proof is referred to as a ‘trick’ in the recent highly acclaimed book Proofs from THE BOOK [1, p. 27], where the surprising statement is made that ‘This trick … isn’t even good enough to prove that e2 is irrational’. In this article we show that there are nevertheless two similar simple proofs that e2 is irrational. The first proof is in fact well known [2, p. 12]. The second, though elementary, is less obvious and certainly not well known but, together with the idea in the first proof, it implies the even stronger property that e4 is irrational.

One admires and applauds the enterprise of anyone who uses Gauss’s 1801 Disquisitiones arithmeticae as the starting point for mathematical exploration. I enjoyed McKeon and Sherry’s description of their journey [1] and the challenge of their conjectures. They drew attention to a class of polynomials that satisfy what they called the double angle condition ((1) below). Unfortunately, their failure to work with an appropriate definition of cyclotomic polynomials seriously handicapped their computer-aided attempt to classify double angle polynomials. Once this is remedied, a pleasant classification emerges, at least for polynomials with rational coefficients, without recourse to a computer. The main aim of this article is to present this classification. A brief final section considers McKeon and Sherry’s conjectures about irreducible double angle polynomials.

In this paper graphs that contain unique Hamiltonian cycles are introduced. The graphs are of arbitrary size and dense in the sense that their average vertex degree is greater than half the number of vertices that make up the graph. The graphs can be used to generate challenging puzzles. The problem is particularly challenging when the graph is large and the ‘method’ of solution is unknown to the solver.

In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s.

Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given. In [2] Wetzel asked when a given triangle fits into a given square. In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s. For the sake of convenience, let α, β, γ denote the angles opposite sides a, b, c respectively, and we may assume without loss of generality that a ≥ b ≥ c, which implies that α ≥ β ≥ γ.

The other day I received a long tube from a Canadian stranger containing a large poster featuring over a hundred polyhedra, including ‘all 92 Johnson polyhedra’. This term, though probably unfamiliar this side of the pond, was not completely unknown to me; it means convex polyhedra, excluding the regular and Archimedean ones, all of whose faces are regular polygons. Of course, as usual, we have to exclude those naughty polyhedra whose faces go around in pairs collecting squares (prisms) or equilateral triangles (antiprisms) and don’t know when to stop. The word convex is vital, otherwise there would be another infinite set. A lot of them are rather trivial, like sticking pyramids on the faces of a dodecahedron, but they include the deltahedra and many other interesting members. But they have at least one imitator who didn't quite make the grade. Trying to discover why, and how to coach him so that he would, I found that my spherical trigonometry was getting rather rusty so I set out to make one and see what was happening. I thought perhaps other readers would like to share this piece of antiresearch.

When identical copies of a solid can be packed together indefinitely to cover the whole of space and leave no gaps it is said to be space-filling. Usually only convex solids are considered in this context since, as Grünbaum and Shephard point out, non-convex space-filling solids can be generated from convex examples by making compensating additions and reductions, analogous to Escher’s modifications of plane tessellations. Nevertheless some non-convex space-filling solids are not obvious derivatives, and are interesting in their own right. Although the examples considered here have all been generated in the same way (to be described), their relation to other space-filling objects will also be considered.

Probability theory abounds in counterintuitive results, perhaps the most celebrated being the answer to the birthday problem: what is the least value of n such that p (n, 2) > ½ where p (n, 2) denotes the probability that at least two out of n randomly chosen people have the same birthday? The question assumes birthdays are uniformly and independently distributed with leap years being ignored. The solution, n = 23, never fails to startle beginning students, and very often triggers an interest in the subject of probability. It is derived from the well-known observation that by the principle of complementation p (n, 2) is one minus the probability that no two have the same birthday, i.e.

Dedicated to Professor S. Kurepa on the occasion of his 73rd birthday. We are familiar with many trigonometric formulas (identities)

leading to five more identities

and so on, where #x211D; is the set of reals.