It is well known that the sum of an absolutely convergent series is invariant under rearrangement of its terms. On the other hand, a conditionally convergent series, that is one which converges but the sum of whose absolute values is unbounded, can be rearranged to have any sum whatsoever, or diverge in any desired manner (see for example [1, §44]). A simple examplS of a conditionally convergent series is the alternating harmonic series (AHS), . In [2], the following theorem on rearrangement of the AHS was proved:

Theorem A: The AHS remains convergent under a simple rearrangement (i.e. the sub-sequence of its positive terms and the sub-sequence of its negative terms are in their original order) when p of its positive terms alternate throughout with q of its negative terms, and the alteration in sum is (p/q).

A ferry starting on one side of a river is to cross to the adjacent point on the opposite bank travelling at a constant speed f but moving against a variable river sneed We will ignore starting and stonninc difficulties which only operate for a short interval of time. The ferry moves so that it is always aiming at the target position T. The river is assumed to be straight and the speed of the river is given by the function r(x) which is non-negative, continuous and differentiable for 0 ≤ x ≤ a, where a is the width of the river. We shall show that the solution reduces to two integrations and the condition which determines whetIer the derry can reach its destination will be given.

The identity in question really is simple: it says, for u ≠ −1,

We describe two types of definite integral that look quite formidable, but dissolve into a much simpler form by an application of (1) in a way that seems almost magical.

Both types, or at least special cases of them, have been mathematical folklore for a long time. For example, case (10) below appears in [1, p. 262], published in 1922 (we are grateful to Donald Kershaw for showing us this example). However, they do not seem to figure in most books on calculus except possibly tucked away as an exercise The comprehensive survey [2] mentions the second type on p. 253, but only as a lemma on the way to an identity the authors call the ‘master formula’ We come back to this formula later, but only after describing a number of other more immediate applications.

The Original problem

Take a right-angled triangle ABC, where ∠C = 90°, and fold along UV so that C lies on AB, as shown in Figure 1, where C' is the new position of C.

Problem: Where should one fold in order to minimise the area of the folded triangld CUV?

This problem was discussed by Hirschhom in [1], where he gives its origins and provides a solution, describing the answer as ‘quite remarkable’. However, he does not relate the answer to the geometry of the configuration. We adopt a more geometrical approach and use some simple folding ideas—there is a close relationship between paper folding and geometry which deserves to be more widely known (see [2] for example).

A uniform ladder leaning against a wall, or sliding down against a wall, is a familiar theme in introductory mechanics and calculus courses, made popular in recent years by several papers pertaining to its various intriguing aspects and important limitations. This configuration is particularly well-known in the context of static equilibrium where the friction exerted by the floor keeps the ladder from sliding [1]. The situation involving the ladder sliding down instead of being static is a staple topic of calculus courses, although usually only from a geometric point of view. The physics of the sliding ladder was addressed in [2, 3, 4], none of which took friction into consideration. In the present work we study in detail a situation involving a ladder sliding between a rough wall and a frictionless floor. This is a situation where in spite of the presence of friction the ladder is still destined to slide down, no matter how large the friction coefficient between the wall and the ladder is. As we will establish in this paper, in spite of the friction present, the ladder must always be speeding up while sliding down, until it breaks off the wall at some critical angle.

This paper is dedicated to the memory of George Boole FRS (1815-1864), first Professor of Mathematics at University College Cork.

A relation on a non-empty set K is any subset W of the cartesian product K × K. If (x, x) ∈ W for every x ∈ K, W is said to be reflexive on K. If for x, y, z ∈ K, (x, y) ∈ W implies that (y, x) ∈ W, W is said to be symmetric on K. If for x, y, z ∈ K (x, y) ∈ W and (y, z) ∈ W together imply that (x, z) ∈ W, W is said to be transitive on K. Finally, if the relation W is reflexive, symmetric and transitive on K, W is said to be an equivalence relation on K. We contend that equivalence relations are of central importance in many areas of mathematics. Many texts on algebra (for example Topics in Algebra [1]) contain as an exercise the following fallacious argument which the reader is challenged to refute.

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

The Lucas-Lehmer (LL) test is the most efficient known for testing the primality of Mersenne numbers, i.e. the integers Ml = 2l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range, namely l = 2,3,5,7, 13, 17, 19,31,67, 127 and 257 [1]. Mersenne's list turned out to be incorrect, omitting the prime-producing l = 61, 89 and 107 and including the composite-producing l = 67 and 257, although this was not finally confirmed until 1947, using both the LL test and contemporary mechanical calculators [2]. The LL test is based on the following theorem.

Problems concerning the set

of numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identity

which shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.