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).

During my year as President of The Mathematical Association the government began a review of the curriculum across all subjects taught in school. Given the constant tinkering with the education system by every government, this is probably a sentence that any MA president could write during their tenure. But 2013 saw government really trying to layout their vision for what education should deliver in the coming decade.

In English, an aim of the curriculum review is to develop a student's love of literature through widespread reading for enjoyment. Students will get to read Shakespeare, romantic poetry, the great novels of the nineteenth century together with seminal works of world literature. It is hoped that this will expose students to the great works of literature and allow them to acquire an appreciation of ‘our rich and varied literary heritage’ [1, p. 13].

In science, by building on fundamental ideas and concepts, the curriculum review aims to foster in students ‘a sense of excitement and curiosity about natural phenomena’ [1, p. 99]. Students will study the fundamental mechanics of the cell, learn about stem cells, photosynthesis and genomics; they will be exposed to radiation (in a theoretical sense) and the evidence for the Big Bang. Such fundamental ideas will give students the scientific knowledge to be able to understand the uses and implications for the impact of science on society.

There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1]. This article is based on other integer sequences. It sets out to determine other pairs of such sequences that have the same relation as the Fibonacci and Lucas have to each other. So we shall be concerned with second order recurrence relations with constant coefficients:

and pairs of sequences (un) and (vn) that each satisfy it. We seek a condition that ensures the pair of sequences behave as the Fibonacci-Lucas pair behave.

Albrecht Dürer's mysterious engraving Melencolia I (Figure 1) has always intrigued both art critics and mathematicians. Among art critics, according to Campbell Dodgson [1], “The literature on the Melancholia is more extensive than that on any other engraving by Dürer” (he adds “the statement would probably remain true if the last two words were omitted”). Mathematicians, if disconcerted by the association between mathematics and melancholy, have been fascinated by the objects appearing in the print, such as the polyhedron occurring on the left of the engraving and—the subject of this note—the magic square in the upper right-hand corner (Figure 2).

The Mathematical Association was established in 1871 as the Association for the Improvement of Geometrical Teaching (AIGT). Gradually through the nineteenth century, the AIGT broadened its range of mathematical concerns beyond those of an essentially anti-Euclid pressure group. Its forms of professional support developed to include meetings, presentation of papers and publications, and its own journal, The Mathematical Gazette, was launched in 1894 [1]. By this time the AIGT had also started to build a small collection of books and periodicals, which was initially boosted by the donation of mainly geometrical texts from its first President, T. A. Hirst (1830-1892). By 1897 the AIGTcouid comfortably change its name to the Mathematical Association (MA), advertised as ‘an association of teachers and students of elementary mathematics’ [2].

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.

Born in London, Reuben Louis Goodstein (1912-1985) completed his secondary education at St Paul's School and in 1931 proceeded to Magdalene College, Cambridge, with a Major Open Scholarship to read mathematics. He graduated in 1933 having taken firsts in Parts I and II of the Mathematical Tripos. From 1933 to 1935 his research on transfinite numbers was supervised by Professor J. E. Littlewood. He took a MSc and left Cambridge in 1935 to take up an appointment as lecturer in pure and applied mathematics at Reading University, a position he held until late 1947. While undertaking a strenuous teaching load at Reading his research interests were developing and for this work he received a PhD from the University of London in 1946, which was supervised by the philosopher Ludwig Wittgenstein.

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.

Older readers will remember using a slide rule to carry out multiplication and division in the days before calculators. There was also a graphical tool called a nomogram, which was used for evaluating certain functions. Although such devices have little practical use nowadays, I was interested in investigating which classes of functions with two independent variables could be calculated using this type of device.

Figure 1 shows how a slide rule was used to multiply 3 by 4 (or equivalently to divide 12 by 4). To do this, you align the 1 on the upper scale with the 3 on the lower scale and then read off the value below the 4 on the upper scale to get 12 (or, for division, you line up the 4 above the 12 and read off the answer 3 on the lower scale).

Like Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers are a fertile ground for creativity and exploration. They also have interesting applications to combinatorics [1], especially to the study of lattice paths [2, 3], as we will see shortly.

Pell numbers Pn and Pell-Lucas numbers Qn are often defined recursively [4, 5]:

where n ≥ 3. They can also be defined by Binet-like formulas:

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).

In an earlier note by W. J. A. Colman [1], the reader was treated to an interesting discussion concerning a BBC programme about the earth's radius in the context of endeavours to determine it by an 11th century Persian mathematician, Al-Biruni. At the same time a modem but approximate formula that might not have been available to the ancients was proposed for the radius and, following that account, it is the purpose here to explore just how accurately one might determine the earth's radius using this formula together with a simple measurement apparatus (not a sophisticated astrolabe) that might be constructed from materials found in the garage of any DIY handyman, or indeed, in the laboratory of any school science department. A schematic for the configuration of an observer P at a height h above the earth (of radius R) where the deflection angle to the horizon is denoted by θ, is shown in Figure 1. Also shown in the figure is a ‘sighting tube’ of length L about which more will be said later.

The motivation to write this paper arose out of the following problem which was posed in a recent mathematical olympiad:

Given a polynomial P(X) with integer coefficients, show that there exist non-zero polynomials Q(X), R(X) with integer coefficients such that P(X)Q(X) is a polynomial in X2 and P(X)R(X) is a polynomial in X3.

For instance, if P(X) = 2 − 5X + 3X2 + 12X3, then we notice that Q (X) = 2 + 5X + 3X2 − 12X3 serves the purpose for the first part, viz.

A moment's thought makes it fairly evident that this trick easily solves the first part of the problem for a general polynomial P(X).

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.

For any polyhedron (3-polytope) P in ordinary three-dimensional space let S(P) denote its surface, that is, the union of its (closed) 2-faces. If P1 and P2 are given polyhedra, they are said to be clones of each other if there exists an isometry which maps S(P1) onto S(P2). In practical terms, this means that if one makes a paper model of one of the polyhedra, say P1 then, ignoring the ‘creases’ corresponding to the edges of P1, S(P1) can be changed into S(P2) without any stretching or cutting.

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.

Archimedes developed a method for calculating the area of a segment of a parabola. This result was outstanding in mathematical antiquity. The idea and the method were not developed further until nearly 1800 years later. In our article his method is presented in modem notation using modem concepts. We show that the method generalises to curves of higher order. Many of Archimedes' arguments can be used without any adjustments and thus show the strength of his original approach.

The mathematician from Syracuse, Archimedes (born about 287 BC, died 212 BC), developed a method for the calculation of the area of a segment of the parabola.

For a fixed positive integer m, factorial m is defined by

The problem of finding a formula extending the factorial m! to positive real values of m was posed by D. Bernoulli and C. Goldbach and solved by Euler. In his letter of 13 October 1729 to Goldbach [1], Euler defined a function (which we denote as Γ (x + 1)) by means of

and showed that Γ (m + 1) = m! for positive integers m. After that, Euler found representations for the so-called gamma function (1) in terms of either an infinite product or an improper integral. We refer the reader to the classical (and short) treatise [2] for a brief introduction and main properties of the gamma function.