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.

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].

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.

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.

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.

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.

In [1], the following theorem is stated and proved as Proposition XXXII, using the Principle of Inclusion and Exclusion:

The number of ways of deranging a series of n terms so that no term may be followed by the term which originally followed it is

(Note that the conventional definition of a derangement of a series of terms is an arrangement such that no term occupies its original position. However, the usage in this paper allows some terms to remain in their original positions.) This may most easily be visualised by considering a string of n distinct characters, abc … n say. This string has a sequence of n − 1 pairs of consecutive characters, namely, ab, bc, … mn. This article derives three theorems that can be deduced from the above theorem. The first theorem determines the number of permutations of a character string that are free from a random subset comprising a given number m of the n − 1 pairs, but contain all the remaining pairs. The second theorem sums the results of the first theorem for m, m + 1, …, n − 1 pairs.

“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1]

Vincenzo Viviani, a 17th century mathematician, proved that the sum of the (perpendicular) distances from a point within an equilateral triangle to its sides is constant as shown in Figure 1. The theorem, named after him, generalises to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel. Viviani's theorem is easily proved by summing the areas of triangles APB, BPC and CPA, equating to the area of the triangle ABC, and then simplifying to obtain h1 + h2 + h3 = H, where H is the altitude of ABC.

Suppose n (fair) dice each having m faces marked with numbers 1 to m, are thrown at random. The problem of determining the number of ways in which the sum of the numbers exhibited by the dice will be equal to a given number k has a very long history. In particular, the three dice problem (i.e. the case where m = 6, n = 3), goes back to the 13th century (see [1]). Later, it was Cardano and Galileo who solved it (see [2, 3]). The general case was stated without proof in A. de Moivre's first work on probability, De Mensura Sortis, (1712, p.220). De Moivre was the first who published a proof (using generating functions) in Miscellanea Analytica (1730, p. 196). Furthermore, Montmort had also the solution (via inclusion-exclusion) by 1713, and independently of De Moivre (see [4, 2]).

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. Thus

This constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].

Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.

For non-negative integers n, the recurrence relation defined by

with initial conditions

can be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

When I was teaching calculus recently, an exercise in the textbook [1, p. 687, qn. 3] asked students to find the Taylor series of cosx centred at π/4. My students were able to find the series

but wondered how to take care of the sign changes, in order to express the series in I.-notation. They were quite puzzled after finding that the answer given in [I] is

The exponent n (n + 1)/2 seemed a rabbit out of the hat! This article is the result of trying to answer their question satisfactorily, and to have some serious fun. Readers are invited to try two teasers first before reading for their solutions.

Let T be a triangle and let A be the area of T. Given a point p inside the triangle and a number r that satisfies 0 < r ≤ ½, we seek to count the number N (p, r) of straight lines that pass through p and cut T into two pieces so that one piece has area rA. As the point p and the ratio r vary, the value of N (p, r) ranges from 0 to 6. Given a ratio r, we want to determine the regions within T for which the function N (p, r) assumes various integer values. For example, a region inside T for which N (p, r) = 6 only exists for values of r between and ½, and we would like to quantify the area of this region (as a fraction of the total area of the triangle) as r varies.