Consider two related tasks:

A) The diagram below displays a fragment of a pattern known to extend indefinitely and to be generated by an endlessly repeating pattern. What kind of shading will the 137th cell have? What number cell will the 137th white cell be in?

B) What is the next term in the sequence 3, 6, 11, 18, 27, …?

Tasks like the first can provide excellent experience of detecting and expressing generality as one route to algebra, but only if learners are first asked to describe a structural relationship which generates the succeeding items. Tasks like the second feature in intelligence tests, but it is well known that such tasks are mathematically incomplete.

It is an elementary fact that a particle sliding on a rough horizontal table experiences a deceleration of magnitude μg, where μ is the coefficient of friction between the particle and the table. Thus, if the initial speed u of the particle and the braking distance d are both known, then μ can be determined by the formula

The idea of using braking distances to determine the coefficient of friction led the second author to ask the following question:

A particle slides on the interior surface of a rough hemispherical bowl starting from rest. If the starting point of the particle and the point at which it comes to instantaneous rest after its slide are both known, is it possible to determine the coefficient of friction between the particle and the bowl?

Every tetrahedron has a set of quadritangent spheres, that is, spheres that touch the four planes containing its faces. In this article I discuss three questions about them.

1. How many quadritangent spheres does a given tetrahedron have?

2. How can we find their sizes and positions with as little calculation as possible?

3. What relations exist between the vertices and points of contact that lie on anyone of the planes?

Many students can get a lot of answers in A-level statistics right. But some of the ideas behind the calculations are somewhat subtle. Some of the subtleties are occasionally brought out in examination questions, usually those requiring verbal or discursive answers; unfortunately answers to these questions tend to be poor. Such questions are often dismissed as ‘waffle’ by strong students, while weaker ones generally try to avoid issues of understanding by memorising what they hope will be relevant slogans. Even respected textbooks are open to criticism. This is all very sad as I think it is here that much of the interest of the subject lies; ignoring the subtleties risks reducing the subject to a series of meaningless formulae. There should be time to focus on what is actually happening, as well as on mere technique. This article looks at some common problem areas.

In the autumn of 2011 the movie Sherlock Holmes: A Game of Shadows appeared (see [1]). It was a sequel to the Holmes movie from two years earlier. Unlike the first movie, A Game of Shadows Holmes' arch enemy Professor Moriarty featured more prominently than Holmes himself.

One scene, in particular, has Holmes sizing up his opponent by making a visit to Moriarty at his university office. But in the background sits Moriarty's blackboard, filled completely with mathematical equations, symbols, drawings, and formulas. One of these deserves our special attention.

Recall that for integers n ≥ 2, ζ (n) is defined by

Of course, ζ (1) is not defined, since is divergent. A well-known particular value is ζ (2) = π2/6: numerous alternative proofs of this fact have been presented in the Gazette, e.g. the recent notes [1], [2].

The following three inequalities hold:

The first inequality is trivial. The second one was proved without calculating aids in note [1], and the third along similar lines in note [2]. The author of note [2] also suggested an extension to the relation between

How best to continue the sequence of inequalities is not obvious and we return to that point shortly. Before doing so, we note that an interesting geralisation is to replace π by a variable x, and to determine the precise interval of x in which the regularity of the ‘alternation of inequality signs’ is maintained. We need no longer consider particular properties of π.

In the course of a coffee-table conversation with my friends regarding the nature of static equilibrium of different solid objects the situation involving a uniform hemisphere came up. Intuition (and perhaps experience) tells that a uniform hemisphere as shown in Figure 1 resting on a flat surface will be at stable equilibrium, and so will an oblate hemispheroid as shown in Figure 2. Things get complicated when we move to a prolate hemispheroid like the one shown in Figure 3, for the nature of its equilibrium is less obvious. The intuition does come to mind though that if the prolate hemispheroid is made indefinitely taller, keeping its equatorial radius fixed, then the equilibrium should eventually become unstable. Intrigued, we decided to probe into the matter quantitatively.

Given two triangles ΔABC and ΔDEF, if each side of ΔDEF contains a vertex of ΔABC, then we call ΔDEF an outscribed triangle of ΔABC. Given ΔABC, let ΦΔABC be the set of all outscribed equilateral triangles of ΔABC. Clearly ΦΔABC is non-empty. In the following we will determine the area of the largest member of ΦΔABC when each angle of ΔABC is smaller than 120° and show that this largest member can be constructed by ruler and compass from ΔABC. The corresponding problem on quadrilaterals has been considered in [1].

Given positive numbers a > b, consider the ‘agm iteration’ given by a0 = a, b0 = b and

At each stage, the two new numbers are the arithmetic and geometric means of the previous two. It is easily seen that bn < an, (an) is decreasing, (bn) is increasing and an + 1 − bn + 1 < ½(an − bn), and hence that (an) and (bn) converge to a common limit, which is called the arithmetic-geometric mean of a and b. We will denote it by M (a, b).

We start from two simple identities:

For any p > 0 and 0 ≥ b ≥ a, now let

Can we formulate statements about Gp(a, b) that generalise (1) and (2)? We cannot hope for equalities, but perhaps we can establish inequalities which somehow reproduce (1) when p = 1 and (2) when p = 2. For (1), this might mean an inequality of the form Apap ≤ Gp (a, b) ≤ Bpap for certain constants Ap and Bp, and for (2) a similar statement with ap replaced by ap + bp. However, these are not the only possibilities, as we shall see.