In my professional life, I have repeatedly had cause to return to the question of

It's time I began to frame some responses! In doing so, I am very aware that membership of The Mathematical Association is broad church, ranging from those with an amateur interest in mathematics (and mathematics education), through non-specialist teachers, to professional mathematicians. As such it is a rich community, but one in which it is challenging to address such questions in a way which is meaningful to all. However, I shall attempt to do so and I hope that the balance will in a small way also prove illuminating to the reader. In particular, I have a hope that it will point to the breadth and depth of what we expect from our teachers of mathematics, at all levels, and to a valuing of that.

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relation

generates a cycle of period 5 for almost all values of u1 and u2 [1].

The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is then

Lyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.

No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].

This question can be written in the form of a game: you win if a triangle can be formed from the three numbers interpreted as side lengths, you lose if they do not.

We let ℕn = {1, 2, 3, …, n}, and see there are two variants of the game:

• • From ℕn draw at random three numbers sequentially. At each stage do not replace the drawn number.

• • From ℕn a number is drawn at random and is recorded. In three stages three numbers are drawn. Replace the number at each stage.

In this paper we are going to generalise some standard results about the ellipses of least or greatest area, respectively circumscribing or inscribed in a triangle, to the corresponding ellipses for a convex quadrilateral.

Given ΔABC let D, E, F be the midpoints of BC, CA, AB respectively; then the medians AD, BE, CF concur at the centroid G of ΔABC. There is an ellipse with centre G, touching the sides of ΔABC at their respective midpoints. It is known as the midpoint ellipse or Steiner inellipse of ΔABC, and it is the ellipse of greatest area inscribed in ΔABC. Since AG : GD = 2 : 1, and similarly for the other medians, a central dilation (an enlargement) with centre G and scale factor –2 (or, equivalently, 2) applied to the Steiner inellipse produces a similar ellipse, also with centre G, and passing through A, B and C. This is the Steiner circumellipse of ΔABC, and it is the ellipse of least area circumscribing ΔABC. See Figure 1. For more detail, see [1, 2] and the references given there.

We will prove a generalisation to our Theorem 10 about duality between orthodiagonal quadrilaterals and equidiagonal quadrilaterals in [1, p. 134]. These are quadrilaterals with perpendicular diagonals and diagonals of equal lengths respectively. Unfortunately we made a careless mistake in the proof of Theorem 10 (ii) in [1], which was found by Zoltán Szilasi at the University of Debrecen in Hungary. When reviewing that theorem, we realised that it's just a special case of a more general theorem. The equilateral triangles in Theorem 10 can be replaced by almost any regular polygons.

The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?

We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties of convex bisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.

We can use a spirograph in different ways to draw curves. Do these different modes of drawing necessarily yield different curves, or is it also possible to get the same curve by using two different modes of drawing? We address this question geometrically, not using the parametric equations of the curves.

Tangential quadrilaterals are defined to be quadrilaterals in which a circle can be inscribed that is tangent to all four sides. It is well known and easy to prove that a convex quadrilateral is tangential if, and only if, the angle bisectors of all four vertex angles are concurrent at a point, which is the centre of the inscribed circle (incircle). The most well-known and in problem solving useful characterisation of tangential quadrilaterals is Pitot's theorem, which states that a convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d satisfy the relation a + c = b + d [1, pp. 64-67]. If you want to have more background information about characterisations of tangential quadrilaterals, then we recommend you to check out the lovely papers [2, 3, 4], as well as our previous contributions on the subject [5, 6, 7]. These six papers together include about 30 characterisations that are either proved or reviewed there with references.

Optimisation problems are among the most practical applications of calculus to everyday life, and a survey of exercises in various calculus textbooks will provide a teacher with many interesting scenarios for framing intriguing questions on this topic. Whether it is finding a container's dimensions that yield the least surface area for a given volume, or finding that ideal movie ticket price which will maximise a theatre's revenue, students can usually relate to these problems. Pólya in his book Plausible reasoning makes the following remarks about the attraction of extrema problems:

Catalan's constant, named after E. C. Catalan (1814-1894) and usually denoted by G, is defined by

It is, of course, a close relative of

The numerical value is G ≈ 0.9159656. It is not known whether G is irrational: this remains a stubbornly unsolved problem. The best hope for a solution might appear to be the method of Beukers [1] to prove the irrationality of ζ (2) directly from the series, but it is not clear how to adapt this method to G.

In an earlier issue of the Mathematical Gazette, Nick Lord established the familiar formula for the sum of the first n squares using a physical argument based on the centroid of a configuration of masses in the plane [1]. In [2] we demonstrate an alternative configuration that gives the same result. This article is a follow-up to these papers, in which we find physical derivations of the formula for

for each k ∈ {1, 2, 3, 4, 5}. Let us first summarise the required theory. Take any region X ⊆ ℝ2 with uniform density and total area A. The centroid of is the arithmetic mean position of all of the points in X. If a region has a line of symmetry then the centroid will be located on that line.

In 1979, Douglas Hofstadter published a Pulitzer prizewinning classic, Gödel, Escher, Bach: an eternal golden braid [1], henceforward referred to as GEB. I have presented a course at La Salle University based on GEB eleven times during the last twenty-one years. From these experiences, I present below some useful strategies for instructors who are interested in teaching a course where at least half of GEB is covered in some depth.

As usual, write

In [1], it is shown, by an elegant method, that for all n ⩾ 2, Hn is not an integer. The method can be traced back at least to [2, Exercise 251, p. 159]. Actually, rather more is shown: if Hn is expressed as a fraction an/bn in its lowest terms, then for all n ⩾ 2, bn is even. One way of looking at this is to say that once the term has entered the sum, the factor 2 persists in the denominator from then on. This suggests the following question:

(Q1) for primes p ⩾ 3, is bn a multiple of p for all n ⩾ p?

and more generally:

(Q2) for any prime p, is bn a multiple of pm for all n ⩾ pm?

In other words, once 1/pm has entered the sum, does the factor pm persist in the denominator?

Let dn denote the lowest common multiple of 1, 2, … , n. This number can be described as follows: for each prime p ⩽ n, let mp be the largest integer m such that pm ⩽ n. Then . Hence a positive answer to (Q2) would imply that bn simply equals dn.

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

We consider here the positive integers with respect to their unique decimal expansions, where each n ∈ ℕ is given by for some non-negative integer k and digit sequence αkαk-1 … α0. With slight abuse of notation, we also use n to denote αkαk-1 … α0. For such sequences of digits (as well as for the numbers represented by the corresponding expansions) we write x ⊲ y if x is a subsequence of y, which means that either x = y or x can be obtained from y by deleting some digits of y. For example, 514 ⊲ 352148. The main problem is as follows: Given a set S ⊂ ℕ, find the smallest possible set M ⊂ S such that for all s ∈ S there exists m ∈ M with m ⊲ s.

A continued fraction is an expression of the form

The expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

There have been numerous attempts to model the governing dynamics between the two ostensibly competing concepts of brilliance and mechanical steadiness. One interesting study is given by the English mathematician G. H. Hardy in his model [1] describing two characteristically different golfers playing a match against each other. The model challenges the apparently accepted doctrine of the ‘Brilliant player’ having the advantage over the ‘Steady player’ in a long series of golf matches by holes. Hardy defines ‘brilliance’ as the capacity to produce ingenious results as well as the capacity to make mistakes, compared to ‘steadiness’ being completely mechanical producing the same average result all the time. The two players in his model are equal in performance on average, only the brilliant player has a higher standard deviation whereas the consistent player has a standard deviation of 0.

Infinite series are an important topic in mathematical analysis and convergence is the most crucial concept in the theory of infinite series. The speed at which the sequence of the partial sums of a series approaches its limiting sum has been a subject of investigation for many a mathematician. Euler [1], Kummer [2] and Markoff [3] all developed techniques for accelerating the convergence of slowly converging series. Euler's method only works for alternating series, Kummer's and Markoff's are suitable for general series with a special form.

Gosper [4] illustrates how the rate of convergence of infinite series can be accelerated by a suitable splitting of each term into two parts and then combining the second part of the n th term with the first part of n + 1 th the term and leaving the first part of the first term. Repeated application of this process yields a new series which approaches 0 and the series of the left out first parts (‘orphans’) that converges faster than the original series.