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 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.

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 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.

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.

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.

TantrixTM tiles are black hexagons imprinted with three coloured paths [1] joining pairs of edges. There are three different kinds of path. One is a straight line going from an edge to the opposite edge, one a circular arc joining adjacent edges and one an arc of larger radius joining alternate edges (or two apart). Tiles can be rotated but, since they are opaque, they cannot be turned over. A careful enumeration would indicate that, identifying tiles under rotation but not under reflection, there are 16 such tiles. However, the two tiles consisting of three straight lines (meeting at the centre of the hexagon) are not part of the set, so actually there are only 14 different tiles. The game is played by matching tiles to connect paths of the same colour; the goal is to create loops or long paths of a single colour This easy to learn yet hard to master game has inspired research on strategy (e.g. [2]) and complexity (e.g. [3]).