Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.

This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:

(a)

(b) , also for complex values of x;

(c) x → exp (x) is the unique solution to the initial value problem [4]

(d) x → exp (x) is the inverse of

(e)x → exp (x) is the unique continuous function satisfying the

functional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7];

(f) Define dr for rational r, and then use a continuity/density argument [8].

All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.

Two friends, David and Hamish, were discussing on the telephone whether they should play a forthcoming croquet match at David's club or at Hamish's. Hamish said that he would toss a coin and David could call. David pointed out that this required a substantial degree of mutual trust, so Hamish suggested how they might use the published results of the following night's National Lottery draw. Hamish's proposal was, in fact, unfair and this article shows why and proposes a satisfactory alternative. There is no suggestion that the alternative proposed is a sensible way to solve David and Hamish's problem, but it does take us on a route that reinforces some important lessons. In particular it illustrates some of the advantages and disadvantages of the use of simulation, a technique widely used in more difficult problems. It also emphasises once more the importance in probability of not taking claims at face value.

We consider the packing of rectangles of dimension m x (m + 1) — where m is a natural number — into a larger rectangle. More specifically, we consider the following problem: What is the smallest area of a rectangle into which rectangles of dimensions 1 x 2, 2 x 3, 3 x 4,…, n x (n + 1) will fit without overlap?

Unlike the corresponding problem for squares of areas 12, 22, 32, …, n2(see [1]), where there is no known non-trivial example of an exact fit into a rectangle, in many cases we can achieve an exact fit for our set of m x (m + 1) rectangles. Intuitively, this is because each m x (m + 1) rectangle has two possible orientations, which considerably increases the chances of an exact fit. As in [1], we make the (possibly unnecessary) assumption that the sides of each m x (m + 1) rectangle are parallel to the sides of the bounding rectangle, whose dimensions are integral. For any given n, we consider two solutions to our problem to be distinct only if the bounding rectangles have different dimensions (but equal area).

In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.

This article describes an investigation into Kiepert lines, and leads to some surprising and little-known relationships between the Fermat, Napoleon and Vecten points of a triangle.

If we draw similar isosceles triangles A'BC, B'CA and C'AB outwards on the sides of a given scalene triangle ABC as in Figure 1, Kiepert's theorem tells us that the lines A'A, B'B and C'C meet in a single point - a Kiepert point [1, Chapter 11]. Since its position depends on the common base angles θ of the isosceles triangles, I label it K(θ), taking θ as the parameter of this point.

We live in a golden age regarding the opportunities to explore Euclidean geometry. The access to dynamic geometry computer programs for everyone has made it very easy to study complex configurations in a way that has never been possible before. This is especially apparent in the study of triangle geometry, where the flow of new problems, properties, and papers is probably the highest it has ever been in the history of mathematics. Even though it has increased a bit in recent years, the interest in quadrilateral geometry is significantly lower. Why are triangles so much more popular than quadrilaterals? In fact, we think it would be more logical if the situation were reversed, since there are so many classes of quadrilaterals to explore. This is the primary reason we think that quadrilaterals are a lot more interesting to study than triangles.

What is the likelihood P(n, T) that at least two people in a gathering of n people are born within a given time T of each other? In particular, for T = 24 hours what is the smallest n for which P(n, T) is at least 50%? The well-known classic version of the birthday problem asks for the smallest number of people needed to give a better than 50% chance of at least one birthday match with the assumptions that the birthdays are independently selected from a discrete uniform distribution over 365 days. Using the calculus tool of the limit, we refine two characterisations for P(n, T) and show that they give consistent results with each other and with the classic birthday problem as well. At first thought P(n, 24 hours) should answer the classic birthday question. Yet consider this experiment: suppose Andrea was born at noon on June 1, then potential matching birth times (for the other n - 1 people) with her birth time extend from noon on May 31 to noon on June 2, a period of 48 hours! What this period suggests is that P(n, 24hours) exceeds the classic answer. But, by how much? Read on.

We shall use the word ‘polyhedron’ to mean a connected, simply-connected 3-polytope of positive (non-zero) volume. The idea of the net of a polyhedron P is well known. For example, the regular octahedron has eleven distinct nets, three of which are shown in Figure 1. A net consists of three parts:

• (a) A plane connected and simply-connected polygon Q (denoted by heavy lines in the diagrams), known as the carrier of the net;

  (b) A set of lines known as, fold-lines in the interior of Q;

  (c) A labelling of the edges of Q.

  If one cuts Q out of paper or similar material, folds it along the fold-lines, and then pastes together edges with matching labels, one obtains a model of the polyhedron P. We say that a net is convex if, and only if, its carrier Q is convex.