On December 25th, 2021, NASA, in collaboration with the European and Canadian space agencies, achieved a remarkable feat by successfully launching the James Webb Space Telescope (JWST), marking it as the largest and most powerful telescope ever constructed.* The JWST, a monumental collaborative project spanning over two decades, involved scientists and engineers from 300 universities and 14 countries, with a total investment of ten billion dollars.

In 1742 the Swiss mathematician Gabriel Cramer gave the following seemingly innocuous problem to the young Italian mathematician, Jean de Castillon:

The Cramer-Castillon problem: Given a circle and three points A, B and C, inscribe in the circle a triangle DEF whose sides (extended) pass through each of these points.

The goal of this paper is to derive a series expansion for standard normal probabilities F(b), where b ≥ 0 and the upper half of the cumulative distribution function (or CDF) of the standard normal is given by(1) See [1, Chapter 4] for information and results about the normal distribution. It is known that this integral is not elementary, and the usual approach is to use the Taylor series expansion for ez in order to evaluate the integral. We know that and therefore that (2).

In the long history of the study of the refraction of light, Willebrord Snell is credited with being the first to show that the ratio of the sines of the angles of incidence and reflection is constant for any given pair of media.

In [1], the authors presented a class of sequences generated by powers of the kth-order Fibonacci recurrence relation and showed that the sequences converge. Precisely, the following result was shown:

Suppose that k ∈ Z+ and 0 < p < 1. Let x1, x2, …, xk be positive numbers. For n > k, let (1) Then {xn} converges to .

Here we are interested in the familiar identity but only in as far as it provides the solution (1, 2, …, n) of the Diophantine equation (1) where n and the aj are positive integers. In fact, not only is (1, 2, …, n) a solution of (1), it has the remarkable property that, up to a permutation of its entries, it is the only solution of (1) in which the aj are distinct (see Section 4). Thus any other solution will necessarily contain repetitions among the aj and, for this reason, a solution must be defined to be a sequence (which allows repetitions) rather than a set. However, we shall not distinguish between two solutions whose entries are permutations of each other.

In [1], the following problem appears on page 205.

12. A right-angle tool can be used to draw a line through two points and erect a perpendicular to a given line at a given point. Use this tool to drop a perpendicular from a given point to a given line.

In geometry, we sometimes prove that a certain configuration Q has property P. If it turns out that Q is the only configuration that has property P, then we say that P is a characterisation of Q. For example, the centre of a circle is a point in the plane which is equidistant from every point on the circumference of the circle. In fact, the centre is the only point in the plane which is equidistant from every point on the circumference of a given circle. Thus we say that the centre of a circle is characterised by this property.

The all-real-roots configurations of the univariant cubic equation allow an easy visualisation of the roots as the orthogonal projection onto the real axis of the ends of three vectors of equal magnitude, radiating symmetrically in the XY plane from the origin of the reduced form [1].

The Routh Array is known to be a very convenient way to determine the number of polynomial roots that lie in the right half of the complex plane using only the polynomial coefficients. Extensive use of this is made particularly in control system stability studies [1]. We now show that the same real-coefficient Routh algorithm can be used to obtain this information for polynomials with complex coefficients, which are prominent in areas of engineering such as electrical inductance theory [2]. The method is easy to apply and is seen to be more straightforward than existing methods for this task [e.g. 3,4]. If required, it also enables the use of appropriate root-finding methods for real-coefficient polynomials to obtain the roots of complex- coefficient polynomials.

The recent discovery of the aperiodic monotiles known as the hat and the spectre has sparked a surge of interest by the general public in exactly what it means to tile the Euclidean plane in an aperiodic fashion. Explaining a mathematical breakthrough to those without background knowledge can be difficult. However, in this case, communicating the ideas has been facilitated by the highly visual nature of what has been found, much to the delight of the editors of national newspapers and magazines, and also those who love to blog on social media. News of the first tile, the hat, broke in March 2023, with a second tile, the spectre, announced two months later. A festival, hatfest, took place a couple of months after that; the two day event featured lectures for established and aspiring mathematicians, a display of associated art, and an open-to-the-public evening lecture, all at the Mathematical Institute at the University of Oxford. Figure 1 shows the two tiles at the centre of the festivities. To tile the plane aperiodically the hat requires the use of its mirror tile but the spectre, with a minor modification to its edges, is the first true monotile; so adjusted, it, like the hat, can tile the plane aperiodically and not periodically, but without the need for its congruent mirror.

There has long been a recognition of the role history can play in supplementing the mathematics curriculum, perhaps to make the mathematics “easier, clearer, or more interesting” [1], though not all view this positively [2]. Beyond this, history of mathematics is often taught in its own right at university.

The Chebyshev polynomials: some basic facts

The Chebyshev polynomials Tn, named after the Russian mathematician P. L. Chebyshev (1821-1894), are defined by(1)

$${\text{cos }}nt = {T_n}({\text{cos }}t).$$

When limnologists (scientists who study lakes) wish to find the volume of a lake they find depth contours, where the depth of the lake is the same at any two points on the given contour. Let A (x) be the area contained in the contour at depth x. The function A (x) is assumed to be strictly decreasing with A (D) = 0 where D is the maximum depth of the lake. The standard formula for the volume that they use is

$$V \approx \sum\limits_{k = 1}^n {\frac{{{A_{k - 1}} + {A_k} + \sqrt {{A_k}_{ - 1}{A_k}} }}{2}\Delta x,{\text{ where }}{A_k}{\text{ = }}} A\left( {\frac{{kD}}{n}} \right)$$

over n + 1 equally-spaced depth contours with $\Delta x{\text{=}}\frac{D}{n}{\text{.}}$ This approach is very well analysed in [1].

I was a member of the team Death and Mayhem, which won the 2017 MIT Mystery Hunt. My team's reward was to write the 2018 MIT Mystery Hunt. By the way, some people consider it a punishment rather than a reward. As one colleague said to me on our win, ‘Congratudolences!’

Consider a complex polynomial $$f(z) = \sum\nolimits_{r = 0}^n {{a_r}{z^r}} $$. Define

$$M(f) = {\rm sup}\left\{ {\left| {f(z)} \right|:\left| z \right| \le 1} \right\}.$$