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.

According to Mitchelmore [1], generalisations are the cornerstone of school mathematics, covering various aspects like numerical generalisation in algebra, spatial generalisation in geometry and measurement, as well as logical generalisations in diverse contexts. The process of generalising lies at the heart of mathematical activity, serving as the fundamental method for constructing new knowledge [2, 3]. In this paper we will generalise an interesting geometry problem that appeared in the 1995 edition of the International Mathematical Olympiad (IMO) [4].

Let ABC be a triangle with sides a, b, c, semiperimeter s, circumradius R, inradius r and area Δ. We introduce

$$Q\, = \,{(a - b)^2}\, + \,{(b - c)^2}\, + \,{(c - a)^2}$$

$$M\, = \,{(\left| {a - b} \right|\, + \left| {b - c} \right|\, + \,\left| {c - a} \right|)^2}.$$

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.

At university I was intrigued by a historian friend and his constant quest for gobbets – short, apposite quotations that he avidly collected for possible use in future essays. It strikes me that there might be analogous nuggets of mathematics and I offer the following five short, self-contained items both for amusement and for possible classroom use. Each contains a surprising element and I have deliberately left a few loose ends to encourage further exploration.

The Chebyshev polynomials: some basic facts

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

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

It is a well known and easily verifiable fact that not all integer triangles have integer areas. Consider the triangles with sides {9, 10, 17}, {13, 14, 15}, {5, 7, 8} and {6, 7, 9} with respective areas 36, 84, and . The first two, whose areas are integers, are called Heronian triangles. The second triangle also has the additional property that its sides are consecutive integers and is an example of a Brahmagupta triangle, named after an Indian mathematician, born in AD 598. These are called Super-Heronian triangles in [1] and a method is developed there for generating examples of such triangles.

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].

Self-working card tricks have been a staple for budding magicians and entire books have been written about them, e.g. Fulves [1]. Lately selfworking card tricks have become common fodder for social media. In this note, we generalise two such card tricks (from [2] and [3]) that are based on the Australian shuffle.

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!’

The centre of mass of a plane, uniform, lamina in the shape of a regular n-gon coincides with the centre of mass of n unit masses, one at each of its vertices. As many n-gons fail to have this property it is remarkable that all triangular laminas, whether regular (i.e. equilateral) or not, do have this property. Here we discuss this difference and we assume throughout that each lamina has unit density (so, disregarding units, its mass is the same as its area). We shall refer to the centre of mass of the lamina as its centroid, and the centre of mass of unit point masses, one at each vertex, as its vertex-centroid. Also, we shall say that a polgonal lamina is vertex-equivalent if its centroid coincides with its vertex-centroid, i.e. if, and only if, .

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

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