My Mathematical Association Presidential Address focused on maths education in the context of the current environment in England. However, many of the issues are relevant to other UK nations and maths education more widely.

Close friends Rebecca and Aisha are taking an integral calculus class. With their permission, we listen in on a conversation that they are having about an integration problem.

Well-known shuffles, such as the riffle shuffle and the standard over- hand shuffle [1], are meant to randomise the order of the cards, making it difficult to predict the appearance of specific cards in games like poker or blackjack. However, shuffles have a different purpose in self-working card tricks: they reorder the cards so that the position of the target card (the card revealed at the end of the trick) is known to the magician, but not to the observer. This Article generalises three self-working card tricks that are based on the milk and Monge shuffles and have appeared on social media.

In Euclidean geometry, a tangential polygon or a circumgon is a polygon that contains an inscribed circle which is also called an incircle and the radius of its incircle is called the inradius. A tangential polygon may be concave or convex. If it is concave, it is the extension of some of its sides that is tangent to the incircle. In this Article we restrict ourselves to convex tangential polygons, so that their corresponding incircles are tangent to each of its sides. Every regular polygon is a circumgon and there are also irregular polygons that are circumgons but not all irregular polygons are circumgons. In what follows, we shall use circumgon or tangential polygon interchangeably according to the situation.

Mathematics and physics are fascinatingly connected with each other. Since [1] first pointed out the relationship between the number π and the total number of collisions between two blocks and a wall, it has generated wide interest among the general public [2, 3, 4, 5]. A carefully made video presents an elegant and crystal clear geometric explanation [6], which is followed by a slightly more advanced explanation based on linear algebra, matrix formalism in particular [7]. References [2, 3, 6] attribute the geometric approach to different individuals, but the relationship actually appeared much earlier in a physics journal which focused on the number of collisions rather than its connection with the number π [8].

Figure 1 shows an arbitrary triangle Z1Z2Z3, the base triangle, with similar isosceles triangles Z1Z2V1, Z2Z3V2 and Z3Z1V3 drawn outwards on the respective sides Z1Z2, Z2Z3 and Z3Z1, all with base angles (the angles at Z1, Z2 and Z3) equal to α. We call ΔV1V2V3 the outer α-triangle of ΔZ1Z2Z3. Note that, if ΔZ1Z2Z3 is positively oriented (that is, if the labels go around the triangle in anticlockwise order), then so is ΔV1V2V3, but the triangles Z1Z2V1, Z2Z3V2 and Z3Z1V3 are negatively oriented (that is, the labels go around the triangle in clockwise order).

In a previous article [1] I showed that the probability of a difference of t units arising when n numbers from a uniform distribution are rounded before, as opposed to after, they are added is(1)Here n is a non-negative integer and t is a positive or negative integer with, the maximum possible error.

This Article is an exposition of two very old theorems. One was first proved by A. A. Markov in 1889 [1], the other by S. N. Bernstein in 1912 [2]. Both Markov (1856–1922) and Bernstein (1880–1968) were authors of numerous pioneering works: there are many other results designated Markov’s theorem or Bernstein’s theorem (or Bernstein’s inequality) in their respective subject areas. “Markov chains” have become a standard concept in probability theory, and “Bernstein polynomials” a standard one in approximation theory.

I show that a neat expression emerges for the number of spheres in each shell of an octahedron composed of close packed spheres, and that from this expression we can compute the value of the packing density for an infinite array of such spheres.

Baillie’s identity is (see [1] or [2], p. 240). Its integral analogue, , is not difficult to prove (see Lemma 1, below). In this Article, we prove a generalisation of the latter result (see Theorem 1). Theorems 2 and 3 are extensions, involving in addition, powers of cosines in the integrand. Theorem 4 answers a question raised after the proof of Theorem 1, and Theorem 5 collects together the preceding results in the form of three identities between trigonometric integrals. Theorem 6 gives a further generalisation.

A Rascal triangle is a Pascal-type numeric triangle developed in 2010 by three middle-school students, Alif Anggoro, Eddy Liu and Angus Tulloch [1]. During their mathematics classes they were challenged to provide the next row for the following sequence:The expected answer was the one that matches Pascal’s triangle, i.e. 1 4 6 4 1, obtained by applying the recurrence rule of binomial coefficients South = East + West. Instead, the young students suggested that the next row should be 1 4 5 4 1. In contrast to Pascal’s triangle rule South = East + West, they produced the new row using the relation they called the diamond formulaBy applying the diamond formula (1), the students produced an original triangular sequence known as the Rascal triangle, see Table 1.

Let ABC be a triangle with incentre I, circumcentre O, orthocentre H, centroid G and symmedian point K. In standard notation, the triangle ABC has sides a, b, c, semiperimeter s, circumradius R and inradius r. Euler’s well-known result that the incentre I is always within the orthocentroidal disc DGH, the disc with diameter GH, is probably the first result about the location of the incentre in some disc formed by triangle centres. Investigating the location of the incentre I in other discs, in [1] we proved that the incentre is interior to the Brocard disc DOK, that is, the disc with diameter OK. The disc is named after the French military meteorologist and geometer Henri Brocard (1845-1922), known in triangle geometry for the Brocard points and the Brocard angle (see [2]).

In this Article the author hopes to present an unusual view of the Riemann zeta landscape when Re (s) > 0, to show how the prime numbers are discoverable by the altered configuration, and to show a correlation amongst the primes, the regular polygons and a set of complex numbers with real part${1 \over 2}$.

For convenience, throughout this Article, we shall use the standard notations for any triangle ABC, where:

1. 1) the lengths of the sides BC, CA, AB are denoted by a, b, c, respectively,

2. 2) the area of the triangle is denoted by S,

3. 3) the semiperimeter of the triangle is denoted by s,

4. 4) the circumradius and inradius of the triangle are denoted by R and r, respectively,

5. 5) the lengths of the medians corresponding to vertices A, B, C are denoted by ma, mb, mc, respectively,

6. 6) the lengths of the angle bisectors corresponding to vertices A, B, C are denoted by la, lb, lc, respectively,

7. 7) the lengths of the altitudes corresponding to vertices A, B, C are denoted by ha, hb, hc, respectively.