The following equations relate y only implicitly to x:(1)(2) In both equations, y is a function of x for a continuous range of (x, y) values in the real x-y plane. (1) represents an ellipse. (2) has been designed by the author to have a solution in the real x-y plane at (−1, 2), and because the function on the left-hand side of (2) meets certain conditions regarding continuity and partial differentiability there must be a line of points in the real x-y plane satisfying (2) and passing continuously through (−1, 2) [1, pp. 23-28].

A good friend of mine asked, if mathematicians are not proud of their subject, then who on earth will be? No longer must our subject be seen as irrelevant, boring and a joke. And it starts in schools − full stop! Let’s call it as it is. Mathematics is Indisputably the Greatest Subject in the World.

On the 3rd of January 2023, the UK Prime Minister, Rishi Sunak, first advocated that Mathematics should be a compulsory subject for all students to the age of 18. He was supported in principle by the following statement from the President of the Royal Society, Professor Sir Adrian Smith, issued on the 4th of January 2023: “Maths, data, statistics and numeracy are essential skills for a modern world, whether for the workplace or for playing an active role in society. If we want our economy to thrive and young people to be prepared for well paid jobs, we need a radical overhaul of our education system that will include all young people doing some level of maths to 18 years of age. The PM understands this and today’s announcement is welcome.

Mathematical modelling has several valuable properties that address a number of pedagogical issues. As such, it makes for an excellent classroom exercise. First and foremost, it reinforces the insight that mathematics is nearly ubiquitous. It is all around us, hiding in plain sight. Secondly, in contrast to textbook problems which are often highly artificial, mathematical modelling is decidedly real. Next, it serves to un-silo the curriculum. As mathematics students advance, they populate their toolbox with more and more tools. However, these tools are generally course specific. Algebra tools in algebra class, calculus tools in calculus class, and so on. In attacking a modelling problem, the student needs to decide what tool to pick up and how to apply it. Finally, modelling is empowering in that it allows, indeed it requires, the student to decide for herself what the salient features of a problem are and which features are extraneous and can safely be suppressed.

The idea of using the generalised inverse of a singular matrix A to solve the matrix equation Ax = b has been discussed in the earlier papers [1, 2, 3, 4] in the Gazette. Here we discuss three simple geometric questions which are of interest in their own right, and which illustrate the use of the generalised inverse of a matrix. The three questions are about polygons and circles in the Euclidean plane. We need not assume that a polygon is a simple closed curve, nor that it is convex: indeed, abstractly, a polygon is just a finite sequence (v1, …, vn) of its distinct, consecutive, vertices. It is convenient to let vn + 1 = v1 and (later) Cn + 1 = C1.

Consecutive squares are, of course, not equally spaced: the gap increases by 2 each time. However, it is quite possible to select three equally spaced squares, for example 1, 25, 49. Actually, such triples correspond to Pythagorean triples in a pleasantly simple way, which we will describe.

The aim of this paper is to give a solution to three conjectures from Euclidean geometry concerning the location of the Mittenpunkt. The first two are solved without dependence on computer technology and with only a moderate amount of calculations. They were initially tackled by heavy calculations using computer algebra systems.

In number theory, we frequently ask if there are infinitely many prime numbers of a certain type. For example, if n is a natural number:

1. (i) Are there infinitely many (Mersenne) primes of the form 2n − 1?

2. (ii) Are there infinitely many primes of the form n2 + 1?

These problems are often very difficult and many remain unsolved to this day, despite the efforts of many great mathematicians. However, we can sometimes comfort ourselves by asking if there are infinitely many composite numbers of a certain type. These questions are often (but not always) easier to answer. For example, echoing (i) above, we can ask if there are infinitely many composites of the form 2p − 1 with p a prime number but (to the best of our knowledge) this remains an unsolved problem. Of course, it must be the case that there are either infinitely many primes or infinitely many composites of the form 2p − 1 and it seems strange that we currently cannot decide on either of them.

Readers familiar with numbers and their patterns will most likely know of the game-playing exercise where one is invited to spot the next number in a sequence, only to find that after giving an answer there can be a multiplicity of possibilities, each of which is valid according to its own rule of sequence construction. This is an instance of partial sequence matching and the purpose of this note is to show how a simple method of constructing a partial sequence match can be achieved. It requires only a knowledge of polynomials and matrix algebra, which are topics within the grasp of an undergraduate, and will be instanced against the backdrop of some wellknown sequences.

Our goal is to find new constructions and properties of parabolas. Our strategy is to display the steps in a known construction or property, and then to take the dual of the steps in order to create a new construction or property.

Given a real differentiable function f we say that a point x0 is a stationary point of f if f′ (x0) = 0.

In any standard single-variable calculus class, students learn how to determine the nature of a stationary point by checking the sign of f(x) in intervals to the left and to the right of the stationary point. In doing so, they are performing the first derivative test.

In the special theory of relativity, there is an unusual formula for addition of velocities. We will use this formula to generate Pythagorean triples. We define a Pythagorean triple in the usual manner as a triple of positive integers (a, b, c) such that a2 + b2 = c2. The numbers a and b are called legs, and c is called the hypotenuse. Later, we will allow b to take on the value of any integer. A primitive Pythagorean triple is a Pythagorean triple for which a, b and c are relatively prime.

When first learning about infinite series, students typically are shown some examples for which the partial sums can be simplified by taking advantage of telescoping sums. In this paper, we present many examples of such series, all involving the inverse tangent function and most of which involve the Fibonacci and Lucas numbers. Most of the series presented here have appeared in various papers (see the references), but the authors are usually working in an abstract setting which makes it difficult for students to follow the basic ideas. We seek to make these results accessible to a wider audience.

One fine spring morning, at some point, I was simply tearing a few sheets of paper towel off a fresh roll. As the sheets came off and the roll rotated about the vertical axle of the holder (Figure 1), I lazily pondered the geometry lurking in the situation. I realised that sometimes I would let the roll rotate as the unwound part moved straight until tearing, while at other times, I might hold the roll in place and unwind the sheets around the roll. More often, it would be some spontaneous combination of the two modes of unwinding.

There is a very interesting mathematical puzzle involving the geometrical configuration in the book Mathematical Curiosities [1, 2] by Alfred Posamentier and Ingmar Lehmann. It is shown in Figure 1.

We give a geometric proof of the binomial identity for all natural n and real a,b. This work was inspired by the book [1], where the binomial identity for n = 3 and a,b > 0 is proved by breaking a cube C of size (a + b) × (a + b) × (a + b) into eight rectangular boxes and counting their volumes as follows.

This Article is a follow-up to a recent Gazette Article about a probabilistic betting game studied by Abdin et al. [1]. We examine the speed of convergence of the probability needed to investigate this game by giving concrete examples, using the large deviation, which is a valuable tool for estimating probabilities of repeated trials (see [2], [3, Chapter 6], [4, Section 5.11]). Moreover, to get a deep understanding of the game, we study fairness when it is repeated infinite times. Let us call it fairness in the sense of infinity, whose exact definition will be given in the final section.

In Euclidean geometry, a regular polygon is equiangular (all angles are equal in size) and equilateral (all sides have the same length) polygon. So regular polygons should be thought of as special polygons.