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.

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 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.

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.

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.

Long live the vis-viva equation. There is sometimes more than one way of telling a story with the same ending, and this is particularly true in the field of applied mathematics where there are often different ways of obtaining the same result to a given problem, so that what might be lost or not appreciated in one approach can be found and appreciated in another. It is the purpose here to illustrate this by presenting a well-known example drawn from the field of orbital dynamics, namely the development of what is called the Vis-viva equation. This equation is simply an expression relating the square of the velocity of an orbiting object, for example a planet orbiting a sun, to orbit parameters and scientific constants. It is a standard workhorse equation that is used extensively today by orbit control specialists wishing to determine and affect velocities of spacecraft orbiting significantly larger masses, and it was developed from work carried out centuries ago by Gottfried Leibniz (1646 – 1716). The first story will simply acknowledge the equation and how it arose. The second story will present an alternative approach based on calculus, trigonometry, algebra and computations set against the backdrop of an ellipse geometry. This second story leads to the vis-viva equation in disguise, so to speak, and examples of the speeds of two planets, Earth and Mercury, orbiting the Sun will be discussed. Where relevant, the nomenclature of orbit dynamics will be acknowledged. The discussion will involve primarily a planet’s orbit velocity of translation and detailed considerations will not be given to effects due to its speed of rotation.

Independence is a key concept in probability. Conceptually, we think of two events as being independent if the outcome of one event doesn’t affect the outcome of the other and vice versa. Mathematically, we say that events A and B are independent if the probability that both occur is the product of the probabilities that each occurs. More precisely, P (A ∩ B) = P (A) (P (B) in which P () denotes the probability of the given event. Alternatively, we say that A and B are independent if the conditional probability that A occurs given that B has occurred, P (A | B), satisfies P (A | B) = P (A). That is, whether or not B occurs does not affect whether or not A occurs.

An essential feature of the Fibonacci sequence (Fn): 1, 1, 2, 3, 5, 8, … is an explicit expression for its nth term Fn (Binet’s formula) and the corollary that is the golden ratio and {x} is the nearest integer to x.

In 2001 Simons [1] discovered a curious binomial identity which can be written as(1)

In this paper we discuss sudoku-solving strategies and how graph theory can be used to explain some of the advanced techniques. There are many websites that provide tutorials on solving sudoku puzzles. The sites [1] and [2] discuss the xy-chain technique, and the two explanations are quite different. We will define xy-chains as paths in a graph, and properties of the paths show why the technique works.

The term ‘Asymmetric Propeller’ and studies on it appeared first in [1] by Bankoff, Erdos and Klamkin, it was in [2] by Alexanderson, and more recently in [3] by Gardner. The original propeller theorem refers to three congruent equilateral triangles that share the same vertex.

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal then the corresponding sides have equal lengths. That is to say if P is the incentre of ΔABC and if BP and CP meet the sides AC and AB at B′ and C′, respectively, then

An elegant proof of this theorem appeared in [1] and is reproduced in [2].