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.

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.

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.

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.

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.

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.

Consider the sequence of numbers an defined by the iteration (1) for n ≥ k, where pr 1 ≤ r ≤ k) are non-negative numbers with and the starting values are given. So an is a weighted average of the previous k terms.

Number theory and especially primes are an infinite source of inspirational ideas in elementary mathematics. Many of these ideas can be understood and some also explained with very basic mathematical knowledge. In our paper we explain the cycle which appears between the first six multiples of the decimal expression of and explore an inspiringly simple relation between primes, which somehow makes them gather into mysterious groups. And while exploring these simple questions, we are led to unveil some aspects of the power and limits of human mind and those of a computer.

Several problems that were composed and/or solved by the tenth century Islamic mathematician Abu Sahl al-Kuhi reached us via the writings of Abd al-Jalil al-Sijzi, another tenth century Islamic mathematician who, according to [1], was presumably a student of al-Kuhi’s. Twelve of these (sets of) problems and theorems are discussed in [1] and are referred to as The Fragments. The theorem of al-Kuhi alluded to in the title is Fragment #9, which is presented below, together with Fragment #5 and their proofs, in Section 2. Section 3 is devoted to the generalisation referred to in the title. Section 4 describes a relation to angle trisection and Sections 5 and 7 a relation too a configuration of Serenus. Section 6 contains a speculation on what motivated Fragment #9.

Given two vectors $\overrightarrow u= {({u_1},\,\,{u_2},\,{u_3})^t}$ and $\overrightarrow y= {({v_1},\,{v_2},\,{v_3})^t}$ in ${{\mathcal{R}}^3}$, the cross product $\overrightarrow u\times \overrightarrow v $is defined as follows (see [1] or [2]):

$$Algebraic{\rm{ }}definition{\rm{:}}\,\,\overrightarrow u\times \overrightarrow v \, = \,\left[ {\matrix{ {{u_2}{v_3}} \hfill &-\hfill & {{u_3}{v_2}} \hfill\cr{{u_3}{v_1}} \hfill &-\hfill & {{u_1}{v_3}} \hfill\cr{{u_1}{v_2}} \hfill &-\hfill & {{u_2}{v_1}} \hfill\cr} } \right].$$

The aim of this Article is to present a natural generalisation of a well-known result from plane geometry, called Stewart's Theorem. We will present our result in the Euclidean space , but we will prove it only for n = 3. Since its proof for n > 3 is involved, our intention is to give the details in another Article. A search in the relative literature returns many articles which generalise Stewart's Theorem in different directions. We first refer the reader to [1], where a collection of generalisations of Stewart's theorem is presented. In [2], two results are given, which describe a relation between k points in the Euclidean space and their weighted average. If we apply these theorems to the plane, we straightforwardly get Stewart's Theorem. Another result is given in [3], which is also reproduced in [1]. This result considers n + 1 points in which belong to a hyperplane U of dimension n − 1 and another point A ∉ U , and presents an interesting relation that combines distances between each of these points and A, hypervolumes of simplices and powers of points with respect to hyperspheres. Another relative result which involves convex quadrilaterals on the plane is proved in [4]. Lastly, in [5], a generalisation of [4] is proved for 2k points in , and a relation is found between distances of these points and the distance of two corresponding averages (of these points). These articles support the fact that there can be many interesting generalisations of Stewart's Theorem in different directions.

Integer linear combinations of cube, fourth, or sixth roots of unity form lattices in the complex plane . In contrast, integer linear combinations of fifth roots of unity do not form a lattice; in fact, they are dense in . Nevertheless, the geometry for fifth roots of unity has considerable structure. Here we consider only sums of distinct fifth roots of unity, and show that 20 of these sums are orthogonal projections of the vertices of a regular dodecahedron. Pentagonal symmetry here is only to be expected, but it is a little surprising to encounter a plane projection of a polyhedron with much richer dodecahedral symmetry.

If you were a candidate in an election, would you prefer more support from voters or less? Put another way: would you prefer that your campaign staff persuade more voters to vote for you, or fewer? These questions seem silly, because of course you would want more support from voters. Surprisingly, when using certain voting methods it is actually possible for more voter support to produce a worse result for a candidate; such an outcome is a type of monotonicity paradox. In [1], we searched for various types of monotonicity paradoxes in 1079 single-transferable vote (STV) elections from a database of Scottish local government elections. The purpose of this article is to present in detail two of the most interesting elections revealed by our search. These two elections are arguably the most paradox-riddled real-world political ranked choice elections ever documented, perhaps rivalled only by four single-winner examples from the United States: the 2009 mayoral election in Burlington, Vermont ([2, 3]); a 2021 city council race in Minneapolis, Minnesota [4]; the 2022 Special Election for US House in Alaska [5]; and the 2022 District 4 School Director election in Oakland, California [6]. The first election we present is the 2017 council election in the Buckie Ward of the Moray Council Area, which demonstrated the most extreme instance of a committee size monotonicity paradox ever observed in an actual election. The second election is the 2012 council election in the Steòrnabhagh a Tuath Ward of the Nah-Eileanan Siar Council Area, which demonstrated upward and downward monotonicity paradoxes, as well as a no-show paradox. To contextualise these elections, as part of our discussion we indicate how often these kinds of paradoxes occur in Scottish local government elections.