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.

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.

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.

The Greek architect Kostas Vittas published in 2006 a beautiful theorem ([1]) on the cyclic quadrilateral as follows:

Theorem 1 (Kostas Vittas, 2006): If ABCD is a cyclic quadrilateral with P being the intersection of two diagonals AC and BD, then the four Euler lines of the triangles PAB, PBC, PCD and PDA are concurrent.

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.

It is almost twenty years since Branko Grünbaum lamented that the ‘original sin’ in the theory of polyhedra is that from Euclid onwards “the writers failed to define what are the ‘polyhedra’ among which they are finding the ‘regular’ ones” ([1, p. 43]). Various definitions of ‘regular’ can be found in the literature with a condition of convexity often included (e.g. [2, p. 301], [3, p. 77], [4, p. 47], [5, p. 435], [6, p. 16]). The condition of convexity is usually cited to exclude regular self-intersecting polyhedra, i.e. the Kepler-Poinsot polyhedra, such as the ‘great dodecahedron’ consisting of twelve intersecting pentagonal faces shown in Figure 1 with one face shaded. Richeson also notes ([4, pp. 47-48]) that, for a particular definition of ‘regular’, convexity is needed to exclude the ‘punched-in’ icosahedron shown in Figure 2.

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.

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]):

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

Euler’s polynomial f (n) = n2 + n + 41 is famous for producing 40 different prime numbers when the consecutive values 0, 1, …, 39 are substituted: see Table 1. Some authors, including Euler, prefer the polynomial f (n − 1) = n2 − n + 41 with prime values for n = 1, …, 40. Since f (−n) = f (n − 1), f (n) actually takes prime values (with each value repeated once) for n = −40, −39, …, 39; equivalently the polynomial f (n − 40) = n2 − 79n + 1601 takes (repeated) prime values for n = 0, 1, …, 79.

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.

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

If we talk about the centre of a triangle, what might we be referring to? Any triangle has many different points that could regarded as its centre; in fact, Encyclopedia of Triangle Centres lists over 70 000 possibilities. Three of the most famous centres, that every triangle will possess (although they may coincide), are the incentre (where the three angle bisectors meet), the centroid (where the three medians meet) and the orthocentre (where the three altitudes meet). Proofs that these centres are well-defined and exist for every triangle are simple and satisfying, good examples of reasoning (if we are teachers) for our students. Proving the three altitudes of a triangle share a point using the scalar product of vectors is a wonderful demonstration of the power of this idea.

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.

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.

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.

The 2 × 2 identity matrix, $${I_2} = \left( \begin{gathered}{\rm{1 \,\,\,0}} \hfill \\{\rm{0 \,\,\,1}} \hfill \\ \end{gathered}\right)$$, has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples.