This short article is about our Giessener Mathematikum, a hands-on minds-on science centre which is unique in the world as an institution solely directed to reaching out to the general public in order to show that mathematics is an interesting and enjoyable subject for everybody. Given many unjustified prejudices against this subject from lay people, it was a challenge to create a large science centre that is attractive to everybody and aims to convince that the beauty, depth and usefulness of mathematics is comprehensible to all, independent of a particular education and particular scientific inclination. We believe we have succeeded, and my purpose is to outline the history and present operation of the Mathematikum, and its various ways of meeting the mentioned challenge.

The pressures currently on teachers of mathematics at every level seem greater than ever, whether that is university mathematics lecturers, school teachers or those who work with very young learners. So, I began my Presidential Address with the hope that no one would leave the Joint Conference of Mathematics Subject Associations 2023 feeling as though they were not good enough and must do more. I did not want anyone to go home with a lengthy to-do guilt list of additional things they must try to incorporate into their practice. Instead, I advocated giving attention to the ancient proverb “less is more”, versions of which are found within many cultures; e.g., “brevity is the soul of wit” (Hamlet), “Sometimes diminishing a thing adds to it; Sometimes adding to a thing diminishes it” (Tao Te Ching). I offered five aspects of mathematics teaching and learning that might benefit from ‘less of’ something, in each case trying to see how this might create space for more of something else. In this article, I present these five suggestions for improving by removing.

In this paper we explore some of the geometry that lies behind the real linear, second order, constant coefficient, recurrence relation(1)where a and b are real numbers. Readers will be familiar with the standard method of solving this relation, and, to avoid trivial cases, we shall assume that ab ≠ 0. The auxiliary equation of t2 = at + b of (1) has two (possibly complex) solutionsand the most general solution of (1) is given by

1. (i) when are real and distinct;

2. (ii) when

3. (iii)

We are all familiar with the formulae for finding the sums of the powers of the first n natural numbers. For example,

This paper focuses on the mathematics behind the repayment of financial debt. The availability of credit, the inevitable accompaniment of which is debt, is an essential component of a monetised economy. Without it most people would not be able to purchase large items like homes, cars and other expensive consumer durables. Businesses would not expand and prosper without access to credit. However, the debt incurred by the availability of credit can also bring huge stress, and indeed distress, from personal and commercial insolvencies to the inhumanities of debt bondage associated with modern-day slavery and human trafficking. The historical origins of debt lie in antiquity and even pre-date the existence of monetised economies [1].

There is a long tradition of mathematicians valuing collections of proofs, e.g. [1] is a famous recent collection. In the past, professional mathematicians have also written many different proofs of single important theorems, e.g. [2]. Proofs of the Pythagorean Theorem were collected by [3], and [4] discusses mathematical style through proofs of the irrationality of . The recent work of [5] is a discussion of mathematical style via a comparison of 99 different ‘proofs’ the following ‘theorem’.

The prolific output of G. H. Hardy included a number of inequalities, each known, in its own context, simply as ‘Hardy’s inequality’. Here we give an account of one of them, together with some applications and generalisations. It relates to averages.

Throughout this Article, we use the following notations for the triangle ABC

• a, b and c are the lengths of the sides BC, CA and AB, respectively,

• Δ denotes the area of triangle ABC,

• ha, hb and hc are the lengths of the altitudes through the vertices A, B and C, respectively,

• ma, mb and mc are the lengths of the medians through the vertices A, B and C, respectively.

Gerrymandering has affected U. S. politics since at least 1812. A political cartoon that year decried this tactic by then Massachusetts Governor Elbridge Gerry. (Gerrymandering is manipulating the boundaries of districts to benefit a group unfairly.)

The happy function S of each positive integer x is defined to be the sum of the squares of the decimal digits of x. For example, S(2) = 4 and S(123) = 12 + 22 + 32 = 14. It is well known that for any , there exists such that , where S(n) is the n-fold composition of S. In addition, if and for some , then x is called a happy number.

Recall that for a 2π-periodic function f , the Fourier coefficients on the interval (–Π, Π) are

and the Fourier series for f is

This article, like our previous one [1], combines known and new characterisations of parallelograms. Both can be thought of as additions to Martin Josefsson’s series on ‘characterisations of’ and ‘properties of’ various types of quadrilaterals – a series that does not include parallelograms. Josefsson’s publications can be found listed in [2], [3] and [4]. For the importance of characterisations in geometry, see [5].

This Article is a follow-up to a recent Gazette Article about the lost boarding pass problem by Grimmett and Stirzaker [1]. According to their book [2, 1.8.39, p. 10], it seems that they recognised this lovely problem in 2000 or earlier. We quote it with suitable minor changes.

Desargues’ Involution Theorem is a powerful problem solving tool to anyone interested in projective geometry and its contemporary applications. To give a better understanding of this fundamental result, we present the history of the idea and we illustrate several direct applications.

On 21 December 2020 the night sky offered a beautiful astronomical treat for stargazers worldwide. An exceptionally rare conjunction of Jupiter and Saturn brought them to 0.1 degrees of angular separation – a fifth of the full moon’s diameter. It marked their closest approach since 1623 and the closest visible conjunction since 1226 (almost 800 years ago!). Astronomy enthusiasts crossed their fingers for clear skies and waited eagerly for this event. The internet and social media were inundated with pictures and news reports, celebrating the great conjunction of our solar system’s two most extensive and majestic planets.

When studying properties of the circumcircle of an integer triangle, it quickly becomes evident that the radius of such a circle (circumradius) need not itself be an integer. When it is not an integer, the circumradius can still be rational but it can also be irrational, as exemplified in the following examples. It is left to the reader to verify that the triangle with sides 10, 24, 26 has circumradius 13, and that the corresponding values for the triangles with sides 13, 14, 15 and 1, 1, 1 are 65/8 and respectively. It is shown in Theorem 1 below that a necessary condition for the circumradius to be an integer is that the area of the triangle is itself an integer (Heronian triangle) but this condition is not in itself sufficient. A simple counterexample is given by the 13, 14, 15 triangle above which has area 84. However, as a consequence of Theorem 1, we can restrict ourselves to consideringHeronian triangles, and relevant properties of such triangles, proved in [1], are given in Theorems 2 and 3 below. We also need to quote some well-known results involving the sums of two squares (see e.g. [2]) and these are listed in Lemma 3. In all that follows, we will use the convention that if T is a triangle with sides a, b, c and z > 0, then zT will denote the triangle, similar to T, with sides za, zb, zc.

The basic inequalities

The inequalities in question are(1)

and(2)

for x > 1.

Figurate numbers are numbers that can be represented by a regular and discrete geometric pattern of evenly-spaced points. Their study has attracted the attention of many mathematicians and scientists since the dawn of mathematical history, including Pythagoras of Samos (582 BC-507 BC), Diophantus of Alexandria (200/214-284/298), Fibonacci (1170-1250), Pierre de Fermat (1601-1665), Leonhard Euler (1707-1783), Waclaw Franciszek Sierpińshi (1882-1969) [1]. Although it is not one of the basic topics, it serves many fields of mathematics, such as Number Theory and Geometry. Many special numbers are related to figurate numbers. Polynomial values, some theorems and solutions of Diophantine equations are expressed in figurate numbers and studied [2, 3, 4]. Two-dimensional figurate numbers are known as polygonal numbers. In the early 1990's, polygonal numbers were expressed and visualised with the help of computers [5]. Richard K. Guy asked the question ‘Every number is expressible as the sum of how many polygonal numbers?’ [6]. As can be understood from his study, it is seen that most of the studies conducted are about ordinary polygonal numbers. But Euler proved the ‘generalized pentagonal number theorem’ about partitions [7].

What is it that makes us judge two proofs of the same theorem to be the same or different? This is not an idle question: one central aspect of judging mathematics is the novelty of the mathematics presented. This is important everywhere, from the peer-review system, to assigning international prestige, to funding agencies’ grant decisions. It even matters to some extent in examinations, to avoid accusations of collusion. Surprisingly, philosophers of mathematics have not paid the question of novelty much attention. In this Article, we will consider the appealing conjecture that the main ideas that make up the proof, the essence of a proof, can indeed be identified and that very different styles of proofs can share common main ideas. Further, that a particular theorem can be proved using quite different, independent main ideas. As a means of exploring whether this is plausible, we will present a number of novel proofs of the following theorem.