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.

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.

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.

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.

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].

Cryptography, the mathematics of protecting secret or sensitive information, is a continuously evolving area of interest. A concrete example of this is the fact that worldwide spending on information security and risk management technology and services has been estimated to reach over $150 billion in 2021. Modern cryptography is actually not a single separate domain of mathematics but an advanced encryption scheme can be based on applications of results in number theory, such as the Euler–Fermat Theorem, or involve the discrete logarithm problem using either a primitive root of a large prime or an element of an elliptic curve over a prime field. In the background, probability theory, statistics, studies on computational models and finite geometries, etc. play a major role. Recent research has considered even DNA-based molecular cryptography systems.

Many readers are familiar with the celebrated Finsler-Hadwiger and Weitzenböck’s inequality in triangle geometry. On the other hand, Wolstenholme’s inequality is not widely known, but is equally important and in an indirect way can be used to derive these two. We will introduce it and then show its strength.

Several years ago, Professor Martin Josefsson told the first-named author, in a private communication, that his ‘characterisations of’ and ‘properties of’ series of papers that comprised tangential, extangential, bicentric, orthodiagonal, equidiagonal, and bisect-diagonal quadrilaterals, and rhombi and trapezoids, did not include parallelograms because the many characterisations of these figures are well known and are easily accessed via the net, and that nothing interesting can be added. In this Article, we present some characterisations of parallelograms that are very likely unknown and that will hopefully appeal to the readers of the Gazette. Other interesting characterisations that we have obtained and that could not find their place here are intended to form the material of another Article.

When children are first taught about multiplication of the natural numbers, it is usually presented as repeated addition. Later, learning about raising to an exponent is presented as repeated multiplication. Then the following analogy is obvious: addition is to multiplication as multiplication is to raising to an exponent. An interesting question is to ask what happens if we go in the other direction. That is, multiplication is to addition as addition is to what? In this paper, we will answer this question, and show that there are several possible operations that could be used to answer the question. Some of them will be presented in connection with algebraic semirings. A semiring is a set, together with two operations, semiring addition and semiring multiplication, satisfying certain properties. We will define several semirings, such that the semiring multiplication is ordinary addition. In each case, the semiring addition will then be an answer to the question in the title of this paper.

In order to state our primary results, we must first establish some notation. Let S-1 (x) = sin x and C-1 (x) = cos x, then for each non-negative integer n, let

these are the remainders of the Maclaurin series for sine and cosine, respectively. Note that for each and for each . It is known that

See [1] for several different proofs of the well-known fact that

the values of αn and βn for then follow rather easily using induction and integration by parts. (Details are provided in the Appendix.)

In 1879, the Oxford mathematician Charles Dodgson (1832-1898) published Euclid and his modern rivals. This book compared the work of contemporary geometers to that of the Greek master from two millennia earlier. Dodgson, best known by his pen name of ‘Lewis Carroll’, much preferred Euclid. For him, the ancients outshone the moderns.