Workshop 1

**Non-constructive Theorems in
Problem Solving**

**Prof. Sava Grozdev**

Bulgarian Academy of Sciences** **

** **

**Abstract**

The Curriculum of Mathematical Olympiads comprises the so-called non-constructive theorems and we shall discuss them in more details during the Workshop. These are theorems of existence but they do not provide an algorithm to find what is claimed to exist. We shall give the theorems without proof but with illustrations of their application, formulating problems and discussing their solutions. Among the theorems to be considered are: the Fundamental Theorem of Algebra, Bolzano and Weierstrass theorems on continuous functions, Lagrange growth theorem, Cantor theorem, Chinese remainder theorem, Thue Lemma, Dirichlet theorem for infinite arithmetic progression, Bachet theorem, Dirichlet principle.

Workshop 2

**The Passage from Recreational
Mathematics to Pure Mathematics**

**Professor Michael Lambrou**

Department of Mathematics, University of Crete

**Abstract**

In the workshop we will discuss the idea of making mathematics attractive by using material from recreational mathematics. In particular, we will discuss various intriguing "impossibility proofs" drawn from recreational mathematics and build up to a similar topic in serious mathematics.

Workshop 3 (80 minutes)

** **** Introduction into
elementary coding theory **

**Professor Dr. Guenter Toerner**

**Professor ****Törner Anne**

Universitaet Duisburg-Essen

**Abstract**

Coding theory is the study of methods for efficient and accurate transfer of information from one place to another. The mathematical theory behind is about 60 years old, however its questions and preliminary answers can be traced back to ancient times. Many fields of mathematics such as linear algebra, ring theory, geometry and algebraical geometry have contributed essentially to the still engrowing theory within discrete mathematics. There are numerous applications developing methods for e.g. computer science, information theory, astronomy and astronautics.

The workshop (80 minutes) will introduce into this topic and will reveal the high importance of coding theory for modern communication. First models can be understood even at school level. So the lecturer is anticipating an interesting auditorium and is welcoming all participants.

Workshop 4

**Flexagons**

**Professor Peter Kortesi**

University of Miskolc** **

**Abstract**

The flexagons can be considered as a type of
topologiocal model. They are figures made from a sheet of paper, but end up
having surprising properties, like many faces appearing by flexing them. The
first flexagon description - the* tri-tetrafleaxgon *- appeared in 1897
(Albert Hopkins: Magic Stage Illusions and Scientific Diversions) and it was
very popular. The creation of the *hexa-hexaflexagon* in 1934 by Arthur
Stone initiated mathematical interest in these figures. Three friends B.
Tuckerman, R. Feyhman, and J. Tukey studied the properties of the
hexa-hexaflexagons, and developed a complete mathematical theory for them. A
number of scientific papers appeared (M.Gardner in Scientific American, 1956,
O.C. Oakley and R.T. Wisner in The American Mathematical Monthly, 1957). The
tetra-flexagons are rectangular flexagons. The simplest of them is the
tri-tetraflexagon, where tri- indicates the number of faces, and tetra the form
of the figure. We will study the contruction of the tetra-tetraflexagons, and
hexa-tetraflexagons. Recently in the article:*The combinatorics of all
regular flexagons,* (McLean et all. , European Journal of Combinatorics Volume 31
Issue 1, January, 2010) a regular flexagon is given as one that contains 9n
equilateral triangular regions on a straight strip of paper. The flexagon is
said to have order 3n because you can color 3n of the faces with 3n different
colors. An other family of regular flexagons to be studied are the
hexaflexagons, we will experiment the hexa-hexaflexagons, nona-and
dodeca-hexaflexagons, and discover a bit of their mathematics.

Workshop 5: 60 min

**The role of counter examples in
overcoming the errors in mathematics**

**Gagatsis Athanasios, University of
Cyprus**

**Philippou Andreas, CMS**

**Timotheou Savas, CMS**