TITTLE: Mathematics on the chessboard AUTHOR: Jiří Šperl DEPARTMENT: The Department of mathematics and the teaching of mathematics SUPERVISOR: RNDr. Antonín Jančařík, Ph.D. ABSTRACT: The main subject of my thesis is mathematical problems on the chessboard using chess pieces. The work aims to demonstrate how a secondary school student would approach and solve several typical mathematical tasks of this nature. Consequently, it outlines ways to incorporate chessboard mathematical problems and exercises in mathematical classes. Moreover, the thesis includes a compact collection of solved problems on the chessboard that can serve as an inspiring source of unconventional mathematical tasks in conventional mathematical education. My own mathematical research forms a major part of the thesis. The research was conducted as a series of tests in three school classes. In order to achieve a high de- gree of objectivity classes of students with different specializations were selected to take part in the tests. The participating classes were also of different age groups. The theoretical part of the thesis takes a look at the past of the subject and presents several interesting historical problems concerning the mathematics on the chess- board. Last but not least, the thesis contains a discussion of solutions of the...

This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SGn,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation provides a positive answer to this question in the case k = 2. In this case it is also shown that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SGn,2. Part two of this dissertation studies simplicial complexes that arise from non-attacking rook placements on a subclass of Ferrers boards that have ai rows of length i where ai > 0 and i ≤ n for some positive integer n. In particular, enumerative properties of their facets, homotopy type, and homology are investigated.

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The contents of this dissertation is informing readers about invariant use in solving tasks from various fields of elementary mathematics. Individual tasks are devided into thematic wholes according to the ways of solution and they are arranged in order of difficulty. The dissertation should be used as a study material for mathematics talented pupils respectively a methodical handbook for teachers.

This master thesis discusses various mathematical problems related to the placement of chess pieces. Solutions to the problems are mostly elementary (yet sometimes quite inventive), in some cases rely on basic knowledge of graph theory. We successively focus on different chess pieces and their tours on rectangular boards, and then examine the "independence" and "domination" of chess pieces on square boards. The text is complemented with numerous pictures illustrating particular solutions to given problems.