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Glavni cilj ovog diplomskog rada bio je prezentirati matematički aspekt popularne Rubikove kocke. Pritom je stavljen naglasak na teoriju grupa, a sporedno je obrađena i kombinatorička problematika povezana s Rubikovom kockom. Koristeći činjenicu da skup svih poteza na Rubikovoj kocki čini algebarsku strukturu grupe, predočena su neka strukturalna svojstva kao što su asocijativnost, egzistencija neutralnog i inverznog elementa, ali i dva osnovna svojstva inverznog elementa koji su u kontekstu ove teme nazvani svojstvima suprotnih poteza na Rubikovoj kocki. Također, prikazujući efekte konjugata i komutatora na Rubikovoj kocki kao i poteze koji za posljedicu imaju određene cikluse, predočeni su složeniji pojmovi teorije grupa na intuitivan i razumljiv način. Po pitanju kombinatorike, u radu se ukazalo na broj svih mogućih stanja Rubikove kocke kao i na minimalan broj poteza dovoljnih za slaganje kocke koja se nalazi u proizvoljnom stanju. Tom prilikom posredno je korišten i pojam metrike iz koje promatramo poteze na Rubikovoj kocki te je time ukazano da je definiranje poteza na Rubikovoj kocki kao okreta jedne njene strane upravo za 90◦, samo jedna moguća definicija koja generalno ovisi o odabiru metrike (u ovom slučaju riječ je o tzv. četvrt-metrici). U zadnjem dijelu rada, dan je osvrt na grupu koja se može uočiti na popularnoj slagalici 15-puzzle. Cilj osvrta bio je komparativne prirode pa su tako istaknute neke sličnosti u procesima rješavanja te slagalice i slaganja Rubikove kocke. The main objective of this thesis was to represent the mathematical aspects of the popularly known Rubik’s cube. Thereby we elaborated on group theory and analogously on combinatoric problems tied to Rubik’s cube. Using the fact that the set of all moves on a Rubik’s cube generates an algebraic structure of a group, certain structural properties are demonstrated such as associativity, identity, and invertibility, as well as two basic properties of invertibility which in the context of this thesis are called properties of opposite moves on the Rubik’s cube. Furthermore, by displaying effects of the conjugate and commutator on the Rubik’s cube as moves which have assigned cycles as a consequence, complex concepts from group theory are presented with an intuitive and reasonable approach. On the issue of combinatorics, the thesis touched upon the number of all possible combinations of the Rubik’s cube, as well as the minimum number of moves sufficient for solving a cube given an arbitrary position. In this case, a parameterized metric was used indirectly for observing moves on a Rubik’s cube, which demonstrated that defining a move on the Rubik’s cube as a turn of one of its sides by exactly 90 degrees is only one of the possible definitions of a move which generally depends on the selection of its metric (In this case, the selection is of the so-called quarter metric). In the last chapter, a review was given of the algebraic group which can be derived from the popular fifteen piece sliding puzzle game. The purpose of this review was of a comparative nature, and as such certain similarities are highlighted between the processes of solving the fifteen piece puzzle and Rubik’s cube. |
