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This thesis examines the algebraic structure of the Rubik’s Cube—focusing on both the classic 3×3×3 model and its generalization to an n×n×n model—through the application of group theory. It delineates the fundamental group-theoretic characterizations of the Rubik’s Cube and establishes necessary and sufficient conditions for its solvability. Utilizing these conditions, formulas are derived for the number of solvable configurations of the Rubik’s Cube across all sizes.

This thesis deals with an evolutionary solving of the Rubik's cube. The worldwide known puzzle has been for several decades not only a toy for children and adults, but also almost a lifestyle for crowds of fans and definitely a big challenge in the world of computation, where scientists seek to find an effective automated solution. The potential for its solution could also be borne by evolutionary algorithms. The author of this thesis has developed an application employing, apart from genetic algorithms, also many advanced technics, such as linear genetic programming or local search. The goal of this special technics is to make the evolutionary process more effective. There have also been made tests of the crossover, the population size and the mutation probability influence. All the tests have been statistically evaluated.

Concrete is the work, which sets its goal to make connection between digital game and interactive art. Visuals of the work rely on brutalism architecture. Game mechanics is based on options offered by the aesthetics of the architecture. Visual content has been created within game engine Unity and it is controlled thru the custom controller. The controller is based on mechanics of the Rubik's cube. Visually it is adapted to the content of the work. The work is presented as a working prototype.

Every knapsack problem may be relaxed to a cyclic group problem. In 1969, Gomory found the subadditive characterization of facets of the master cyclic group problem. We simplify the subadditive relations by the substitution of complementarities and discover a minimal representation of the subadditive polytope for the master cyclic group problem. By using the minimal representation, we characterize the vertices of cardinality length 3 and implement the shooting experiment from the natural interior point. The shooting from the natural interior point is a shooting from the inside of the plus level set of the subadditive polytope. It induces the shooting for the knapsack problem. From the shooting experiment for the knapsack problem we conclude that the most hit facet is the knapsack mixed integer cut which is the 2-fold lifting of a mixed integer cut. We develop a cutting plane algorithm augmenting cutting planes generated by shooting, and implement it on Wong-Coppersmith digraphs observing that only small number of cutting planes are enough to produce the optimal solution. We discuss a relaxation of shooting as a clue to quick shooting. A max flow model on covering space is shown to be equivalent to the dual of shooting linear programming problem.

This thesis deals with solving of the Rubik's cube. It describes the Rubik's cube and the famous methods for its composition. The main goal of this work is to propose an evelutionary method that for any configuration of blocks will lead to its composition. The theis describes the problem encoding, the proposed evelutionary algorithm and its proper configuration and deployment. The achieved results are commented on and compared with other known mehtods in conclusions.