Algebraic Modeling of Rubiks Cube with Group Theory

Transitioning from the practical manipulation of the Rubiks Cube to its theoretical abstraction presents significant challenges, compounded by a scarcity of foundational resources facilitating this shift. To address this gap and establish a theoretical basis for abstract Rubiks Cube analysis, this paper provides the fundamental methodology and key conclusions for constructing an algebraic model of the standard 3x3x3 Rubiks Cube using group theory. Cube operations are defined using the definition of a group, and the concept of the Rubiks Cube group is formally introduced. Employing knowledge of permutation groups, it is demonstrated that any state of the cube can be represented by an element within the set denoted by the direct product of four special groups. Subsequently, by employing group actions, this paper successfully integrates the Rubiks Cube group with the state space thereby completing the core algebraic modeling of the cube. Utilizing this model, the equivalence condition for solvable cube configurations: the sign of permutation in position of corner and edge are the same, and there was no single edge cube or corner cube artificially flipping. Finally, based on this equivalence, the number of solvable states for a standard 3x3x3 Rubiks Cube is found.