Effect System and Reference Frames in Rubik’s Cube and Megaminx

Abstract

The two basic puzzles in Rubik’s cube can be summarized as recognizing solvable configurations and finding a common solution for valid configurations. Indeed, in the process of solving the first puzzle, we are required to come up with a common solution. The mathematical structure of moves in Rubik’s cube is also famous. This paper clarifies the difference between process and essence through an original concept: effect, and redefines the operation group G in a strict way. It also corrects a widespread mistake in pinpointing the mapping type between the operation group G and permutation groups, which appears in some former papers. The effect system grasps the essence of operations, introduces reference frames in physics through pure mathematics, and specifically facilitates the solution for the basic puzzles in Rubik’s cube and megaminx. The effect system has the potential to be utilized in physically symmetric structures with moves that permute mathematically analyzable configurations.