Mathematical Foundations and Group Theory Applications in the CFOP Method for Solving Rubik’s Cube

Yiran Hu

ABSTRACT

This paper explores the mathematical principles underlying the CFOP method for solving the Rubik's Cube, with a particular focus on the concepts of conjugation and commutators from group theory. The CFOP method, developed by Jessica Fridrich, is one of the most efficient techniques for solving the Rubik's Cube, breaking the solution process into four distinct stages: Cross, F2L (First Two Layers), OLL (Orientation of the Last Layer), and PLL (Permutation of the Last Layer). We begin by providing a historical overview of the Rubik's Cube, followed by definitions of key concepts in group theory. We then demonstrate that the set of all possible Rubik's Cube moves forms a group under the operation of composition, satisfying the properties of closure, associativity, identity, and inverse. Through detailed examples, we illustrate how each step of the CFOP method can be achieved using conjugation and commutators, highlighting their role in simplifying the solving process and maintaining the overall structure of the cube.

While the focus is primarily on the CFOP method, the paper acknowledges other solving methods and advanced techniques that were not covered in detail. Future work could explore these areas further, expanding on the foundational concepts presented here. By mastering the principles of group theory, cubers can enhance their solving strategies, achieve faster solve times, and gain a deeper appreciation for the mathematical beauty of the Rubik's Cube.

Mathematical Foundations and Group Theory Applications in the CFOP Method for Solving Rubi.

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