The Legal Rubik's Cube Group

This project describes the group structure found in the popular children’s toy, the 3x3 Rubik’s Cube. We begin with an introduction to notation specific to the Rubik’s Cube, leading to the first fundamental theorem of cube theory which describes a unique state of the cube. We then build up the illegal cube group which encompasses all possible combinations of states, including ones not achievable with the physical limits of the basic moves. This group is then reduced to the legal cube group described in the second fundamental theorem of cube theory. Investigating the legal group we find the center, its order, how many moves of prime order it has, and how far from solved a move can be. [An accessible MathML version of the paper is included with this page as a supplementary file.]