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The Rubik's cube was invented in 1974 by Erno Rubik, who had no idea of theincredible popularity and mathematical fascinations his toy would bring.Through the years of study on the mathematical properties of the cube, theRubik's Cube group was introduced to represent all possible moves one couldperform on the cube. In this paper, we define a planar analogue to the Rubik'scube, which we dub the Rubik's Square, and prove that the Rubik's square iscomplete in the sense that given any two configurations there is a sequence ofmoves which changes one to the other. The Rubik's cube does not have thisproperty. We then abstract the concept of the Rubik's Square to a Rubik's Shapeand analyse the completeness in this more general setting.

The Completeness of 2D Rubik's Shapes