The Completeness of 2D Rubik's Shapes

TL;DR

This paper introduces the Rubik's Square, a planar analogue of the Rubik's cube, and proves its completeness property, then generalizes to Rubik's Shapes to analyze their configurational reachability.

Contribution

It defines the Rubik's Square and proves its completeness, extending the concept to Rubik's Shapes for broader analysis of configurational transformations.

Findings

Rubik's Square is complete, allowing any configuration to be reached from any other.

The Rubik's cube is not complete in this sense.

The concept of Rubik's Shapes generalizes the analysis of configurational completeness.

Abstract

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