Permutations of bounded degree generate groups of polynomial diameter
Information Processing Letters
The complexity of finding minimum-length generator sequences
Theoretical Computer Science
The (n2-1)-puzzle and related relocation problems
Journal of Symbolic Computation
A real-time algorithm for the (n2 − 1)-puzzle
Information Processing Letters
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
On the diameter of permutation groups
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Polynomial-time algorithms for permutation groups
SFCS '80 Proceedings of the 21st Annual Symposium on Foundations of Computer Science
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The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic structure. Specifically, we show that the n×n×n Rubik's Cube, as well as the n×n× 1 variant, has a "God's Number" (diameter of the configuration space) of Θ(n2/ log n). The upper bound comes from effectively parallelizing standard Θ(n2) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically optimal algorithm for solving a general Rubik's Cube in the worst case. Given a specific starting state, we show how to find the shortest solution in an n×O(1)×O(1) Rubik's Cube. Finally, we show that finding this optimal solution becomes NP-hard in an n×n× 1 Rubik's Cube when the positions and colors of some cubies are ignored (not used in determining whether the cube is solved).