# Algorithms for Solving Rubik's Cubes

@article{Demaine2011AlgorithmsFS,
title={Algorithms for Solving Rubik's Cubes},
author={Erik D. Demaine and Martin L. Demaine and Sarah Eisenstat and Anna Lubiw and Andrew Winslow},
journal={ArXiv},
year={2011},
volume={abs/1106.5736}
}
• Published 28 June 2011
• Computer Science, Mathematics
• ArXiv
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…
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## References

SHOWING 1-10 OF 46 REFERENCES
NxN Puzzle and Related Relocation Problem
• Computer Science
J. Symb. Comput.
• 1990
The ( n 2-1 )-Puzzle and Related Relocation Problems
This paper extends the 8-puzzle and the 15puzzle to an n xn board and shows that finding a shortest solution for the extended puzzle is NP-hard and is thus believed to be computationally infeasible.
Computers and Intractability: A Guide to the Theory of NP-Completeness
• Computer Science
• 1978
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge
The complexity of satisfiability problems
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
A Real-Time Algorithm for the (n²-1)-Puzzle
• I. Parberry
• Computer Science, Mathematics
Inf. Process. Lett.
• 1995
God's number is 20
There are many different algorithms for solving the Cube, varying in complexity and number of moves required, but those that can be memorized by a mortal typically require more than forty moves.
A Survey of NP-Complete Puzzles
• Education
J. Int. Comput. Games Assoc.
• 2008
This article surveys NP-Complete puzzles in the hope of motivating further research in this fascinating area, particularly for those puzzles which have received little scientific attention to date.
Polynomial-time algorithms for permutation groups
• Mathematics
21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
• 1980
It is demonstrated that the normal closure of a subgroup can be computed in polynomial time, and that this proceaure can be used to test a group for solvability.