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A257401
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God's number for a Rubik's cube of size n X n X n (using the half turn metric).
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2
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OFFSET
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1,2
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COMMENTS
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"God's Number" is the maximum number of turns required to solve any scrambled cube. The "Half turn metric" considers a 90- or 180-degree turn of any side to be a single turn. The number is not known for cubes of size larger than 3 X 3 X 3.
God's number has been proved using a brute-force attack for the 2 X 2 X 2 and 3 X 3 X 3 cubes. For the 4 X 4 X 4 cube, it has been proved only that the lower bound is 31, while the most probable value is considered to be 32; solving this by brute force would require checking all the A075152(4) possible permutations of the "Master Cube". - Marco Ripà, Aug 05 2015
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LINKS
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Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw, and Andrew Winslow, Algorithms for Solving Rubik's Cubes, in: C. Demetrescu and M. M. Halldórsson (eds.), Algorithms - ESA 2011, 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011, Proceedings, Lecture Notes in Computer Science, Vol. 6942, Springer, Berlin, Heidelberg, 2011, pp. 689-700; arXiv preprint, arXiv:1106.5736 [cs.DS], 2011.
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FORMULA
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a(n) = Theta(n^2/log(n)) [Demaine et al.].
Conjecture: a(n) ~ (1/4)*log(24!/4!^6) * n^2/log(n).
(End)
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CROSSREFS
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KEYWORD
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nonn,hard,more,bref
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AUTHOR
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STATUS
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approved
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