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A257401 God's number for a Rubik's cube of size n X n X n (using the half turn metric). 2
0, 11, 20 (list; graph; refs; listen; history; text; internal format)



"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


Table of n, a(n) for n=1..3.

Jerry Bryan, God's Algorithm for the 2x2x2 Pocket Cube.

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.

Joseph L. Flatley, Rubik's Cube solved in twenty moves, 35 years of CPU time.

Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge, The Diameter Of The Rubik's Cube Group Is Twenty, SIAM J. of Discrete Math, Vol. 27, No. 2 (2013), pp. 1082-1105.

Jaap Scherphuis, Mini Cube, the 2×2×2 Rubik's Cube.

Speedsolving.com, Rubik's Cube Fact sheet.

Wikipedia, Optimal solutions for Rubik's Cube.


From Ben Whitmore, May 31 2021: (Start)

a(n) = Theta(n^2/log(n)) [Demaine et al.].

Conjecture: a(n) ~ (1/4)*log(24!/4!^6) * n^2/log(n).



Cf. A256573 (quarter turn metric), A054434 (possible positions), A075152 (possible permutations).

Cf. A079761, A079762, A080601, A080602.

Sequence in context: A129909 A174976 A003284 * A283903 A063589 A102815

Adjacent sequences:  A257398 A257399 A257400 * A257402 A257403 A257404




Peter Woodward, Apr 21 2015



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