A080601 Number of positions of the Rubik's cube at a distance of n moves from the solved state, in the half-turn metric.

1, 18, 243, 3240, 43239, 574908, 7618438, 100803036, 1332343288, 17596479795, 232248063316, 3063288809012, 40374425656248, 531653418284628, 6989320578825358, 91365146187124313

Offset: 0

Author

N. J. A. Sloane, Feb 25 2003

Keywords

• nonn
• fini

Comments

The half-turn metric counts both quarter-turns and half-turns as 1 move.

This is the number of positions that can be reached in n moves from the start, but which cannot be reached in fewer than n moves.

The total number of positions is (8!*12!/2)*(2^12/2)*(3^8/3) = 43252003274489856000. - Jerry Bryan, Mar 03 2003

Relationship with A080583: 243 = 262 - 18 - 1, 3240 = 3502 - 262, 43239 = 46741 - 3502, ...

References

• Robert G. Bryan (Jerry Bryan), posting to Cube Lovers List, Jul 10, 1998.
• Rokicki, Tomas. Thirty years of computer cubing: The search for God's number. 2014. Reprinted in "Barrycades and Septoku: Papers in Honor of Martin Gardner and Tom Rogers", ed. Thane Plambeck and Tomas Rokicki, MAA Press, 2020, pp. 79-98. See Table 9.4.
• Rokicki, T., Kociemba, H., Davidson, M., & Dethridge, J. (2014). The diameter of the rubik's cube group is twenty. SIAM REVIEW, 56(4), 645-670. See Table 5.1.

Links

• Alan Bawden, Cube Lovers Archive, Part 25
• Jerry Bryan, God's Algorithm...
• David Dijon, The insanely large number of Rubik's cube permutations | MegaFavNumbers, video (2020)
• Mark Longridge, God's Algorithm Calculations for Rubik's Cube...
• Tomas Rokicki, God's Algorithm out to 13f*
• Tomas Rokicki, God's Number is 20
• T. Rokicki, Twenty-two moves suffice for Rubik's Cube, Math. Intell. 32 (1) (2010) 33-40.
• T. Rokicki, 15f* in the Face Turn Metric.
• T. Rokicki, God's Algorithm out to 14f*
• 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.

Crossrefs

Cf. A080638, A005452, A080602.

Extensions

a(11) (from Jerry Bryan, 2006) and a(12) (from Tom Rokicki, 2009) added by Herbert Kociemba, Jun 24 2009

a(13) added by Tomas Rokicki, Jul 25 2009

a(14) (from Thomas Scheunemann) and a(15) (from Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki) added by Tomas Rokicki, Jul 29 2010

Name edited by Charles R Greathouse IV, Jan 19 2016

Name edited by Ben Whitmore, Jul 31 2024