A080602 Number of positions of the Rubik's cube at a distance of n moves from the solved state, in the quarter-turn metric.

1, 12, 114, 1068, 10011, 93840, 878880, 8221632, 76843595, 717789576, 6701836858, 62549615248, 583570100997, 5442351625028, 50729620202582, 472495678811004, 4393570406220123, 40648181519827392, 368071526203620348

Offset: 0

Author

N. J. A. Sloane, Feb 25 2003

Keywords

• nonn
• fini

Comments

The quarter-turn metric counts quarter-turns as 1 move and half-turns as 2 moves.

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

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

References

• Robert G. Bryan (Jerry Bryan), postings to Cube Lovers List, Feb 04, 1995 and Oct 26, 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 9.5.

Links

• Alan Bawden, Cube Lovers Archive, Part 15
• Alan Bawden, Cube Lovers Archive, Part 26
• Mark Longridge, God's Algorithm Calculations for Rubik's Cube...
• Tomas Rokicki, God's Algorithm out to 15q*. Posted Sep 26 2009. - _Tomas Rokicki_, Jul 14 2010
• Thomas Scheunemann, God's Algorithm out to 16q*. Posted Jul 09 2010. - _Tomas Rokicki_, Jul 14 2010
• Thomas Scheunemann, God's Algorithm out to 17q*. Posted Jul 09 2010. - _Tomas Rokicki_, Jul 14 2010
• Tomas Rokicki, God's Algorithm out to 18q*. Posted Jul 19 2014.

Crossrefs

Cf. A005452, A080583, A080601, A080638.

Extensions

Added a(14) and a(15) from my earlier investigations, confirmed by Scheunemann, and also added his result for a(16). - Tomas Rokicki, Jul 14 2010

Added a(17) from Thomas Scheunemann, a(18) from my God's Number investigations, corrected some links. - Tomas Rokicki, Sep 01 2014

Name edited by Charles R Greathouse IV, Jan 19 2016

Name edited by Ben Whitmore, Aug 02 2024