A054434 Number of possible positions in an n X n X n Rubik's cube reachable from the starting position.

1, 88179840, 43252003274489856000, 177628724197557644876978255387965784064000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000

Offset: 1

Author

Antreas P. Hatzipolakis

Keywords

• nonn
• nice

Comments

The sequence counts possible positions of the Rubik's cube considering the positions which are related through rotations of the cube as a whole (there are 24 of those) as distinct. At odd n, the orientation of the cube as a whole is usually considered fixed by the central squares of each face (i. e., the cube as a whole cannot be rotated) so there is a difference compared to A075152 only in the case of even n. - Andrey Zabolotskiy, Jun 07 2016

Examples

			From _Andrey Zabolotskiy_, Jun 24 2016 [following Munafo]: (Start)
a(4) = 8! * 3^7 * 24! * 24! / 4!^6 is constituted by:
8! permutation of corners
× (12*2)! permutation of edges
× (6*4)! permutation of centers
× 1 (combination of permutations must be even, but we can achieve what appears to be an odd permutation of the other pieces in the cube by "hiding" a transposition within the indistinguishable pieces of one color)
× 3^8 orientations of corners
/ 3 total orientation of corners must be zero
× 1 (orientations of edges and centers are determined by their position)
/ 4!^6 the four center pieces of each color are indistinguishable
(End)
		

Links

• Francocube forum, [4x4x4] Les maths du 4x4x4
• Georges Helm, Rubik's Cube
• M. E. Larsen, Rubik's Revenge: The Group Theoretical Solution, Amer. Math. Monthly 92, 381 (1985), DOI:10.2307/2322445.
• Christopher Mowla, Math 3900
• Robert Munafo, Rubik's Cube and other Cuboid Puzzles
• Philippe Picart, Le Rubik's cube
• E. Rubik, Rubik Cube Site
• Jaap Scherphuis, Puzzle Pages
• Xavier Servantie, All about Rubik's cube
• Author?, Rubik's Cube
• Index entries for sequences related to Rubik cube

Crossrefs

See A075152, A007458 for other versions.

Programs

Mathematica

f[1]=1; f[2]=24*7!3^6; f[3]=8!3^7 12!2^10; f[n_]:=f[n-2]*24^6*(24!/24^6)^(n-2); Table[f[n], {n, 1, 10}] (* Herbert Kociemba, Dec 08 2016 *)

Formula

From Andrey Zabolotskiy, Jun 24 2016: (Start)

a(n) = A075152(n)*24 if n is even,

a(n) = A075152(n) if n is odd.

a(2) = Sum(A080629) = Sum(A080630). (End)

a(1)=1; a(2)=24*7!*3^6; a(3)=8!*3^7*12!*2^10; a(n)=a(n-2)*24^6*(24!/24^6)^(n-2). - Herbert Kociemba, Dec 08 2016

Extensions

a(4) and a(5) corrected and definition clarified by Andrey Zabolotskiy, Jun 24 2016