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Number of possible positions in an n X n X n Rubik's cube reachable from the starting position.
14

%I #44 Jun 26 2017 03:45:27

%S 1,88179840,43252003274489856000,

%T 177628724197557644876978255387965784064000000000,

%U 282870942277741856536180333107150328293127731985672134721536000000000000000

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

%C 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

%H Francocube forum, <a href="http://forum.francocube.com/viewtopic.php?t=3212">[4x4x4] Les maths du 4x4x4</a>

%H Georges Helm, <a href="http://cube.helm.lu/myweb/cubeold.htm">Rubik's Cube</a>

%H M. E. Larsen, <a href="http://www.jstor.org/stable/2322445">Rubik's Revenge: The Group Theoretical Solution</a>, Amer. Math. Monthly 92, 381 (1985), DOI:10.2307/2322445.

%H Christopher Mowla, <a href="https://docs.google.com/file/d/0Bx0h7-zg0f4NTUM1MWUtaDJrbms">Math 3900</a>

%H Robert Munafo, <a href="http://mrob.com/cube/cube-puzzles.html">Rubik's Cube and other Cuboid Puzzles</a>

%H Philippe Picart, <a href="http://trucsmaths.free.fr/rubik.htm">Le Rubik's cube</a>

%H E. Rubik, <a href="http://www.rubiks.com/">Rubik Cube Site</a>

%H Jaap Scherphuis, <a href="http://www.jaapsch.net/puzzles/">Puzzle Pages</a>

%H Xavier Servantie, <a href="http://web.archive.org/web/20030621195327/http://www-ensimag.imag.fr/eleves/Xavier.Servantie/rubik/index.eng.html">All about Rubik's cube</a>

%H Author?, <a href="http://rubiks.cube.resolve.free.fr/chiffres.htm">Rubik's Cube</a>

%H <a href="/index/Ru#Rubik">Index entries for sequences related to Rubik cube</a>

%F From _Andrey Zabolotskiy_, Jun 24 2016: (Start)

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

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

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

%F 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

%e From _Andrey Zabolotskiy_, Jun 24 2016 [following Munafo]: (Start)

%e a(4) = 8! * 3^7 * 24! * 24! / 4!^6 is constituted by:

%e 8! permutation of corners

%e × (12*2)! permutation of edges

%e × (6*4)! permutation of centers

%e × 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)

%e × 3^8 orientations of corners

%e / 3 total orientation of corners must be zero

%e × 1 (orientations of edges and centers are determined by their position)

%e / 4!^6 the four center pieces of each color are indistinguishable

%e (End)

%t 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 *)

%Y See A075152, A007458 for other versions.

%K nonn,nice

%O 1,2

%A Antreas P. Hatzipolakis

%E a(4) and a(5) corrected and definition clarified by _Andrey Zabolotskiy_, Jun 24 2016