cp's OEIS Frontend

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

%I A054434 #44 Jun 26 2017 03:45:27
%S A054434 1,88179840,43252003274489856000,
%T A054434 177628724197557644876978255387965784064000000000,
%U A054434 282870942277741856536180333107150328293127731985672134721536000000000000000
%N A054434 Number of possible positions in an n X n X n Rubik's cube reachable from the starting position.
%C A054434 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 A054434 Francocube forum, <a href="http://forum.francocube.com/viewtopic.php?t=3212">[4x4x4] Les maths du 4x4x4</a>
%H A054434 Georges Helm, <a href="http://cube.helm.lu/myweb/cubeold.htm">Rubik's Cube</a>
%H A054434 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 A054434 Christopher Mowla, <a href="https://docs.google.com/file/d/0Bx0h7-zg0f4NTUM1MWUtaDJrbms">Math 3900</a>
%H A054434 Robert Munafo, <a href="http://mrob.com/cube/cube-puzzles.html">Rubik's Cube and other Cuboid Puzzles</a>
%H A054434 Philippe Picart, <a href="http://trucsmaths.free.fr/rubik.htm">Le Rubik's cube</a>
%H A054434 E. Rubik, <a href="http://www.rubiks.com/">Rubik Cube Site</a>
%H A054434 Jaap Scherphuis, <a href="http://www.jaapsch.net/puzzles/">Puzzle Pages</a>
%H A054434 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 A054434 Author?, <a href="http://rubiks.cube.resolve.free.fr/chiffres.htm">Rubik's Cube</a>
%H A054434 <a href="/index/Ru#Rubik">Index entries for sequences related to Rubik cube</a>
%F A054434 From _Andrey Zabolotskiy_, Jun 24 2016: (Start)
%F A054434 a(n) = A075152(n)*24 if n is even,
%F A054434 a(n) = A075152(n) if n is odd.
%F A054434 a(2) = Sum(A080629) = Sum(A080630). (End)
%F A054434 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 A054434 From _Andrey Zabolotskiy_, Jun 24 2016 [following Munafo]: (Start)
%e A054434 a(4) = 8! * 3^7 * 24! * 24! / 4!^6 is constituted by:
%e A054434 8! permutation of corners
%e A054434 × (12*2)! permutation of edges
%e A054434 × (6*4)! permutation of centers
%e A054434 × 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 A054434 × 3^8 orientations of corners
%e A054434 / 3 total orientation of corners must be zero
%e A054434 × 1 (orientations of edges and centers are determined by their position)
%e A054434 / 4!^6 the four center pieces of each color are indistinguishable
%e A054434 (End)
%t A054434 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 A054434 See A075152, A007458 for other versions.
%K A054434 nonn,nice
%O A054434 1,2
%A A054434 Antreas P. Hatzipolakis
%E A054434 a(4) and a(5) corrected and definition clarified by _Andrey Zabolotskiy_, Jun 24 2016