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A054434 Number of possible positions in an n X n X n Rubik's cube reachable from the starting position. 14
1, 88179840, 43252003274489856000, 177628724197557644876978255387965784064000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=1..5.

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


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


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



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 *)


See A075152, A007458 for other versions.

Sequence in context: A003825 A114259 A234981 * A164850 A253269 A227654

Adjacent sequences:  A054431 A054432 A054433 * A054435 A054436 A054437




Antreas P. Hatzipolakis


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



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Last modified November 30 09:26 EST 2021. Contains 349419 sequences. (Running on oeis4.)