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A075152 Number of possible permutations of a Rubik cube of size n X n X n. 20
1, 3674160, 43252003274489856000, 7401196841564901869874093974498574336000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

More precisely, order of group of n X n X n Rubik cube, under assumptions not-s, not-m, not-i.

The three possible assumptions considered here are the following:

s (for n odd) indicates that we are working in the "supergroup" and so take account of twists of the face centers.

m (for n > 3) indicates that the pieces are marked so that we take account of the permutation of the identically-colored pieces on a face.

i (for n > 3) indicates that we are working in the theoretical invisible group and solve the pieces on the interior of the cube as well as the exterior. It is assumed that the M and S traits apply to the interior pieces as if they were on the exterior of a smaller cube.

REFERENCES

Dan Hoey, posting to Cube Lovers List, Jun 24, 1987.

Rowley, Chris, The group of the Hungarian magic cube, in Algebraic structures and applications (Nedlands, 1980), pp. 33-43, Lecture Notes in Pure and Appl. Math., 74, Dekker, New York, 1982.

LINKS

Robert Munafo, Table of n, a(n) for n = 1..27 (first 10 terms from Robert G. Wilson v)

Answers.com, Rubik's Cube.

Isaiah Bowers, How To Solve A Rubik's Cube.

Richard Carr, The Number of Possible Positions of an N x N x N Rubik Cube

Cube Lovers, Discussions on the mathematics of the cube

Cube Lovers Archive, Mailing List

Cube20.org, God's Number is 20

Christophe Goudey, Information. [From Robert G. Wilson v, May 23 2009]

Jaap Scherphuis, Puzzle Pages

Eric Weisstein's World of Mathematics, Rubik's Cube . [From Robert G. Wilson v, May 23 2009]

WikiHow, How to Solve a Rubik's Cube with the Layer Method . [From Robert G. Wilson v, May 23 2009]

Wikipedia, Rubik's Cube . [From Robert G. Wilson v, May 23 2009]

Wikipedia, Professor's Cube . [From Robert G. Wilson v, May 23 2009]

FORMULA

a(1)=1; a(2)=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

MAPLE

f := proc(n) local A, B, C, D, E, F, G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := 1; C := 1; D := 0; E := (n+1)*(n-3)/4; G := (n-1)*(n-3)/4; else A := n/2; F := 1; B := 1; C := 0; D := 0; E := n*(n-2)/4; G := (n-2)^2/4; fi; (2^A*((8!/2)*3^7)^B*((12!/2)*2^11)^C*((4^6)/2)^D*(24!/2)^E)/(24^F*((24^6)/2)^G); end;

MATHEMATICA

f[n_] := Block[{a, b, c, d, e, f, g}, If[OddQ@ n, a = (n - 1)/2; b = c = 1; d = f = 0; e = (n + 1) (n - 3)/4; g = (n - 1) (n - 3)/4, a = n/2; b = f = 1; c = d = 0; e = n (n - 2)/4; g = (n - 2)^2/4]; Ceiling[(2^a*((8!/2)*3^7)^b*((12!/2)*2^11)^c*((4^6)/2)^d*(24!/2)^e)/(24^f*((24^6)/2)^g)]]; Array[f, 10] (* Robert G. Wilson v, May 23 2009 *)

f[1]=1; f[2]=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 *)

PROG

(Maxima) A075152(n) := block( if n = 1 then return (1), [a:1, b:1, c:1, d:1, e:1, f:1, g:1], if mod(n, 2) = 1 then (  a : (n-1)/2,  f : 0,  b : 1,  c : 1,  d : 0,  e : (n+1)*(n-3)/4,  g : (n-1)*(n-3)/4 ) else (  a : n/2,  f : 1,   b : 1,   c : 0,   d : 0,   e : n*(n-2)/4, g : (n-2)^2/4  ), return ( (2^a * ((factorial(8)/2)*3^7)^b * ((factorial(12)/2)*2^11)^c * ((4^6)/2)^d * (factorial(24)/2)^e) / (24^f * ((24^6)/2)^g) ) )$ for i:1 thru 27 step 1 do ( sprint(i, A075152(i)), newline() )$ // Robert Munafo, Nov 12 2014

CROSSREFS

See A007458, A054434, A074914, A080656-A080662 for other versions.

Cf. A079761, A079762, A152169 (sums give a(2)), A080601, A080602 (sums give a(3)).

Sequence in context: A217673 A271027 A080657 * A080658 A080656 A074914

Adjacent sequences:  A075149 A075150 A075151 * A075153 A075154 A075155

KEYWORD

nonn

AUTHOR

Warren Power, Sep 05 2002

EXTENSIONS

Entry revised by N. J. A. Sloane, Apr 01 2006

Offset changed to 1 by N. J. A. Sloane, Sep 02 2009

STATUS

approved

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Last modified February 22 23:44 EST 2020. Contains 332157 sequences. (Running on oeis4.)