A075152 Number of possible permutations of a Rubik cube of size n X n X n.

1, 3674160, 43252003274489856000, 7401196841564901869874093974498574336000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000

Offset: 1

Author

Warren Power, Sep 05 2002

Keywords

• nonn

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
• David Joyner, Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, 2008. See Example 4.5.1. p. 73.
• Naren Ramesh, Generalising the configurations of an N X N X N Rubik's Cube, Parabola (2023) Vol. 59, Issue 3. See p. 22.
• Jaap Scherphuis, Puzzle Pages
• Eric Weisstein's World of Mathematics, Rubik's Cube
• WikiHow, How to Solve a Rubik's Cube with the Layer Method
• Wikipedia, Rubik's Cube
• Wikipedia, Professor's Cube

Crossrefs

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

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

Programs

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 *)
f[1]=1;f[n_]:=7!3^6(6*24!!)^(s=Mod[n,2])24!^(r=(n-s)/2-1)(24!/4!^6)^(r(r+s)); Array[f,5] (* Herbert Kociemba, Jul 03 2022 *)

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 */

PARI

A075152(n)=ceil(3674160*(11771943321600)^(n%2)*620448401733239439360000^floor((n-2)/2)*(3246670537110000)^floor(((n-2)/2)^2)) \\ Davis Smith, Mar 20 2020

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

a(n) = ceiling(3674160*11771943321600^(n mod 2)*620448401733239439360000^floor((n - 2)/2)*3246670537110000^floor(((n - 2)/2)^2)). - Davis Smith, Mar 20 2020

Extensions

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

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