A080660 Order of group of n X n X n Rubik cube, under assumptions s, m, not-i.

1, 3674160, 88580102706155225088000, 707195371192426622240452051915172831683411968000000000, 5289239086872492808525454741861751983960246149231077646632506991757159229816832000000000000000

Offset: 1

Author

N. J. A. Sloane, Mar 01 2003

Keywords

• nonn

Comments

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

• Alan Bawden, Cube Lovers Archive, Part 6

Crossrefs

See A007458, A054434, A075152, A074914, A080656-A080662 for other versions.

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 := 1; E := (n+1)*(n-3)/4; G := 0; else A := n/2; F := 1; B := 1; C := 0; D := 0; E := n*(n-2)/4; G := 0; 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[1]=1;f[2]=7!3^6;f[3]=8!3^7 12!2^10 4^6/2;f[n_]:=f[n-2]*24!(24!/2)^(n-3);Table[f[n],{n,1,10}] (* Herbert Kociemba, Dec 08 2016 *)

Formula

a(1)=1 ;a(2)=7!*3^6; a(3)=8!*3^7*12!*2^10*4^6/2; a(n)=a(n-2)*24!*(24!/2)^(n-3). - Herbert Kociemba, Dec 08 2016