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 A075152 Number of possible permutations of a Rubik cube of size n X n X n. 23
 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. Cube Lovers, Discussions on the mathematics of the cube Cube Lovers Archive, Mailing List Cube20.org, God's Number is 20 Christophe Goudey, Information Jaap Scherphuis, Puzzle Pages Eric Weisstein's World of Mathematics, Rubik's Cube Wikipedia, Rubik's Cube Wikipedia, Professor's Cube 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(36410*11771943321600^(n mod 2)*620448401733239439360000^floor((n - 2)/2)*3246670537110000^floor(((n - 2)/2)^2)). - Davis Smith, Mar 20 2020 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; f=7!3^6; f=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 (PARI) A075152(n)=ceil(3674160*(11771943321600)^(n%2)*620448401733239439360000^floor((n-2)/2)*(3246670537110000)^floor(((n-2)/2)^2)) \\ Davis Smith, Mar 20 2020 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 November 27 16:35 EST 2021. Contains 349394 sequences. (Running on oeis4.)