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Number of possible permutations of a Rubik cube of size n X n X n.
23

%I #87 Feb 13 2024 11:50:02

%S 1,3674160,43252003274489856000,

%T 7401196841564901869874093974498574336000000000,

%U 282870942277741856536180333107150328293127731985672134721536000000000000000

%N Number of possible permutations of a Rubik cube of size n X n X n.

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

%C The three possible assumptions considered here are the following:

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

%C 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.

%C 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.

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

%D 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.

%H Robert Munafo, <a href="/A075152/b075152.txt">Table of n, a(n) for n = 1..27</a> (first 10 terms from Robert G. Wilson v)

%H Answers.com, <a href="http://www.answers.com/topic/rubik-s-cube">Rubik's Cube</a>.

%H Isaiah Bowers, <a href="https://fr.scribd.com/doc/219723/How-To-Solve-A-Rubik-s-Cube">How To Solve A Rubik's Cube</a>.

%H Richard Carr, <a href="http://www.ws.binghamton.edu/fridrich/Richard/Cubes.pdf">The Number of Possible Positions of an N x N x N Rubik Cube</a>

%H Cube Lovers, <a href="http://forum.cubeman.org/">Discussions on the mathematics of the cube</a>

%H Cube Lovers Archive, <a href="http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/">Mailing List</a>

%H Cube20.org, <a href="http://cube20.org">God's Number is 20</a>

%H Christophe Goudey, <a href="http://cubeland.free.fr/infos/ernorubik.htm">Information</a>

%H Naren Ramesh, <a href="https://www.parabola.unsw.edu.au/2020-2029/volume-59-2023/issue-3/article/generalising-configurations-ntimes-ntimes-n-rubik%E2%80%99s-cube">Generalising the configurations of an N X N X N Rubik's Cube</a>, Parabola (2023) Vol. 59, Issue 3. See p. 22.

%H Jaap Scherphuis, <a href="http://www.jaapsch.net/puzzles/">Puzzle Pages</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RubiksCube.html">Rubik's Cube</a>

%H WikiHow, <a href="http://www.wikihow.com/Solve-a-Rubik&#39;s-Cube-with-the-Layer-Method">How to Solve a Rubik's Cube with the Layer Method</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Rubik%27s_Cube">Rubik's Cube</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Professor&#39;s_Cube">Professor's Cube</a>

%F 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

%F a(n) = ceiling(3674160*11771943321600^(n mod 2)*620448401733239439360000^floor((n - 2)/2)*3246670537110000^floor(((n - 2)/2)^2)). - _Davis Smith_, Mar 20 2020

%p 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;

%t 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 *)

%t 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 *)

%t 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 *)

%o (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

%o (PARI) A075152(n)=ceil(3674160*(11771943321600)^(n%2)*620448401733239439360000^floor((n-2)/2)*(3246670537110000)^floor(((n-2)/2)^2)) \\ _Davis Smith_, Mar 20 2020

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

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

%K nonn

%O 1,2

%A Warren Power, Sep 05 2002

%E Entry revised by _N. J. A. Sloane_, Apr 01 2006

%E Offset changed to 1 by _N. J. A. Sloane_, Sep 02 2009