cp's OEIS Frontend

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

%I A075152 #95 Feb 20 2025 03:23:00
%S A075152 1,3674160,43252003274489856000,
%T A075152 7401196841564901869874093974498574336000000000,
%U A075152 282870942277741856536180333107150328293127731985672134721536000000000000000
%N A075152 Number of possible permutations of a Rubik cube of size n X n X n.
%C A075152 More precisely, order of group of n X n X n Rubik cube, under assumptions not-s, not-m, not-i.
%C A075152 The three possible assumptions considered here are the following:
%C A075152 s (for n odd) indicates that we are working in the "supergroup" and so take account of twists of the face centers.
%C A075152 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 A075152 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 A075152 Dan Hoey, posting to Cube Lovers List, Jun 24, 1987.
%D A075152 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 A075152 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 A075152 Answers.com, <a href="http://www.answers.com/topic/rubik-s-cube">Rubik's Cube</a>.
%H A075152 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 A075152 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 A075152 Cube Lovers, <a href="http://forum.cubeman.org/">Discussions on the mathematics of the cube</a>
%H A075152 Cube Lovers Archive, <a href="http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/">Mailing List</a>
%H A075152 Cube20.org, <a href="http://cube20.org">God's Number is 20</a>
%H A075152 Christophe Goudey, <a href="http://cubeland.free.fr/infos/ernorubik.htm">Information</a>
%H A075152 David Joyner, <a href="http://www.sandal.tw/upload/Adventures%20in%20Group%20Theory_%20Rubik&#39;s%20Cube,%20Merlin&#39;s%20Machine,%20and%20Other%20Mathematical%20Toys%20by%20David%20Joyner.pdf">Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys</a>, 2008. See Example 4.5.1. p. 73.
%H A075152 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 A075152 Jaap Scherphuis, <a href="http://www.jaapsch.net/puzzles/">Puzzle Pages</a>
%H A075152 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RubiksCube.html">Rubik's Cube</a>
%H A075152 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 A075152 Wikipedia, <a href="http://en.wikipedia.org/wiki/Rubik%27s_Cube">Rubik's Cube</a>
%H A075152 Wikipedia, <a href="http://en.wikipedia.org/wiki/Professor&#39;s_Cube">Professor's Cube</a>
%F A075152 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 A075152 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 A075152 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 A075152 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 A075152 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 A075152 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 A075152 (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 A075152 (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 A075152 See A007458, A054434, A074914, A080656-A080662 for other versions.
%Y A075152 Cf. A079761, A079762, A152169 (sums give a(2)), A080601, A080602 (sums give a(3)).
%K A075152 nonn
%O A075152 1,2
%A A075152 Warren Power, Sep 05 2002
%E A075152 Entry revised by _N. J. A. Sloane_, Apr 01 2006
%E A075152 Offset changed to 1 by _N. J. A. Sloane_, Sep 02 2009