A075152 -id:A075152 - cp's OEIS frontend

A309113 Number of possible permutations of a Megaminx of size 2n+1.

1, 100669616553523347122516032313645505168688116411019768627200000000000

Offset: 0

Jianing Song, Jul 13 2019

a(3) has 264 digits and a(4) has 574 digits.

The Megaminx is a dodecahedron-shaped puzzle similar to the Rubik's Cube. The rotational axes of the pieces are perpendicular to the faces. Here only a Megaminx of odd size is considered, see the picture below showing the relationship between the Megaminx of size 2n and 2n+1.

			See the Michael Gottlieb link above.

Amiram Eldar, Table of n, a(n) for n = 0..4
Michael Gottlieb's blogger, Notes on Twisty Puzzles
Jianing Song, Relationship between the Megaminx of size 2n and 2n+1
Wikipedia, Megaminx

Number of possible permutations of: tetrahedron puzzle (without tips: A309109, with tips: A309110); cube puzzle (A075152); octahedron puzzle (without tips: A309111, with tips: A309112); dodecahedron (this sequence).

a(n) = if(n, 30! * 20! * 60!^(n^2-1) * 5!^(-12*n^2+12*n) * 2^(28-n) * 3^19, 1)

a(0) = 1; a(n) = 30! * 20! * 60!^(n^2-1) * 5!^(-12*n^2+12*n) * 2^(28-n) * 3^19 for n > 0.

A080661 Order of group of n X n X n Rubik cube, under assumptions s, not-m, not-i.

Original entry on oeis.org

1, 3674160, 88580102706155225088000, 7401196841564901869874093974498574336000000000, 579319689784815322186097322203443872344325595106656531909705728000000000000000

Offset: 1

N. J. A. Sloane, Mar 01 2003

nonn

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.

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.

Alan Bawden, Cube Lovers Archive, Part 6

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

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 := (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;

A257401 God's number for a Rubik's cube of size n X n X n (using the half turn metric).

Original entry on oeis.org

0, 11, 20

Offset: 1

Peter Woodward, Apr 21 2015

"God's Number" is the maximum number of turns required to solve any scrambled cube. The "Half turn metric" considers a 90- or 180-degree turn of any side to be a single turn. The number is not known for cubes of size larger than 3 X 3 X 3.

God's number has been proved using a brute-force attack for the 2 X 2 X 2 and 3 X 3 X 3 cubes. For the 4 X 4 X 4 cube, it has been proved only that the lower bound is 31, while the most probable value is considered to be 32; solving this by brute force would require checking all the A075152(4) possible permutations of the "Master Cube". - Marco Ripà, Aug 05 2015

Jerry Bryan, God's Algorithm for the 2x2x2 Pocket Cube.
Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw, and Andrew Winslow, Algorithms for Solving Rubik's Cubes, in: C. Demetrescu and M. M. Halldórsson (eds.), Algorithms - ESA 2011, 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011, Proceedings, Lecture Notes in Computer Science, Vol. 6942, Springer, Berlin, Heidelberg, 2011, pp. 689-700; arXiv preprint, arXiv:1106.5736 [cs.DS], 2011.
Joseph L. Flatley, Rubik's Cube solved in twenty moves, 35 years of CPU time.
Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge, The Diameter Of The Rubik's Cube Group Is Twenty, SIAM J. of Discrete Math, Vol. 27, No. 2 (2013), pp. 1082-1105.
Jaap Scherphuis, Mini Cube, the 2×2×2 Rubik's Cube.
Speedsolving.com, Rubik's Cube Fact sheet.
Wikipedia, Optimal solutions for Rubik's Cube.

Cf. A256573 (quarter turn metric), A054434 (possible positions), A075152 (possible permutations).

Cf. A079761, A079762, A080601, A080602.

From Ben Whitmore, May 31 2021: (Start)
a(n) = Theta(n^2/log(n)) [Demaine et al.].
Conjecture: a(n) ~ (1/4)*log(24!/4!^6) * n^2/log(n).
(End)

A339122 Number of elements of the Rubik's Cube group of order A338883(n).

Original entry on oeis.org

1, 170911549183, 33894540622394, 4346957030144256, 133528172514624, 140621059298755526, 153245517148800, 294998638981939200, 55333752398428896, 34178553690432192, 44590694400, 2330232827455554048, 23298374383021440, 14385471333209856, 150731886270873600

Offset: 1

Ben Whitmore, Nov 24 2020

The most common order is 60, with a(33) = 4199961633799421952 elements, or about 9.71% of the group.

The least common order (excluding 1) is 11, with a(11) = 44590694400 elements, or about 0.0000001% of the group. Elements of order 11 are rare because they cannot affect the corner pieces of the cube.

			a(1) = 1 because the only element of order A338883(1) = 1 is the identity element.
a(73) = 51490480088678400 is the number of elements of order A338883(73) = 1260.

Ben Whitmore, Table of n, a(n) for n = 1..73
Tomas Rokicki, SpeedSolving Puzzles Community.

Cf. A057731, A075152, A338883.

pN[p_] := Total[p]!/Times@@p/Times@@Factorial[Flatten[Tally[p]][[2 ;; ;; 2]]]
oddQ[p_] := OddQ[Total[p - 1]]
ord[p_] := LCM @@ p
oriN[p_, o_] := Module[{i, t, a = 0, ns = 0, s = 0, r}, t = ord[p]/p;
  For[i = 1, i <= Length[p], i++,
   If[Mod[t[[i]], o] == 0, a += p[[i]], ns += 1; s += p[[i]]]];
     {If[a == 0, r = o^(s - ns), r = o^a o^(s - ns - 1)], o^(a + s - 1) - r}]
val[p1_, o1_, p2_, o2_] :=
Module[{z}, z = pN[p1] pN[p2];
     {{LCM[ord[p1], ord[p2]],z oriN[p1, o1][[1]] oriN[p2, o2][[1]]},
     {{LCM[ord[p1] o1,ord[p2]],z oriN[p1, o1][[2]] oriN[p2, o2][[1]]}},   {{LCM[ord[p1],ord[p2] o2],z oriN[p1, o1][[1]] oriN[p2, o2][[2]]}},
  {{LCM[ord[p1] o1, ord[p2] o2], z oriN[p1, o1][[2]] oriN[p2, o2][[2]] }}}]
p8 = IntegerPartitions[8]; p12 = IntegerPartitions[12];
ce = Select[p8, ! oddQ[#] &]; co = Select[p8, oddQ[#] &];
ee = Select[p12, ! oddQ[#] &]; eo = Select[p12, oddQ[#] &];
res = {}; max = 0;
For[i = 1, i <= Length[ce], i++,
For[j = 1, j <= Length[ee], j++,
  AppendTo[res, val[ce[[i]], 3, ee[[j]], 2]]]]
For[i = 1, i <= Length[co], i++,
For[j = 1, j <= Length[eo], j++,
  AppendTo[res, val[co[[i]], 3, eo[[j]], 2]]]]
p = Partition[res // Flatten, 2]; c // Clear;
For[i = 1, i <= Length[p], i++,
  If [IntegerQ[c[p[[i, 1]]]], c[p[[i, 1]]] += p[[i, 2]],
   c[p[[i, 1]]] = p[[i, 2]]]; If[p[[i, 1]] > max, max = p[[i, 1]]]];
Select[Table[c[i], {i, 1, max}], IntegerQ[#] &] (* Herbert Kociemba, Jun 30 2022 *)

# See post #11 in SpeedSolving Puzzles Community link.

Sum_{n=1..73} a(n) = 43252003274489856000 = A075152(3).

a(10) corrected by Ben Whitmore, Jun 27 2022

A381367 Number of possible configurations of an n dimensional Rubik's hypercube.

Original entry on oeis.org

43252003274489856000, 1756772880709135843168526079081025059614484630149557651477156021733236798970168550600274887650082354207129600000000000000

Offset: 3

Michel Marcus, Feb 21 2025

nonn

a(4) was calculated by Kamack and Keane (1982) and Velleman (1992). - Amiram Eldar, Mar 03 2025

Amiram Eldar, Table of n, a(n) for n = 3..5
Harry J. Kamack and Tom R. Keane, The Rubik tesseract, unpublished manuscript, 1982; alternative link.
Giovanni Luca Marchetti, Rubik's Abstract Polytopes, arXiv:2502.13518 [math.CO], 2025.
Dan Velleman, Rubik's Tesseract, Mathematics Magazine, Vol. 65, No. 1 (1992), pp. 27-36.
Googology Wiki, Twisty puzzle-related numbers.

Cf. A075152 (for a(3)), A381366.

a[n_] := (1/(If[n >= 5, 1, 3]*2^(2^n + 2*(n-2)))) * Product[(n-i)!^(2^(n-i)*Binomial[n, i])*(2^(n-i) * Binomial[n, i])!, {i, 0, n-2}]; Array[a, 2, 3] (* Amiram Eldar, Feb 21 2025 *)

a(n) = my(c=1); if (n<5, c=3); prod(i=0, n-2, ((n-i)!)^((2^(n-i)*binomial(n,i)))*((2^(n-i)*binomial(n,i))!))/(c*(2^(2^n+2*(n-2))));

See formula (26) on p. 16 of Marchetti.

A080658 Order of group of n X n X n Rubik cube, under assumptions not-s, not-m, i.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 326318176648849198250599213408124182588293120000000000, 6117367460827460912265057790940131872699535863380422035173008779767508369408000000000000000000

Offset: 1

N. J. A. Sloane, Mar 01 2003

nonn

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.

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.

Alan Bawden, Cube Lovers Archive, Part 6

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

f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := (n-1)/2; C := (n-1)/2; D := 0; E := (n+4)*(n-1)*(n-3)/24; G := (n^2-1)*(n-3)/24; else A := n/2; F := 1; B := n/2; C := 0; D := 0; E := n*(n^2-4)/24; G := n*(n-1)*(n-2)/24; 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;

A080659 Order of group of n X n X n Rubik cube, under assumptions s, m, i.

Original entry on oeis.org

1, 3674160, 88580102706155225088000, 31180187340244394380451751732775816935095098996162560000000000, 234260670776288045954071997895225719627421688127737132331392149764072811894713478221812860985540608000000000000000000

Offset: 1

N. J. A. Sloane, Mar 01 2003

nonn

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.

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.

Alan Bawden, Cube Lovers Archive, Part 6

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

f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := (n-1)/2; C := (n-1)/2; D := (n-1)/2; E := (n+4)*(n-1)*(n-3)/24; G := 0; else A := n/2; F := 1; B := n/2; C := 0; D := 0; E := n*(n^2-4)/24; 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;

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

Original entry on oeis.org

1, 3674160, 88580102706155225088000, 707195371192426622240452051915172831683411968000000000, 5289239086872492808525454741861751983960246149231077646632506991757159229816832000000000000000

Offset: 1

N. J. A. Sloane, Mar 01 2003

nonn

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.

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.

Alan Bawden, Cube Lovers Archive, Part 6

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

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;

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 *)

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

A256573 God's number for Rubik's cube size n X n X n (quarter-turn metric).

Original entry on oeis.org

0, 14, 26

Offset: 1

Peter Woodward, Apr 22 2015

"God's Number" is the maximum number of turns required to solve any scramble of cube. "Quarter-turn metric" considers a 90-degree turn of any side to be a single turn, while a 180-degree turn is considered two turns. Cubes of larger size than 3 X 3 X 3 are not solved for God's number.

Jerry Bryan, God's Algorithm for the 2x2x2 Pocket Cube
Tomas Rokicki, Morley Davidson, God's Number is 26 in the Quarter-Turn Metric

Cf. A257401 (half-turn metric), A054434 (possible positions), A075152 (possible permutations).

A330389 Maximal order of an element in the n X n X n Rubik's cube group.

Original entry on oeis.org

45, 1260, 765765, 281801520, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880, 5354228880

Offset: 2

Dmitry Kamenetsky, Dec 13 2019

nonn

For the standard Rubik's cube (n = 3) the longest period 1260 is obtained with the moves RU2D'BD'.

a(n) = a(6) for all n > 6.

Math StackExchange, Determine the highest order of an element of a Rubik's Cube group.
Puzzling StackExchange, Worst way to solve Rubik's in one algorithm.

Cf. A007458, A054434, A075152, A257401.

cp's OEIS Frontend

A309113 Number of possible permutations of a Megaminx of size 2n+1.

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A080661 Order of group of n X n X n Rubik cube, under assumptions s, not-m, not-i.

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A257401 God's number for a Rubik's cube of size n X n X n (using the half turn metric).

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A339122 Number of elements of the Rubik's Cube group of order A338883(n).

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A381367 Number of possible configurations of an n dimensional Rubik's hypercube.

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A080658 Order of group of n X n X n Rubik cube, under assumptions not-s, not-m, i.

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A080659 Order of group of n X n X n Rubik cube, under assumptions s, m, i.

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Maple

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

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