# The entire cube is 2-gen!



## Lucas Garron (Mar 10, 2010)

By which we mean the cube group can be generated by 2 elements:

G = <g1, g2> = <U B L U L' U' B', R2 F L D' R'>

So, any position of the cube can be written as something like \( g1 g2^2 g1^{-1} g2^3 \), except longer.

Apparently this was known in 1980. Joyner cites this set by "F. Barnes" in a book by Bandelow (which I have yet to see).

I believe this, but I'd like to see a proof by generating each of {B, F, U, D, L, R} as an "alg" in <g1, g2>. (Or at least 5 of them.)

ksolve can't quite handle this. Since 4.3*10^19 is about 65 binary digits, we'd expect the length of an alg for a position (like the regular generators) to have a length of about 65, ~32 if we allow inverses.

Does anyone know a nice (algorithmic?) way to generate {B, F, U, D, L, R} using <g1, g2>, or come up with some other nice set of two generators for which we can do this?


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## Stefan (Mar 10, 2010)

You didn't know that?

http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/465

I'll try to find another discussion we had about it in the yahoo group...


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## Johannes91 (Mar 10, 2010)

Lucas Garron said:


> Since 4.3*10^19 is about 65 binary digits, we'd expect the length of an alg for a position (like the regular generators) to have a length of about 65, ~32 if we allow inverses.


Non sequitur? Can you show that almost all of the 2^65 algs produce different positions?


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## vcuber13 (Mar 10, 2010)

are you guys saying that and cube can be solved using 2 gen algs and cube rotations?


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## Stefan (Mar 10, 2010)

vcuber13 said:


> are you guys saying that and cube can be solved using 2 gen algs and cube rotations?



Cube rotations?!? Noob. No cube rotations. Just the two algs.



StefanPochmann said:


> I'll try to find another discussion we had about it in the yahoo group...



I misremembered a bit, this is what I meant:

http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/13973?l=1

Begin at _"Hmm, do you remember that the cube group can be generated by just two different algorithms?"_. It was my attempt to explain how to use two certain algs to generate all positions, embedded in a devil's algorithm thread. But they were different algs, not those that Lucas posted above.


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## Stefan (Mar 10, 2010)

Lucas Garron said:


> Apparently this was known in 1980. Joyner cites this set by "F. Barnes" in a book by Bandelow (which I have yet to see).



I have Bandelow's book (in German). Does Joyner tell the page number or chapter or so?

Edit: Found it, will take a picture. Not sure how much it's explained, though (relies on previous stuff that I haven't checked yet).


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## Lucas Garron (Mar 10, 2010)

StefanPochmann said:


> You didn't know that?
> 
> http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/465


I suppose I never thought about it. And realized it had never been discussed here.



Johannes91 said:


> Lucas Garron said:
> 
> 
> > Since 4.3*10^19 is about 65 binary digits, we'd expect the length of an alg for a position (like the regular generators) to have a length of about 65, ~32 if we allow inverses.
> ...


I was just stating a lightly informed estimate of the depth of a needed search (e.g. why I wasn't expecting ksolve to get it). And no, I can't. But that would only make things a bit worse.

(However, note that the God's number for 3x3x3 is _on the order_ of the log(18, 4.3*10^19).)


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## Lucas Garron (Mar 10, 2010)

Hmm, how hard would it be to find optimal <g1, g2> solutions to the 6 face turns?


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## Stefan (Mar 10, 2010)

vcuber13 said:


> But they were different algs, not those that Lucas posted above.


Not much off, though. The R2 F L D' R' alg Lucas posted is indeed an 11-cycle of edges and a 7-cycle of corners. But the U B L U L' U' B' has a bigger effect than just swapping two edges and two corners.

Apparently from Singmaster, quoted here:

"Frank Barnes observes that the group of the cube is generated by two
moves:
alpha = L2BRD-1L-1
beta = UFRUR-1U-1F-1
Observe that alpha^7 is an 11-cycle of edges and alpha^11 is a 7-cycle
of corners"


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## Stefan (Mar 10, 2010)

Lucas Garron said:


> StefanPochmann said:
> 
> 
> > You didn't know that?
> ...



Not sure you saw it (cause you didn't comment on it): That post tells how to generate each of {B, F, U, D, L, R}.


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## Stefan (Mar 10, 2010)

vcuber13 said:


> But the U B L U L' U' B' has a bigger effect than just swapping two edges and two corners.


Gosh am I blind. It actually does just that, plus orient another edge and corner.

Also, Richard explained in a subsequent post:
http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/480


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## Lucas Garron (Mar 10, 2010)

StefanPochmann said:


> Lucas Garron said:
> 
> 
> > StefanPochmann said:
> ...


Oh, I did see it, thank you very much.


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## cuBerBruce (Mar 10, 2010)

I note that Joyner's book also mentions this Cube Lovers post where it is mentioned that if you pick two elements of the group at random, you seem to have a roughly 50% chance that they will generate the whole group. So such pairs are actually very common.


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## meichenl (Mar 11, 2010)

I had been wondering about the minimum number of generators. Thanks for the discussion.

The claim that roughly half of pairs of algorithms generate the cube is intriguing. Is there an easy way to tell whether two elements of the cube group will generate the cube?


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## rokicki (Mar 11, 2010)

*Random generators*

I'm also keenly interested in this.

Clearly both generators can't be even. So that accounts for 25% of the 50%.

Is there something equally obvious for the other 25%?

Or is the 50% only an approximate answer, and the real answer might be something like 53.1% or some such?

Another unanswered question that has been posed is, what is the shortest pair of generators that generate the cube? (Use HTM).


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## rokicki (Mar 11, 2010)

Wow. If my analysis is correct, then (for instance) just UF and RBL are sufficient to generate the entire group.

Indeed, just restricting yourself to the six generators UFRDBL, there are 720 different ways to select two generators for the first and three for the second such that all five selected are unique (this is an obvious requirement). Of those 720 ways, 96 generate the full cube group.

Can anyone confirm this?


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## Lucas Garron (Mar 11, 2010)

rokicki said:


> Wow. If my analysis is correct, then (for instance) just UF and RBL are sufficient to generate the entire group.


Wow.
Any way you can extend your code to "prove" its results by generating {B, F, U, D, L, R}? That would be enough to confirm.


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## cuBerBruce (Mar 11, 2010)

rokicki said:


> Wow. If my analysis is correct, then (for instance) just UF and RBL are sufficient to generate the entire group.
> 
> Indeed, just restricting yourself to the six generators UFRDBL, there are 720 different ways to select two generators for the first and three for the second such that all five selected are unique (this is an obvious requirement). Of those 720 ways, 96 generate the full cube group.
> 
> Can anyone confirm this?



I confirm.

I note that it is easy to test if a set of generators generates the whole cube group with GAP.



Spoiler





```
gap> U := ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19);
(1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19)
gap> L := ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35);
(1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12)
gap> F := (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11);
(6,25,43,16)(7,28,42,13)(8,30,41,11)(17,19,24,22)(18,21,23,20)
gap> R := (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24);
(3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28)
gap> B := (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27);
(1,14,48,27)(2,12,47,29)(3,9,46,32)(33,35,40,38)(34,37,39,36)
gap> D := (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40);
(14,22,30,38)(15,23,31,39)(16,24,32,40)(41,43,48,46)(42,45,47,44)
gap> G := Group (U,D,L,R,B,F);
<permutation group with 6 generators>
gap> gens := [ U, D, L, R, F, B ];
[ (1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19),
  (14,22,30,38)(15,23,31,39)(16,24,32,40)(41,43,48,46)(42,45,47,44),
  (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12),
  (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28),
  (6,25,43,16)(7,28,42,13)(8,30,41,11)(17,19,24,22)(18,21,23,20),
  (1,14,48,27)(2,12,47,29)(3,9,46,32)(33,35,40,38)(34,37,39,36) ]
gap>
gap> PermNUnpackA := function (n, idx)
>   local i, j, nn, lst;
>   nn := n;
>   i := 1;
>   lst := [ ];
>   while i <= n do;
>     lst[i] := 0;
>     i := i + 1;
>   od;
>   i := nn - 1;
>   while i >= 0 do;
>     lst[i + 1] := idx mod (nn - i);
>     idx := (idx - lst[i + 1]) / (nn - i);
>     j := i + 1;
>     while j < nn do;
>       if lst[j + 1] >= lst[i + 1] then
>         lst[j + 1] := lst[j + 1] + 1;
>       fi;
>       j := j + 1;
>     od;
>     i := i - 1;
>   od;
>   return lst;
> end;
function( n, idx ) ... end
gap>
gap> TryAll720 := function (Lm)
>   local i, z, H, g1, g2, count, Lx;
>   count := 0;
>   i := 0;
>   while i < 720 do
>     Lx := PermNUnpackA (6, i);
>     g1 := Lm[Lx[1]+1]*Lm[Lx[2]+1];
>     g2 := Lm[Lx[3]+1]*Lm[Lx[4]+1]*Lm[Lx[5]+1];
>     H := Group (g1, g2);
>     if H = G then
>       count := count + 1;
>     fi;
>     i := i + 1;
>   od;
>   return count;
> end;
function( Lm ) ... end
gap>
gap> TryAll720 (gens);
96
```


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## rokicki (Mar 11, 2010)

Thanks for the confirmation! (I also used GAP but feared I might
have made a typo or error.)

The sequences U, R2D2BL also work, as do the sequences U, D'LRF.
(You need at least one half turn or counter clockwise turn if you want
to use sequences of length 1 and 4 in the QTM.)


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## Cielo (Mar 21, 2010)

Interesting! I've never thought about this before.


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## cuBerBruce (Mar 21, 2010)

I recalled a post on the Domain of the Cube forum where someone had an example of using GAP to determine an expression for a puzzle state in terms of a set of generators. So I thought I would use GAP to try generating U,F,R,L and D with respect to generators UF and RBL to see what would be produced. The spoiler below contains the result. I edited the GAP output to a somewhat more readable format. I hope I didn't mess anything up in the editing. I haven't verified the expressions.

Let a = UF and b = RBL to interpret the result.



Spoiler



U = ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2b
a'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a5bab'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3b
a2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'a
b'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'
a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'
a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'
a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'aba
b'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'
a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'
b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'
b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'
a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'
a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'
a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'
b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'
a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5b
a'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'
b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'b
a'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5b
a'b'a'8ba'5b'a5ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'a'2bab'a'5ba'2b'aba'
b'aba'b'aba'b'a'3ba5b'a'ba'b'a2ba4b'a5ba6b'a'2bab'a'bab'aba5b'a4ba'b'
a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2ba6b'
a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'a'b
a'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'3aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a2ba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5b'a6bab'a'bab'a'bab'a'ba2
b'a5ba'2b'a'3ba5b'a'bab'a'4ba'b'a2ba2b'a'2bab'a'3ba'b'a2ba'b'a'4ba'b'aba
b'a'b'a'2b3ab'a'b'3a'b'a'b'a'b'a'2b'a'3b'abab'a'bab'a'5ba4b'aba'b'a'4b
a5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'aba
b'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'a'ba'
b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'
b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'
a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2
b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'
a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'
a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3b
a2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'
a'b'4a'5ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'
a'bab'a2bab'a'ba3b'a'5ba2b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5b'a4bab'
a6ba5b'a4ba'b'a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'b
a'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2b
a'b'a'ba'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'2a'b'a2bab'a'ba3b'a'5ba2
b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5b'a4bab'a4ba3b'a'5ba2b'aba'b'a'6ba5
b'a2bab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a2ba'b'a'2ba'3b'a5ba3b'
a'3bab'a'4ba'5b'a9ba'b'a'10bab'a'7ba'b'a'3ba5b'a'2bab'a2ba'b'aba2b'a'ba
b'aba4b'a'4ba'b'a'6ba'b'aba2b'a'ba'b'a'ba'2b'aba'b'a'

F = ba'b'a'ba3b'a'bab'a'bab'a2ba4b'a'3bab'a'bab'a'ba2b'a'5ba'2b'a'3ba5b'
a'2bab'a'7ba5b'a4ba'b'a3ba'3b'a'5ba3b'a2bab'a'2bab'aba'b'a'bab'a'2ba3b'a'3b
a2b'a'ba5b'a'2bab'a'5ba'b'a'4ba'5b'a'4ba'b'a'5ba'4b'a4ba'5b'a2bab'a'ba'2
b'a5ba'3b'aba'b'a'2bab2ababab2a'ba'ba'b'3a'5ba'2ba2b'abab'a2bab'a'9ba5b'
a5ba'4b'a'ba'b'aba'2b'a'bab'a'3bab'aba'3b'a'2ba'6b'a2bab'aba'2b'a'ba'3
b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba5b'a'9ba5b'a'7bab'a'4ba'5b'a'6ba'b'
a'4ba'5b'a'4ba'b'a'5ba'4b'a4ba'5b'a2bab'a'ba'2b'a5ba'3b'aba'b'a'2ba'b'aba
b'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a5b4abab'4a
b'ab2ab'ab2ab'a2ba'3b'aba2b'a3ba2b'abab'abab'a4ba'2b'a3bab'a'2bab'aba
b'a'ba'2b'aba'b'a'bab'a'3ba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'
a'ba'3b'a5ba3b'a'7ba'5b'a7ba5b'a'4ba'b'a'4ba'b'a'4ba4b'aba'b'a'2ba'5b'a7ba
b'a'5ba'2b'aba'b'aba'b'aba'b'ab2a2ba2ba2b2a'b'a'b2a'b'a'ba2b'a'bab'
a6bab'a2ba'3b'a'2ba3b'a'2bab'a3ba'b'a'2ba6b'a2ba3b'a'ba'b'a8ba'b'a10bab'
a'bab2a2ba2b2aba'ba'b'2a'b'aba'bab'a2ba'2b'a'bab'a'bab'a'2ba'b'a'ba'2b'
a2ba'b'a'bab'a4ba'2b'a3bab'a'3bab'aba'3b'a'2ba'6b'a2bab'abab'a'ba'10b'aba'
b'a'5ba'2b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'ba'b'a'bab'a'ba3b'a'5b
a2b'aba'b'a'6ba5b'a'2ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5b
a'b'aba'b'a'ba3ba2bababab3aba'b'3a2baba'b'a'bab'a4bab'a'2bab'a3ba'b'
a2ba'2b'a'2bab'a4ba'b'aba'5b'a3ba2b'a'5ba'2b'aba'b'aba'b'aba'b'a'6ba5b'
a4ba'b'a3ba'3b'a'5ba3b'a2bab'a'2ba'2b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'
b3ababab2a'ba'ba'b'3a'5ba'2ba2b'abab'a2bab'a'9ba5b'a5ba'4b'a'ba'b'aba'2b'
a'bab'a'3bab'aba'3b'a'2ba'6b'a2bab'aba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'
a'5ba5b'a'ba5b'a'9ba5b'a'7bab'a'4ba'5b'a'ba'b'aba'b'a2ba'6b'a'5ba'4b'a'2ba
b'aba'5b'a3bab'a'bab'a'bab'a'ba2b'a5ba'b'a2bab'a'ba3b'a'5ba2b'aba'b'
a'6ba5b'a'5ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'
a'bab'ab4abab'4ab'ab2ab'ab2ab'a2ba'3b'aba2b'a3ba2b'abab'abab'a4ba'2b'
a3bab'a'2bab'abab'a'ba'2b'aba'b'a'bab'a'3ba'2b'a'ba'3b'a'ba2b'a'ba'b'
a'4ba5b'a'5ba5b'a'ba'3b'a5ba3b'a'7ba'5b'a7ba5b'a'4ba'b'a'4ba'b'a'4ba4b'ab
a'b'a'2ba'5b'a7bab'a'5ba'2b'aba'b'aba'b'aba'b'ab2a2ba2ba2b2a'b'a'b2a'
b'a'ba2b'a'bab'a6bab'a2ba'3b'a'2ba3b'a'2bab'a3ba'b'a'2ba6b'a2ba3b'a'ba'
b'a8ba'b'a10bab'a'bab2a2ba2b2aba'ba'b'2a'b'aba'bab'a2ba'2b'a'bab'a'bab'
a'2ba'b'a'ba'2b'a2ba'b'a'bab'a4ba'2b'a3bab'a'3bab'aba'3b'a'2ba'6b'a2ba
b'abab'a'ba'10b'aba'b'a'5ba'2b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'ba'b'
a'bab'a'ba3b'a'5ba2b'aba'b'a'6ba5b'a'2ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'
a4bab'a'ba'4b'a5ba'b'aba'b'a'bab'ab4abab'4ab'ab2ab'ab2ab'a2ba'3b'aba2b'
a3ba2b'abab'abab'a4ba'2b'a3bab'a'2bab'abab'a'ba'2b'aba'b'a'bab'a'3ba'2b'
a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba'3b'a5ba3b'a'7ba'5b'a7ba5b'a'4b
a'b'a'4ba'b'a'4ba4b'aba'b'a'2ba'5b'a7bab'a'5ba'2b'aba'b'aba'b'aba'b'ab2
a2ba2ba2b2a'b'a'b2a'b'a'ba2b'a'bab'a6bab'a2ba'3b'a'2ba3b'a'2bab'a3ba'
b'a'2ba6b'a2ba3b'a'ba'b'a8ba'b'a10bab'a'bab2a2ba2b2aba'ba'b'2a'b'aba'ba
b'a2ba'2b'a'bab'a'bab'a'2ba'b'a'ba'2b'a2ba'b'a'bab'a4ba'2b'a3bab'a'3ba
b'aba'3b'a'2ba'6b'a2bab'abab'a'ba'10b'aba'b'a'5ba'2b'a'ba'2b'a'5ba'2b'ab
a'b'aba'b'abab'a'ba'b'a'bab'a'ba3b'a'5ba2b'aba'b'a'6ba5b'a'2ba5b'a8ba
b'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'a'bab'

R = ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2b
a'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a5bab'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3b
a2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'a
b'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'
a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'
a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'
a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'aba
b'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'
a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'
b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'
b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'
a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'
a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'
a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'
b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'
a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5b
a'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'
b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'b
a'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5b
a'b'a'8ba'5b'a5ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'a'2bab'a'5ba'2b'aba'
b'aba'b'aba'b'a'3ba5b'a'ba'b'a2ba4b'a5ba6b'a'2bab'a'bab'aba5b'a4ba'b'
a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2ba6b'
a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'a'b
a'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'3aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a2ba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5b'a6bab'a'bab'a'bab'a'ba2
b'a5ba'2b'a'3ba5b'a'bab'a'4ba'b'a2ba2b'a'2bab'a'3ba'b'a2ba'b'a'4ba'b'aba
b'a'b'a'2b3ab'a'b'3a'b'a'b'a'b'a'2b'a'3b'abab'a'bab'a'5ba4b'aba'b'a'4b
a5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'aba
b'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'a'ba'
b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'
b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'
a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2
b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'
a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'
a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3b
a2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'
a'b'4abab'a'ba3b'a'5ba2b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5b'a4bab'a6ba5b'
a4ba'b'a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2b
a6b'a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'
a'ba'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'2a'b'a'4ba5b'a'5ba5b'abab'a'ba3
b'a'5ba2b'aba'b'a'ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'aba'b'aba'
b'aba2b'a'5ba3b'a'bab'a'bab'a'7ba5b'a4ba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5
b'a9ba'b'a'2ba'b'a'3ba5b'a'2bab'a2bab'aba'b'abab'a'ba2b'aba'b'a'ba'b'
a2ba'b'aba2b'a'bab'aba4b'a'3ba'b'a3ba'4b'a'3bab'a3ba'b'a'ba'b'a2ba'
b'aba'b'a'ba'b'a'4b'a'bab2ab'ab'a'b'2a'2b'a'2b'3a'ba'b'2abab'2a'2b'a'2b'a'2
b'a'2b'aba'5b'a4ba'b'a2ba'5b'a3bab'a2ba'b'a'3ba2b'a'4ba'2b'a3ba3b'a'3bab'
a3ba'2b'a'b'a'bab'a'b3ab'a'b'

L = b'ab'a'ba'5b2ab'aba'3b'a4ba'3b'a'2bab'a3ba'b'a2ba'b'a3ba'4b'abab'a'ba'2
b'aba'b'a'bab'a'4bab'a'5ba'b'a'4ba'5b'a'3ba2b'a'10ba'2b'a'2ba2ba2ba2ba2b2a'
b'a'b2ab'ab3a2ba2b2aba'ba'b'2a'b'aba4bab'abab'a'bab'a'2bab'abab'a'3ba'
b'a3ba4b'a'3bab'a3ba'4b'a'ba'b'aba'2b'a'bab'a'2bab'abab'a'ba'2b'aba'b'
a'bab'a'ba'b'a'2ba'b'a2ba'5b'a3bab'a2bab'a'9ba5b'a4ba'b'a3ba'3b'a'5ba3b'
a2bab'a'4ba'5b'a7ba'b'aba'b'aba'3b'a5ba'2b'a'bab'a'ba2b2ababab2a'ba'ba'
b'3a'5ba'2ba2b'abab'a2bab'a'9ba5b'a5ba'4b'a'ba'b'aba'2b'a'bab'a'3bab'aba'3
b'a'2ba'6b'a2bab'aba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba5b'a'9b
a5b'a'7bab'a'4ba'5b'a'ba'b'aba'b'abab'a'ba'b'aba'b'a'4ba6b'aba'b'a'7ba'2
b'aba'b'aba'b'aba'b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'a'6ba5b'a8bab'
a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'a'bab'ab4abab'4ab'ab2a
b'ab2ab'a2ba'3b'aba2b'a3ba2b'abab'abab'a4ba'2b'a3bab'a'2bab'abab'a'b
a'2b'aba'b'a'bab'a'3ba'2b'a'ba'3b'a'ba2b'a'ba'b'a'4ba5b'a'5ba5b'a'ba'3
b'a5ba3b'a'7ba'5b'a7ba5b'a'4ba'b'a'4ba'b'a'4ba4b'aba'b'a'2ba'5b'a7bab'a'5b
a'2b'aba'b'aba'b'aba'b'ab2a2ba2ba2b2a'b'a'b2a'b'a'ba2b'a'bab'a6bab'a2b
a'3b'a'2ba3b'a'2bab'a3ba'b'a'2ba6b'a2ba3b'a'ba'b'a8ba'b'a10bab'a'bab2a2b
a2b2aba'ba'b'2a'b'aba'bab'a2ba'2b'a'bab'a'bab'a'2ba'b'a'ba'2b'a2ba'b'
a'bab'a4ba'2b'a3bab'a'3bab'aba'3b'a'2ba'6b'a2bab'abab'a'ba'10b'aba'b'a'5ba'2
b'a'ba'2b'a'5ba'2b'aba'b'aba'b'abab'a'ba'b'a'bab'a'ba3b'a'5ba2b'aba'b'
a'6ba5b'a'2ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5ba'b'aba'b'
a'ba6b'a'6bab'a'bab'a'5ba'b'a'8ba'2b'a5ba'3b'a'ba'b'a'ba'5b'a7ba5b'a'4ba'5
b'a4

D = ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2b
a'5b'a6bab'a'ba'2b'a5ba'3b'aba'b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2
b'aba2b'a5bab'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3b
a2b'a'4ba'b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'a
b'a'b'2a'2b'a'2b'2a'b'aba'b'a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'
a2ba'3b'a2ba3b'a'2ba'b'a'6ba'b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'
a'bab'a'bab'a'ba2b'a5ba'b'a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'
a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'aba
b'a'ba2b'aba'b'a'ba'b'a2ba'b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'
a'ba3b'a'2ba'b'2a'ba'b'2a'ba'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'
b'a'4ba5b'a'2bab'a'bab'a'5ba'b'a'8ba'5b'a2ba'5b'a6bab'a'ba'2b'a5ba'3b'aba'
b'abab'aba'b'a'bab'a'bab'a'ba2b'a5ba2b'aba2b'a5bab'a'ba10b'aba'b'
a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'b'abab'a'2ba2b'abab'
a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2b'a'2b'2a'b'aba'b'
a'10bab'a'8bab'aba'3b'a'2ba'6b'a2bab'a'3ba'b'a2ba'3b'a2ba3b'a'2ba'b'a'6ba'
b'aba'2b'abab'2abab'2a'2b'a'2b'a'2b'2a'bab'a'bab'a'bab'a'ba2b'a5ba'b'
a'7ba5b'a2bab'a'ba'4b'a4bab'a4bab'a4ba'5b'a'7ba5b'a7ba'3b'a'5ba3b'aba'5b'a5b
a'5b'a4bab'aba'2b'aba3b'aba2b'a3ba'b'abab'a'ba2b'aba'b'a'ba'b'a2ba'
b'a'3ba2b'a'4ba'b'a'ba'b'a'ba'2b'a'3ba'2b'a'ba3b'a'2ba'b'2a'ba'b'2a'b
a'b4a'b'a'b'4a'ba'b'abab'a'bab'a'5ba4b'aba'b'a'4ba5b'a'2bab'a'bab'a'5b
a'b'a'13ba5b'abab'a'ba3b'a'5ba2b'aba'b'a'ba'5b'a4bab'a'5ba'2b'aba'b'ab
a'b'aba'b'a'3ba5b'a'ba'b'a2ba4b'a5bab'a3ba2b'aba'2b'abab'a'ba'2b'aba
b'a'2ba'b'a'7ba5b'a'2ba5b'aba'b'a'2ba'3b'a5ba3b'a'3bab'a'4ba'5b'a6bab'a'b
a2b'a5ba'b'a'bab'a'4ba5b'a'2ba2b'a5ba'2b'aba'b'aba'b'a'ba'2b'abab'a'7ba
b'a'ba'b'a'ba'3b'a'4ba5b'a8bab'a5ba'b'aba'b'a2ba'5b'a4bab'a'ba'4b'a5b
a'b'aba'b'a'bab'a2bab'a'ba3b'a'5ba2b'aba'b'a'2ba5b'a'4ba4b'a5bab'a4ba5
b'a4bab'a6ba5b'a4ba'b'a7ba'5b'a9ba'5b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3
b'aba2b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'
a9ba'b'a'2ba'b'a'ba'2b'a2b'a5b3ab'ab'ab'2a'b'a'b'a'b'2a'b'a'4ba5b'a'5b
a5b'abab'a'ba3b'a'5ba2b'aba'b'a'ba'5b'a4bab'a'5ba'2b'aba'b'aba'b'aba'b'
a'3ba5b'a'ba'b'a2ba4b'a5ba6b'a'2bab'a'bab'aba5b'a4ba'b'a7ba'5b'a9ba'5
b'aba'5b'a5ba'5b'a4bab'aba'2b'aba3b'aba2b'a'ba'b'a'2ba6b'a2ba3b'a'ba'
b'a3ba'b'aba2b'a'bab'aba4b'a'5ba'5b'a9ba'b'a'2ba'b'a'ba'2b'a2b'a5b3ab'a
b'ab'2a'b'a'b'a'b'2a'b'a'bab'aba5b'a4ba'b'a7ba'5b'a9ba'5b'a'4ba5b'a4ba
b'a'ba10b'aba'b'a'ba'b'a'2ba6b'a2ba3b'a'ba'b'a3ba'b'a'3ba2b'a'4ba'
b'abab'a'2ba2b'abab'a2ba'b'aba'b'aba2b'a'2ba'b'ab'a'bab2ab'ab'a'b'2a'2
b'a'2b'2a'b'a'2bab'a'bab'a'ba2b'a5ba'b'a'ba2b'a4ba'2b'a3bab'a'4ba'5b'
a7bab'a'2ba'b'a6bab'a'2ba'b'2a'ba'b2a'b'ab'a'b'2a'b3a'b'5a'2b


----------



## rokicki (Mar 22, 2010)

*Wow*

Very nice! I wonder if Lucas would care to actually try one of those
out by hand, and verify it?

Now all we need is a generic optimal solver in *any* metric (that takes the generators to use) and we can see what these "solutions" might look like
in a shortest form.

They'd be pretty long, I suspect, even in optimal form.


----------



## qqwref (Mar 22, 2010)

Someone should make a simulator that lets the player solve the cube using these two generators (and their inverses). It would be entertaining to see if anyone can do it.


----------



## Lucas Garron (Mar 22, 2010)

rokicki said:


> Very nice! I wonder if Lucas would care to actually try one of those
> out by hand, and verify it?
> 
> Now all we need is a generic optimal solver in *any* metric (that takes the generators to use) and we can see what these "solutions" might look like
> in a shortest form.



1) Have you heard of ksolve?

2) Did you read the part of my post where I estimate alg length? I don't think the optimal solution would be that bad.

By hand, this would take a while:


Spoiler



U turn:
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R B L F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' F' U' R B L U F U F U F U F U F U F U F U F U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' F' U' R B L U F U F L' B' R' U F R B L U F L' B' R' U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' U F L' B' R' F' U' R B L U F R B L R B L U F L' B' R' U F L' B' R' F' U' L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F R B L U F L' B' R' F' U' F' U' F' U' R B L F' U' L' B' R' U F U F R B L F' U' F' U' F' U' L' B' R' U F U F R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F L' B' R' L' B' R' U F R B L U F L' B' R' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' F' U' F' U' L' B' R' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' F' U' F' U' F' U' R B L F' U' F' U' L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' L' B' R' L' B' R' F' U' R B L F' U' R B L R B L R B L R B L F' U' L' B' R' F' U' L' B' R' L' B' R' L' B' R' L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F U F U F U F U F U F U F R B L U F L' B' R' U F U F U F U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' F' U' R B L F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' U F U F R B L U F L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F U F U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L F' U' L' B' R' U F U F U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F R B L U F L' B' R' U F R B L F' U' F' U' L' B' R' U F R B L U F U F U F L' B' R' U F R B L U F U F L' B' R' F' U' R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F U F L' B' R' U F U F R B L U F U F U F L' B' R' F' U' R B L F' U' L' B' R' U F U F U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L U F L' B' R' U F R B L U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' U F U F L' B' R' U F U F U F U F U F R B L R B L R B L U F L' B' R' U F L' B' R' U F L' B' R' L' B' R' F' U' L' B' R' F' U' L' B' R' F' U' L' B' R' L' B' R' F' U' L' B' R' U F U F R B L U F L' B' R' F' U' R B L U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' R B L U F U F U F U F L' B' R' U F U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F U F U F U F U F L' B' R' U F U F U F U F R B L U F L' B' R' U F U F U F U F R B L U F U F U F L' B' R' F' U' F' U' F' U' F' U' F' U' R B L U F U F L' B' R' U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' U F U F R B L U F L' B' R' U F R B L F' U' L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F L' B' R' F' U' R B L U F U F L' B' R' U F U F U F U F U F R B L U F U F L' B' R' U F R B L U F U F L' B' R' U F U F R B L F' U' L' B' R' F' U' F' U' R B L F' U' F' U' F' U' L' B' R' U F U F U F U F U F R B L U F U F U F L' B' R' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' F' U' F' U' F' U' F' U' L' B' R' U F U F U F U F U F U F U F U F U F R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L U F L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' R B L U F U F U F U F U F L' B' R' F' U' F' U' R B L U F L' B' R' U F U F R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L U F L' B' R' U F R B L U F U F U F U F L' B' R' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' F' U' F' U' F' U' F' U' F' U' F' U' R B L F' U' L' B' R' U F R B L U F U F L' B' R' F' U' R B L F' U' L' B' R' F' U' R B L F' U' F' U' L' B' R' U F R B L F' U' L' B' R' F' U'



However:





Bruce, you're awesome. 

Edit: Expanded lengths of Bruce's algs: {1804, 1871, 1771, 860, 1497}
I'm willing to bet there are algs under 100 for each.


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## rokicki (Mar 23, 2010)

*ksolve*



Lucas Garron said:


> rokicki said:
> 
> 
> > 1) Have you heard of ksolve?
> ...


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## Stefan (Mar 23, 2010)

ksolve link can be found in the wiki 
http://www.speedsolving.com/wiki/index.php/PC_Software

Direct link: http://www.svekub.se/files/ksolve.zip


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## cuBerBruce (Mar 23, 2010)

I played around with some other generator pairs. Apparently GAP is able to find more efficient solutions if one of the generators has low order. So in the spoiler below, I include what GAP produced for generators a = U D' and b = L F R. This time I added some spaces for additional readability and to prevent forum from inserting spaces at odd places. I also give all 6 face turns this time; with R last because for some reason it was quite a bit longer than the other 5 (at least in terms of text length).



Spoiler



a = U D'
b = L F R

U =
a' bab' abab3 a' b'2 a2 b' a' b' a' b' a' ba
b3 abababab' a' b'2 a' b' a' b' a' bab'3 abab' 
a' b3 a' b' abab'3 ab'2 aba' b' a'2 b3 a'2 b' 
a' bab4 a' b' a' bab' ababab' aba' b' a' b2
a' b'2 a' b' a' b10 abab' a' b'10 aba' b10 a2 b2 a
b'3 a' b'10 ab3 a' b'2 a' b' a' b10 aba' b2 ab'3 a' 
b'20 ab3 a' b'2 ab' a' b2 aba' b' a' b' a' b' a' 
b'10 abababab' a' b'12 ab21 a' b' a' b' a' 
b'10 ababab'10 a' b' a' b' a' b10 ababab' a' 

D =
a2 bab' abab3 a' b'2 a2 b' a' b' a' b' a' ba
b3 abababab' a' b'2 a' b' a' b' a' bab'3 abab' 
a' b3 a' b' abab'3 ab'2 aba' b' a'2 b3 a'2 b' 
a' bab4 a' b' a' bab' ababab' aba' b' a' b2
a' b'2 a' b' a' b10 abab' a' b'10 aba' b10 a2 b2 a
b'3 a' b'10 ab3 a' b'2 a' b' a' b10 aba' b2 ab'3 a' 
b'20 ab3 a' b'2 ab' a' b2 aba' b' a' b' a' b' a' 
b'10 abababab' a' b'12 ab21 a' b' a' b' a' 
b'10 ababab'10 a' b' a' b' a' b10 ababab' a' 

L =
aba' b' a' babab' a2 b2 a'2 b3 a'2 b' a' b'2 ab
a' b' a'2 b' a' bab2 aba' b'3 a'2 b'2 aba' b' 
a' bab4 aba' b' a' ba' b' aba2 bab' a' b3 ab'2
a'2 b'2 a2 bab' a' ba2 b' ab'2 ab' a2 ba' b3 a' b' 
a' b2 a' b' a' bab' ababab' aba' b' a' b2 a' 
b'2 a' b' a' b10 abab10 ab2 ab'3 a' b'10 ab3 a' b'2
a' b' a' b10 ab'7 a' b10 ab'3 a' b20 ab'9 a' b2 ab'10
a' b'2 ab'10 a' b' a' b' a' b10 ababab'10 a' b' 
a' b' a' b'10 ababab9 a' 

F =
ba' b' a' bab' a'2 ba' b'2 a' ba'2 b'5 aba' b' 
a' babab' a2 b2 a'2 b3 a'2 b' a' b'2 aba' b' a'2
b' a' bab2 aba' b'3 a'2 b'2 ab' a' b' a' bab6 ab
a' b' a'2 bab' a' ba'2 b' a'2 b2 aba' b' a' b'2 ab
a' b' a'2 b'10 a2 ba2 b2 ab'3 a' b'10 ab3 a' b'2 a' b9
a' b10 abab2 ab'3 a' b10 ab3 a' b'2 a' b' a' b20 ab
a' b' a' b' a' b' a' bab'10 a' b' abababab'20
a' b2 ab'10 a' b'3 a' b' a' b'10 ababab10 a' b' a' 
b' a' b10 ababab'10 a' b' a' b' a' b10 ababab'10
a' b' a' b' a' b'10 ababab9 a' 

B =
b'2 a'2 b4 a' b2 a' b' abab2 a' b'3 a' b' a' 
b' aba2 b5 a' b5 a' bab3 ab'3 a' b'4 ab'2 ab' a'2 b
a' b2 a' ba'2 b' aba'2 b' a'2 b10 a2 ba' b' a'2 b10
a2 ba'2 b' a' b'20 abab2 ab'3 a' b'10 ab3 a' b'2 a' 
b10 a' b2 ab10 a' b'2 ab9 a' b'10 aba' b2 ab10 a' 
b'2 ab20 a' b' a' b' a' b10 ababab'10 a' b' a' b' 
a' b'10 ababab9 a' 

R =
aba' b' a' b' a' b10 ababab'10 a' b' a' b' a' 
b'10 ababab9 a' b'10 aba' b' a' b' a' b10 ababa
b'11 a' b10 aba' b2 ab10 a' b'2 ab20 a' b' a' b' a' 
b' a' bab10 a' b' abababab' a' b'10 aba' b2 a
b'3 a' b10 ab3 a' b'2 ab' a' b'10 abab' a' b2 ab'2
a'2 b'2 a2 bab' a' b2 ab'2 a'2 b'2 aba' b10 ab' ab3 a
b'2 abab' a' ba' b' a' b' a' ba' b' abab'2 aba
b' a' ba2 b' ab'2 ab' a2 ba' b3 a' b' a'2 b2 a2 b2
a' b'3 aba' b' a'2 b' a' bab' abab' a' b'4 a' 
b' abab' a' b2 a2 b3 ab' a' b'2 a' b' aba2 bab' 
a' b2 aba2 b' a' b2 abab' a' b2 aba' b' a' b' 
a' b' a' bab'2 a' b' abab2 a' b' abab'2 aba' 
b' a' b2 ab3 ab'2 a' b' a' b'3 ab3 a' b'3 a' bab
a' b' a' b' ab'2 ab3 a'2 b' a'2 b' a' ba' b' a' 
b2 a' b2 a'2 b3 ab' a' b3 abab' a' b'2 aba' b' a' 
b2 aba' b' a' b' a' b' a' bab'2 a' b' abab2 ab
a' bab' aba' b'2 a' b'3 ab'2 a2 b'2 a'2 ba' b' a' 
b' abab'6 ab'5 a'


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## rokicki (Apr 11, 2010)

If a=UF and b=RBL, then:

U=a'a'a'b'b'ab'a'a'a'bbaabba'b'a'bba'a'a'baaaaaaba'bab'ab'a'a'
F=aaba'ba'b'ab'a'a'a'a'a'a'b'aaab'b'abab'b'a'a'b'b'aaaba'bbaaaa
R=b'a'babba'b'a'b'a'a'ba'b'a'a'bab'abab'a'a'b'abbbbabababab
D=a'a'baaab'b'aba'a'bab'b'abbbab'a'b'aab'b'a'baaaab'b'aabb
B=b'a'ba'a'a'a'bab'a'a'b'b'a'b'aaba'a'bba'bbaaaaba'ba'bba'b'
L=b'a'a'b'aba'a'b'b'b'a'b'a'b'a'b'a'b'b'b'aaaaaba'b'ababa'b'aba'b

These are shortest solutions. The lengths are 40, 41, 40, 40,
38, and 39 respectively.


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## mrCage (Apr 12, 2010)

This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator 

Per


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## Tim Reynolds (Apr 12, 2010)

mrCage said:


> This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator
> 
> Per



You need at least six generators to rotate all the centers (five could only generate 4^5=2^10 center positions, but there's 4^6/2=2^11 total). So {U,D,F,B,R,L} is a minimal set of generators.


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## rokicki (Apr 12, 2010)

mrCage said:


> This is mostly old stuff. But probably new to many people anyway.



Can you give any citations (beyond those already listed)? I see several
new results in this thread, that I have never seen before.



mrCage said:


> Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator
> 
> Per



I don't know the answer to this, but I'd be surprised if the given generator pairs do not already generate the supergroup that includes center rotations.


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## Lucas Garron (Apr 12, 2010)

rokicki said:


> I don't know the answer to this, but I'd be surprised if the given generator pairs do not already generate the supergroup that includes center rotations.



'cept they don't? <UF, RBL> only gives 16 configurations of center orientations, right?


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## rokicki (Apr 12, 2010)

Lucas Garron said:


> rokicki said:
> 
> 
> > I don't know the answer to this, but I'd be surprised if the given generator pairs do not already generate the supergroup that includes center rotations.
> ...



Very good, I totally missed the obvious. Thanks, Lucas and Tim!


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## JBCM627 (Apr 12, 2010)

Tim Reynolds said:


> mrCage said:
> 
> 
> > This is mostly old stuff. But probably new to many people anyway. Simple question/challenge. How about generating the cube supergroup,i.e. all the face center rotations. How many more generators would be needed? I don't want cube rotation as a generator
> ...



{Rw, Uw, Fw}? But I guess those (and slice turns) would count as cube rotations?


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## cuBerBruce (Apr 12, 2010)

JBCM627 said:


> Tim Reynolds said:
> 
> 
> > mrCage said:
> ...



Haha. That only generates the "keychain" supercube group. And yes, it goes outside the <U,D,L,R,F,B> supercube group. Rw, Uw, and Fw are not in the <U,D,L,R,F,B> group.

EDIT: OK, I realize the elements of the "keychain" group (whether regular cube or supercube) has half the number of elements of the full group. Since a corner cubie serves as a fixed reference (like the fixed centers of <U,D,L,R,F,B>), I believe there is still only a 1:1 mapping between keychain group elements and their corresponding "cube positions." So mapping <Rw, Uw, Fw> elements to cube positions, half of the positions of the cube are generated. So it comes very close to generating all positions, just a factor of 2 short for either the regular cube or supercube.

EDIT2: OK, from using GAP, I conclude <Rw, Uw, FwU> generates all supercube positions. <Rw, Uw, FwD> generates all supercube positions in all 24 cube orientations (the full <U, x, y> group). Of course these generators can be viewed as having implicit cube rotations.


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## rokicki (Apr 13, 2010)

I've found a pair of generators of order 2 and 4 that generate the whole cube.

a=U2F1B1D2F3B3D1
b=F1B2R1L1U2F1

These are the smallest orders possible that generate the whole cube.

The count of positions at levels 1, 2, ... goes 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 ... which I suspect many of you will recognize as the Fibonacci sequence.


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## Stefan (Apr 13, 2010)

rokicki said:


> The count of positions at levels 1, 2, ... goes 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711 ... which I suspect many of you will recognize as the Fibonacci sequence.



Neat! How far does this go?


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## rokicki (Apr 13, 2010)

*Fib through level 32*

It continues through level 32; 33 is the first level where it doesn't hold. Other pairs of generators of orders 2 and 4 may take it further; I'll experiment with some of those.

At level 0 total 1 thislev 1
At level 1 total 4 thislev 3
At level 2 total 9 thislev 5
At level 3 total 17 thislev 8
At level 4 total 30 thislev 13
At level 5 total 51 thislev 21
At level 6 total 85 thislev 34
At level 7 total 140 thislev 55
At level 8 total 229 thislev 89
At level 9 total 373 thislev 144
At level 10 total 606 thislev 233
At level 11 total 983 thislev 377
At level 12 total 1593 thislev 610
At level 13 total 2580 thislev 987
At level 14 total 4177 thislev 1597
At level 15 total 6761 thislev 2584
At level 16 total 10942 thislev 4181
At level 17 total 17707 thislev 6765
At level 18 total 28653 thislev 10946
At level 19 total 46364 thislev 17711
At level 20 total 75021 thislev 28657
At level 21 total 121389 thislev 46368
At level 22 total 196414 thislev 75025
At level 23 total 317807 thislev 121393
At level 24 total 514225 thislev 196418
At level 25 total 832036 thislev 317811
At level 26 total 1346265 thislev 514229
At level 27 total 2178305 thislev 832040
At level 28 total 3524574 thislev 1346269
At level 29 total 5702883 thislev 2178309
At level 30 total 9227461 thislev 3524578
At level 31 total 14930348 thislev 5702887
At level 32 total 24157813 thislev 9227465
At level 33 total 39088158 thislev 14930345 (oops, not a Fibonacci number)


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## cuBerBruce (Apr 13, 2010)

According to my GAP calculations, a minimal generator set for generating all 2,125,922,464,947,725,402,112,000 elements of the <U, x, y> supercube group, using only (single-layer) face turns and double-layer turns in the generators, would be <Uw, LUwRw>.

As for just generating all supercube *positions* (in at least 1 cube orientation each), I believe <Uw, DRw> suffices.

Can anyone confirm these assertions?


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## mrCage (Apr 14, 2010)

Thanf for intersting contributions. But all of them seem to have implicit cube rotations (wide turns). Who can find "pure" generators? Will 3 generators suffice? That's my guess anyway

Per


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## Stefan (Apr 14, 2010)

mrCage said:


> Thanf for intersting contributions. But all of them seem to have implicit cube rotations (wide turns). Who can find "pure" generators? Will 3 generators suffice?



Was Tim's answer not sufficient?


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## rokicki (Apr 15, 2010)

Here's another interesting (2,4) generator pair. This one extends the Fibonacci sequence out to level 41.

U2R3L3D2R1L1D1
R1L2F3B3U2R1

0 1
1 3
2 5
3 8
4 13
5 21
6 34
7 55
8 89
9 144
10 233
11 377
12 610
13 987
14 1597
15 2584
16 4181
17 6765
18 10946
19 17711
20 28657
21 46368
22 75025
23 121393
24 196418
25 317811
26 514229
27 832040
28 1346269
29 2178309
30 3524578
31 5702887
32 9227465
33 14930352
34 24157817
35 39088169
36 63245986
37 102334155
38 165580141
39 267914296
40 433494437
41 701408733


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## mrCage (Apr 15, 2010)

StefanPochmann said:


> mrCage said:
> 
> 
> > Thanf for intersting contributions. But all of them seem to have implicit cube rotations (wide turns). Who can find "pure" generators? Will 3 generators suffice?
> ...


 
Yj

The six face turns answer???? That's trivial. hmmm


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## Stefan (Apr 16, 2010)

mrCage said:


> The six face turns answer???? That's trivial.



If it's oh so trivial, then why did you ask in the first place?


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## mrCage (Apr 16, 2010)

StefanPochmann said:


> mrCage said:
> 
> 
> > The six face turns answer???? That's trivial.
> ...


 
I wanted to know if something better is possible (fewer generators). Duh!

Per


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## jaap (Apr 16, 2010)

mrCage said:


> StefanPochmann said:
> 
> 
> > mrCage said:
> ...



What Tim said is a proof that fewer than 6 generators is not possible for the supergroup.

Let me explain it slightly more than Tim did. Consider the effect of your generators on just the face centres. In other words, take the cube apart, and only use the spider. Your generators generate the supergroup, and so must also generate all 4^6 possible positions of the centres alone. Each generator has at most order 4 when considered as acting only on the centres. The order you apply them also doesn't matter (they form an Abelian group). Therefore you can get at most 4^g positions with g generators, and so you need at least 6. This is why {U,D,R,L,F,B} is minimal.


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## Stefan (Apr 16, 2010)

Also, Tim even explicitly said _"You *need at least six* generators"_ and _"{U,D,F,B,R,L} *is a minimal set*"_ (*). I don't understand how that was not clear. And then Tom and Lucas even went through it again. Btw, I didn't see the solution myself and didn't think much about it, then I saw Tim's answer and thought "darn, of course!".

(*) ikmcmtnstwbidthpuiptnalswc


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## Tim Reynolds (Apr 16, 2010)

Hmm, I said there were only 4^6/2 center positions (that is, that's how many center positions there are for any given (non-super)cube position). I realize now that that /2 doesn't matter. But undercounting in that direction doesn't mess anything up--my proof still effectively works.


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## rokicki (Apr 16, 2010)

*Generators of order 2 and 3?*

Until Jaap posted his explanation, I don't think there was a clear proof of
exactly why six generators were minimal. The key is the effect on the
face centers is Abelian. It may have been clear to some of us, but I bet
not to all.

In any case, moving on to more interesting things---can anyone shed
light on why there is no pair of generators of orders 2 and 3 that
generate the cube? I've been puzzling over this, and not figured it out,
but perhaps it is pretty obvious to someone else out there.


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## qqwref (Apr 17, 2010)

I don't know, but I can prove something about any such generators (if they exist, which I guess they don't) by considering corner permutation. A generator of order 3 must either be one 3-cycle or two 3-cycles; a generator of order 2 must be some number of 2-cycles. The goal is to generate all possible corner permutations. We can't do this if the order-3 generator has just one corner 3-cycle because then the orbit of a piece in that cycle can be at most 6 corners (and it should be 8). So the order-3 generator would have to be two corner 3-cycles (ignoring orientation).

Maybe this will lead to a direct proof by contradiction?


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## rokicki (Apr 17, 2010)

*orders 2 and 3 cannot generate S_8*

I think you're right. I've found a paper by Miller from 1901 that says
generators of orders 2 and 3 cannot generate S_8 so I think that
covers it. I haven't read the Miller paper close enough to finish
it off, but I think you're on the right track.

The 3-generator clearly has to be a pair of three cycles, so we
can call it (1,2,3)(4,5,6) without loss of generalization. The
2-generator must be odd (if it were even, we could only generate
even perms). If the 2-generator was only (a,b), then at least
one piece could not have a full orbit. It can't be (a,b)(c,d)(7,8)
because then the 7 wouldn't have a full orbit. So it must be
(a,b)(c,7)(e,8). The a must be from 1..3 and the c must be
from 4..6, else you couldn't move pieces between these orbits,
so call it (1,4)(c,7)(e,8). There's only a few possibilities left. Who
can finish this off without enumerating them explicitly?


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## qqwref (Apr 17, 2010)

rokicki said:


> So it must be
> (a,b)(c,7)(e,8). The a must be from 1..3 and the *b* must be
> from 4..6, else you couldn't move pieces between these orbits,
> so call it (1,4)(c,7)(e,8). There's only a few possibilities left. Who
> can finish this off without enumerating them explicitly?



Hm, well, c and e cannot be 1, 4, 7, 8, so they are in {2, 3, 5, 6}; and since the positions of 7 and 8 can be swapped WLOG we can assume that c < e. So that leaves just 6 possibilities.

Considering that our other cycle is (1,2,3)(4,5,6) we notice that if (c,e) = (2,5) or (3,6) then the pair (1,4) remains "across" from each other no matter what. We can exchange their positions and/or place them into (7,8) but it is not possible to place them in the same 3-cycle.

I'm not sure about proving the other four cases, but I'm sure GAP will happily tell you that the order of the generated group is less than 8!.


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## mrCage (Apr 17, 2010)

jaap said:


> mrCage said:
> 
> 
> > StefanPochmann said:
> ...


 
That's more clear. Now it seems obvious. A bit sad though that we really need so many "proper" generators for the cube supergroup

Per


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## cuBerBruce (Oct 14, 2011)

Earlier in this thread, I mentioned a generator pair that generates the group consisting of all supercube positions in all 24 orientations of the cube. This pair I believed to minimal if you only use single-layer and double-layer turns to compose the generators. This pair was <Uw, LUwRw>.

If you allow the use of whole cube rotations, then the above pair is not minimal. I have found that <x, Uz> is sufficient to generate this group.

It is easy to see that this indeed generates the whole group. It suffices to find sequences in <x, Uz> that are equivalent to U, x, and y. This is straightforward.

x = x
y = (Uz) x (Uz)'
U = (Uz) z' = (Uz) yxy' = (Uz)2 x (Uz)' x (Uz) x' (Uz)'


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