# 5 gen God's number?



## guysensei1 (Nov 19, 2014)

Title sort of explains it. I know that every normally reachable state on a cube can also be reached when only turning 5 faces of the cube. What is God's number when such a restriction is imposed?


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## martinss (Nov 19, 2014)

guysensei1 said:


> I know that every normally reachable state on a cube can also be reached when only turning 5 faces of the cube. What is God's number when such a restriction is imposed?



God's number is 20. So on a 20-moves-algorithm, we can find 4 lettres (R, L, D, U, B, F) that accurs only 3 times (or less) because int(20/6)=3 and replace one of them by a 13-moves-algorithm(D = R L F2 B2 L' R' U R L B2 F2 L' R'.).
20-3+13+13+13=56
So, God's number when such a restriction is imposed is less or equal than 56.

And -3 because you can always find a 13-moves-algorithm that cancel the next or the previous move (just replace R, F, L or B (don't choose to replace U) and do y, y2 or y' setup before the 13-moves-algorithm (for instance, D = R L F2 B2 L' R' U R L B2 F2 L' R' = y R L F2 B2 L' R' U R L B2 F2 L' R' y' = F B R2 L2 F' B' U B F L2 R2 F' B'  ) ).
So, God's number when such a restriction is imposed is* less or equal than 53.* (less than 54)


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## cmhardw (Nov 19, 2014)

martinss said:


> God's number is 20. So on a 20-moves-algorithm, we can find a lettre (R, L, D, U, B, F) that accurs only 3 times (or less) because int(20/6)=3 and replace it by a 13-moves-algorithm(D = R L F2 B2 L' R' U R L B2 F2 L' R'.).
> 20-3+13+13+13=56
> So, God's number when such a restriction is imposed is *less than 56*.



I like your approach here, but I don't immediately see why God's number in such a case is _less than_ and not _less than or equal to_ 56. Can you explain your reasoning for why you said less than 56?


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## martinss (Nov 19, 2014)

cmhardw said:


> I like your approach here, but I don't immediately see why God's number in such a case is _less than_ and not _less than or equal to_ 56. Can you explain your reasoning for why you said less than 56?



I edited my previous post !


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## TMOY (Nov 19, 2014)

cmhardw said:


> I like your approach here, but I don't immediately see why God's number in such a case is _less than_ and not _less than or equal to_ 56. Can you explain your reasoning for why you said less than 56?



IMHO it's only a language issue. In French, we often say "less than" when we mean "less than or equal to"; if we want to insist on the fact that it's actually less, we say "strictly less than".


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## cuBerBruce (Nov 19, 2014)

I think it needs to be clarified if we are talking about a specific 5-gen group, such as <U, D, L, R, F>, or if we are allowed to pick 5 out of { U, D, L, R, F, B } for each position.


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## GuRoux (Nov 19, 2014)

cuBerBruce said:


> I think it needs to be clarified if we are talking about a specific 5-gen group, such as <U, D, L, R, F>, or if we are allowed to pick 5 out of { U, D, L, R, F, B } for each position.



i don't think that actauly matters.


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## irontwig (Nov 19, 2014)

GuRoux said:


> i don't think that actauly matters.



Surely that makes a huge difference, with the former not allowing for as much reduction of cases due to symmetry?


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## cuBerBruce (Nov 19, 2014)

GuRoux said:


> i don't think that actauly matters.



Sure it does. B, for example, takes 13 moves to solve using <U, D, L, R, F>, but only 1 move if you're allowed to choose the 5 generators.


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## cmhardw (Nov 19, 2014)

GuRoux said:


> i don't think that actauly matters.



How so?

I can see the possibility of a position that is difficult to solve without a D layer turn say, but if you allow a D layer turn you could reduce the overall length of the solution. Do you have a way to argue that such a possibility could be written off as a trivial case?

--edit--
Ninja'd by Bruce


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## GuRoux (Nov 19, 2014)

cmhardw said:


> How so?
> 
> I can see the possibility of a position that is difficult to solve without a D layer turn say, but if you allow a D layer turn you could reduce the overall length of the solution. Do you have a way to argue that such a possibility could be written off as a trivial case?
> 
> ...



can't you just rotate that scramble that's hard on no D layer use to the what ever layer you can't use?

Let's say one of the hardest scrambles for <U,R,L,B,F> is scramble "A." For <D,R,L,B,F>, a scramble that would bring you to the same position as "x2 A x2" would be just as hard.


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## cmhardw (Nov 19, 2014)

GuRoux said:


> can't you just rotate that scramble that's hard on no D layer use to the what ever layer you can't use?



Situation 1) If you are allowed to pick which 5-generator set you would like to use on any scramble, then you can choose whichever side you would like to not turn after having seen the scramble.

Situation 2) If you must pick which side is not to be turned _before_ applying the scramble as well as the solution, then this specific side is always the one that is restricted.

I interpret this post by you to be arguing that God's number is the same under situation 1 as it is under situation 2, and I am disagreeing with that argument.

--edit--



GuRoux said:


> Let's say one of the hardest scrambles for <U,R,L,B,F> is scramble "A." For <D,R,L,B,F>, a scramble that would bring you to the same position as "x2 A x2" would be just as hard.



That argument seems solid to me, can anyone come up with a counterpoint to this?


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## GuRoux (Nov 19, 2014)

cmhardw said:


> Situation 1) If you are allowed to pick which 5-generator set you would like to use on any scramble, then you can choose whichever side you would like to not turn after having seen the scramble.
> 
> Situation 2) If you must pick which side is not to be turned _before_ applying the scramble as well as the solution, then this specific side is always the one that is restricted.
> 
> ...



Oh, misunderstood the claim. i thought it was that God's number is different for <D,R,L,B,F> and <U,R,L,B,F>. i was just saying it's the same.


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## cmhardw (Nov 19, 2014)

GuRoux said:


> Oh, misunderstood the claim. i thought it was that God's number is different for <D,R,L,B,F> and <U,R,L,B,F>. i was just saying it's the same.



I may be getting confused on this too. Looking back at the claims I think my Situation 2 (a fixed non-turnable side) would have a higher God's number than Situation 1 (flexible non-turnable side). This is because I could choose a scramble for a Situation 2 (fixed non-turnable side) sceanario that is particular bad for the non-turnable side I have chosen, and I would be forced to solve the scramble without turning that side.


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## qqwref (Nov 19, 2014)

Of course "use only these 5 sides" could potentially require more moves than "use any set of 5 sides" - you are basically allowing the solver to be color neutral, where they have the freedom to take the easiest of 6 problems, and this can save moves just like doing a color-neutral cross can. But it's also possible that it doesn't save moves... we can't really know yet.

Does anyone have a fast optimal solver that could be relatively easily rewritten to not to D moves? You could make a qcube definition file for this pretty easily but I don't think it could match the performance of a 3x3x3-specialized program.


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## cuBerBruce (Nov 19, 2014)

qqwref said:


> Does anyone have a fast optimal solver that could be relatively easily rewritten to not to D moves? You could make a qcube definition file for this pretty easily but I don't think it could match the performance of a 3x3x3-specialized program.



Cube Explorer can do 5-face (or even 2-face) solving. You need to gray out an edge (or a corner) to get the "incomplete cubes" dialog box to appear, and then you can select not to use some combination of B, D, L, and F.

I note that all the optimal superflip maneuvers (in FTM) use all 6 faces, so 21 is a lower bound.


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## qqwref (Nov 19, 2014)

Maybe it's just my computer, but it takes a while to find solutions that way, even with one or two faces turned off. That and ACube both work but I know there are blazingly fast optimal solvers out there, and I was hoping one of them could come in handy. Perhaps with a clever modification to Kociemba you could do an entire coset.


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## guysensei1 (Nov 20, 2014)

Has anyone found the 5 gen optimal solution for superflip? That should give us a lower bound right?

EDIT: I tried this last night and cubeexplorer took 2.5 hours and it only managed to search depth-16 moves so I gave up.


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## cuBerBruce (Nov 20, 2014)

guysensei1 said:


> Has anyone found the 5 gen optimal solution for superflip? That should give us a lower bound right?
> 
> EDIT: I tried this last night and cubeexplorer took 2.5 hours and it only managed to search depth-16 moves so I gave up.



I concur that the "incomplete cubes" solver in Cube Explorer is much slower than its regular optimal solver. Obviously, it is not creating a "huge" pruning table for use in this mode, or even using the existing huge table when no face restrictions are imposed.

Yes, whatever superflip requires, will be a lower bound. Because of its symmetry, it doesn't matter what orientation of the cube we use (or which face we make restricted). As I said in my previous post, we have that 5-gen superflip gives us a lower bound of 21, but the actual value could be higher, and thus could provide a better lower bound.


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## goodatthis (Nov 20, 2014)

Since this is gods number, so the maximum number of moves, shouldn't we pick the 5-gen group that requires the most amount of moves to solve a specific case?


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## guysensei1 (Nov 20, 2014)

cuBerBruce said:


> I concur that the "incomplete cubes" solver in Cube Explorer is much slower than its regular optimal solver. Obviously, it is not creating a "huge" pruning table for use in this mode, or even using the existing huge table when no face restrictions are imposed.
> 
> Yes, whatever superflip requires, will be a lower bound. Because of its symmetry, it doesn't matter what orientation of the cube we use (or which face we make restricted). As I said in my previous post, we have that 5-gen superflip gives us a lower bound of 21, but the actual value could be higher, and thus could provide a better lower bound.



How did you know that all optimal superflips require the use of all 6 faces?


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## cuBerBruce (Nov 20, 2014)

guysensei1 said:


> How did you know that all optimal superflips require the use of all 6 faces?



Mike Reid determined all optimal superflip maneuvers a long time ago.

These are the only minimal maneuvers for superflip in FTM, up to cyclic shifting, inversion, and conjugation by symmetries of the cube. They both require all six faces.

U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2
U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L

EDIT: I've found a 27-move maneuver for superflip. It's probably not 5-gen optimal.

U R2 F B R B2 R U2 L B2 R' U2 B2 U F B L2 F R2 F' B' U' R L F2 B2 R2

EDIT 2: I've now found a 23-move maneuver for superflip. Optimal 5-gen superflip is now known to be in the range of 21-23.

U2 R U F R2 F B2 L' U F2 R2 B R L' U' L2 B2 R2 B' R2 F' U2 R2

EDIT 3: I've now ruled out any 21f 5-gen solution for superflip. 22 is a lower bound for the 5-gen God's number in FTM.

I am also very doubtful of a 22f solution for superflip. I should have that ruled out soon.


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## cuBerBruce (Nov 26, 2014)

Using Rokicki's RubSolv solver - modified so that it does not use D-layer moves - I have verified that there are no 5-gen solutions for superflip with less than 23 face turns.

So 23 face turns is a lower bound for God's number for solving the cube using only 5 face layers. This lower bound is valid whether or not you're allowed to pick the restricted face separately for each position.

I have found the following 23-move solutions for superflip (unique up to conjugation by cube symmetries, inversion, and cyclic shift):

U2 R U F R2 F B2 L' U F2 R2 B R L' U' L2 B2 R2 B' R2 F' U2 R2
U2 R U F' R2 B U F2 L' U' B L' U' R' L2 F' U' L2 F2 L' B U2 R
U2 R F U F R2 U' R B2 U' F B' L2 B2 R' U' F' L' B' L2 F' B R2
U2 R F' U' F2 L B U B2 R2 U L' F R' L2 U B2 U2 F2 R' L' F L2
R L' U R2 B2 L2 F' B R' F2 U2 R2 L' U F2 R2 L2 B2 U' F' L2 F B
R L' U R2 B2 L2 F' B R' F2 U2 R2 L' U B2 R2 L2 F2 U' F' L2 F B
R L' U R' B2 U' F' R B R L' U' B2 R2 L' F2 U L2 B R2 U' R' F2
R L' U2 F R2 U' F2 R2 L2 B2 U B2 U2 R' F2 B2 L' F' B U2 R2 U' F2
R L' U2 F R2 U' B2 R2 L2 F2 U B2 U2 R' F2 B2 L' F' B U2 R2 U' F2

There could be additional 23f* solutions.


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## qqwref (Nov 26, 2014)

Ooh, cool! A lower bound of 23 is interesting, definitely higher than I expected. Would you be willing to try the same thing (a) in qtm, and (b) with the superflip + 4spot (in 2 orientations) and some asymmetric 20f* position (in 6 orientations)?


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## cuBerBruce (Nov 26, 2014)

qqwref said:


> Ooh, cool! A lower bound of 23 is interesting, definitely higher than I expected. Would you be willing to try the same thing (a) in qtm, and (b) with the superflip + 4spot (in 2 orientations) and some asymmetric 20f* position (in 6 orientations)?



My "database" of 26q* maneuvers has several that are 5-gen. For example:

U U F D D L U' F D' R' L F L' U F' U D' R D R' L' D R F R L

However, all such maneuvers for the superflip plus "equatorial" 4-spot require both U and D layer turns (according to my "database").

Thus, some orientation of superflip + 4-spot will require at least 28q if you're not allowed to pick the restricted face (assuming my "database" is not missing any maneuvers).

As for finding a 5-gen (no D moves) 28q* for the superflip + "equatorial 4-spot position, I can give it a try, as well as your other suggestions. I'm also attaching a ZIP file containing my FTM and QTM versions of rokicki's solver (executables) so you or others can also do so.

EDIT: Some more maneuvers without D-layer moves:

superflip:
U L' U' F' R B L' B L' F' U' R' F B U R U R U L B' L F' R' L U (26q)

"equatorial" 4-spot composed with superflip 

U U F B' R B' U F L' B R' U R L' F R L' U F L F' R B U' R L' F R' (28q)
U U F B' R' F U' B' L F' R U' R' L B' R' L U' B' L' B R' F' U R' L B' R (28q)


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