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 Post subject: The Pure Superflip Quest
PostPosted: Sat May 23, 2009 5:37 pm 
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Hi there,

I recently computed all optimal solutions of the Superflip pattern and was looking for algorithms where no centers are moved.
Unfortunately all 4416 optimal algorithms with the length of 20 moves are twisting either 4 or 5 of the centers.

I figured out that such an optimal solution in ftm metric must be within the range of 23 and 24 moves.
Actually, the search for optimal solutions with more then 20 moves may take a very long time, since with every new step the search space becomes 18 times larger than before.
However, todays home computers may be busy for months or even years when the expected solution has about 23 or 24 moves.

I wonder if such optimal 'Clean Superflip' algorithms in ftm and qtm metric already have been found?


These are the algorithms I found so far:
B2 L2 B2 L' D2 F2 R' D B U L2 F2 R2 L' D' L F D' L D' R' B' U' F' (24 ftm, 32 qtm)
L2 D2 L2 D' B2 R2 U' B L F D2 R2 U2 D' B' D R B' D B' U' L' F' R' (24 ftm, 32 qtm)
B2 L2 B2 D L2 F2 R2 U' R2 L2 B' U2 L2 U2 F' R L F' D L' U D' B D' R' (25 ftm, 36 qtm)
R L' F R2 D2 B U D L F2 B2 R' F B' D2 B2 L2 U2 L2 F2 D R2 F2 U D2 (25 ftm, 38 qtm)

Can someone confirm that 24 moves in ftm are optimal?


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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sat May 23, 2009 6:25 pm 
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Unfortunately, I don't think I know anything that I could help you. :(
CE gave me R L' F B2 R2 U2 B' U' D' R' U2 D2 L F B' D2 B2 R2 D2 R2 F2 D' R2 B2 U (25f) quickly, but it's staying there for a while.

I'd like to throw in another question: what do we get if we consider a 180-degree rotation a "center flip" and ask for a "full superflip" of all three types of pieces?
CE gave me U F D R D B' U' D' F' L B2 D' L B' D2 R L2 U2 F U2 F U2 B2 L2 (24f).
CE gave 24 quite quickly, but I have no idea if it'll hit 23...

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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sat May 23, 2009 7:34 pm 
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A 'Full Superflip' is very similar to the 'Clean Superflip' and I guess it requires the same minimum of moves.
This classic algorithm can't compete with your 24 moves, but it can be written in a much more compact way:
((MR U)4 CR CD)3 (36 ftm, 36 qtm)

If we go a little bit further and do involve corners as well, then we get this:
U R F' B U' D' F U' D F L F' L' U R D F U R L (20* ftm, 20* qtm)
Note, that in this pattern every single piece of the cube is moved in place (edge-flips, corner- and center-twists).
... but amazingly the optimal solution only needs 20 moves in both metrics!


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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sun May 24, 2009 2:19 am 
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Location: Houston/San Antonio, Texas
This is PURE conjecturing here, but I currently believe that the pattern Waran requests, a superflip with all centers oriented correctly is the single most scrambled position for a super 3x3x3. I can't really explain why I think that, I just have a strong hunch. If this is the case, find the fewest number of moves it can be solved in and we know that for a normal 3x3x3 the magic number MUST be at most 1 less. ok fine, only if that number is greater than 20...

What? :lol: I can comment about a hunch with virtually no proof right? 8-)

Peace,
Matt Galla

PS: Note that in the Skewb family, if all pieces require orientation ie Skewb Ultimate, the farthest scramble of 14 moves is the "superflip" in Skewb terms :wink:


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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sun May 24, 2009 5:51 am 
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Location: Koblenz, Germany
Suppose 24 moves can be confirmed:
Then 24 is the current low bound for the diameter of gods algorithm for the Super Cube?
Cool!

And maybe these guys are the right community for such a question:
http://cubezzz.homelinux.org/drupal/


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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sun May 24, 2009 6:32 am 
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I believe there must be a much shorter algorithm in qtm since optimal Superflip solutions require only 4 additional moves.
So maybe this can be achieved for the first time in 28 moves.

Another interesting aspect is that not many picture cubes stay invariant after a Superflip.
One of the few exceptions is the 'Four Color Cube', see the image below.
This is also the perfect cube to check Superflip algorithms and center twists.
The texture remains (visually) unchanged when a pure Superflip has been applied.

You can play an online version here.


Attachments:
File comment: The 'Four Color Cube' is immune to a certain group of Superflips
FourColorCube1.png
FourColorCube1.png [ 12.96 KiB | Viewed 1346 times ]
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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sat Jul 19, 2014 4:15 pm 
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I can confirm now, that an optimal solution for a Pure Superflip (a Superflip pattern where no centers are moved) requires 24 moves in face turn metric (ftm).

My solver was actually running quite a long time (6 weeks, 6 days, 22 hours, 58 minutes and 34.1 seconds) to compute such a solution.
However, the found algorithm also was more efficient in terms of quarter turn moves (qtm): only 28 instead of 32 moves were required:

B' R L2 U' B' D B2 R' L F2 D' F U R2 L' F (U L)2 (R' U-)2 (24 ftm, 28 qtm)


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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sat Jul 19, 2014 5:49 pm 
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Very cool Waran and congrats on providing a lower bound for the Super 3x3x3.

To echo Andreas, I highly recommend you post that result at http://cubezzz.dyndns.org/drupal/

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 Post subject: Re: The Pure Superflip Quest
PostPosted: Sun Jul 20, 2014 8:39 am 
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And since we're at it, the Pure Superflip can be optimally solved in 19 ltm (layer turn metric).
So we do have a new lower bound for this metric too. I found this algorithm in June 18, 2014.

CR2 MF' MR2 MD MF' R U B L U MR F' MR B MR F' L' U' R' U' (19 ltm, 26 ftm, 28 qtm)


What is missing now is a value for the qtm metric.
Does anyone know an algorithm that requires less then 28 moves?


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 Post subject: Re: The Pure Superflip Quest
PostPosted: Mon Jul 21, 2014 3:12 am 
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Hi. I do not believethat the last contribution by Walter could be improved (for qtm).

I would like to post my sequence for the U 4-flip, which does not rotate any center. I know it is optimal in ftm. I also believe it is optimal in qtm. Only 16 as opposed to 14 in the usual turn metric. I am quite sure we could do better for ltm optimality. Does anyone know any sequence better for this than the one I'm posting below?

L- U- B- U R- U2 R B U L U- F U2 F-


Regards,

Per :)

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 Post subject: Re: The Pure Superflip Quest
PostPosted: Mon Jul 21, 2014 2:21 pm 
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@ Per
Your sequence for the U 4-flip is optimal in ftm but also in qtm.
Here is the complete list of all 16 solutions that are optimal in both metrics.
Note, that these algorithms do not rotate any of the centers:

F U2 F' U L' U' B' R' U2 R U' B U L (14 ltm, 14* ftm, 16* qtm)
F U R U' B U2 B' R' U' F' U L' U2 L (14 ltm, 14* ftm, 16* qtm)
F' U2 F U' R U B L U2 L' U B' U' R' (14 ltm, 14* ftm, 16* qtm)
F' U' L' U B' U2 B L U F U' R U2 R' (14 ltm, 14* ftm, 16* qtm)
R U2 R' U F' U' L' B' U2 B U' L U F (14 ltm, 14* ftm, 16* qtm)
R U B U' L U2 L' B' U' R' U F' U2 F (14 ltm, 14* ftm, 16* qtm)
R' U2 R U' B U L F U2 F' U L' U' B' (14 ltm, 14* ftm, 16* qtm)
R' U' F' U L' U2 L F U R U' B U2 B' (14 ltm, 14* ftm, 16* qtm)
B U2 B' U R' U' F' L' U2 L U' F U R (14 ltm, 14* ftm, 16* qtm)
B U L U' F U2 F' L' U' B' U R' U2 R (14 ltm, 14* ftm, 16* qtm)
B' U2 B U' L U F R U2 R' U F' U' L' (14 ltm, 14* ftm, 16* qtm)
B' U' R' U F' U2 F R U B U' L U2 L' (14 ltm, 14* ftm, 16* qtm)
L U2 L' U B' U' R' F' U2 F U' R U B (14 ltm, 14* ftm, 16* qtm)
L U F U' R U2 R' F' U' L' U B' U2 B (14 ltm, 14* ftm, 16* qtm)
L' U2 L U' F U R B U2 B' U R' U' F' (14 ltm, 14* ftm, 16* qtm)
L' U' B' U R' U2 R B U L U' F U2 F' (14 ltm, 14* ftm, 16* qtm)


An optimal solution in Layer Turn Metric (ltm) that does not disturb the centers counts 13 moves.
So, this algorithm can beat yours at least in ltm: :wink:
CR2 MR2 D' MF2 U R2 MF L2 U MF2 D' R2 MF' R2 (13 ltm, 18 ftm, 28 qtm)
SR2 D' MF2 U R2 MF L2 U MF2 D' R2 MF' R2 (13 ltm, 18 ftm, 28 qtm)


If we don't care about centers, then an optimal solution in ltm would require only 10 moves:
MF2 L2 MF L2 U MF2 R2 MF' R2 U' (10* ltm, 14* ftm, 22 qtm)


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