qtm-optimal corner-3-cycles

[Stefan]

I used Herbert Kociemba's new qtm-optimal solver to analyze corner-3-cycles. Always using UFR as the first of the three corners, there are 21 choices for the second and 18 for the third, resulting in 378 cases. Here are some statistics:

8 qtm: 90 cases
10 qtm: 108 cases
12 qtm: 156 cases
14 qtm: 18 cases
16 qtm: 6 cases

378 cases
10.63 average

This would allow you to shoot from a fixed corner to any two corners with 5.3 quarter turns per corner. Using just L-layer setup moves I think the number of algs to learn would drop to around 100, which sounds feasible. Using inverses and cube rotations would reduce the number further.

And yes I know corner-3-cycles can be done intuitively with commutators but it'd be better to just know them. I'm always thinking about not thinking.

 

[malcolm]

One solution to reduce algs to 27 is find a layer which has just two corners on it, there are 3*3=9 orientations, and 3 for the one on the top = 27. You need one setup to move them into the correct position, apply X, then Y is intuitive - just move the next corner where the first one was. This is somewhere between an alg and a commutator, because you learn X for every case, which will only be 3 moves for most of them.

 

[cmhardw]

Daniel Beyer has already optimized algs for all corner 3 cycle cases shooting from one fixed corner. All cases can be done in 12 HTM or less if I remember correctly.

Also, the number of algs to learn is really not that large if you consider them as types of commutators rather than as all distinct cases with an alg to solve them.

For example the alg: R' F U2 F' R F R' U2 R F'
and the alg: F' D L2 D' F D F' L2 F D'

I consider to be basically the exact same case since they use the same type of commutator on the same initial corner, they are just done on different faces. And if you want to be nit-picky yes I realize that this alg is really a cyclic shift of a commutator and not quite a true commutator by itself.

I haven't fully optimized my corner algs yet, but I'm almost done thanks to Daniel. But just so you know Daniel has already optimzed algs for all those cases you're referring to.

Chris

 

[mrCage]

I used Herbert Kociemba's new qtm-optimal solver to analyze corner-3-cycles. Always using UFR as the first of the three corners, there are 21 choices for the second and 18 for the third, resulting in 378 cases. Here are some statistics:

8 qtm: 90 cases
10 qtm: 108 cases
12 qtm: 156 cases
14 qtm: 18 cases
16 qtm: 6 cases

378 cases
10.63 average

This would allow you to shoot from a fixed corner to any two corners with 5.3 quarter turns per corner. Using just L-layer setup moves I think the number of algs to learn would drop to around 100, which sounds feasible. Using inverses and cube rotations would reduce the number further.

And yes I know corner-3-cycles can be done intuitively with commutators but it'd be better to just know them. I'm always thinking about not thinking.

Hi

What's wrong with doing/seeing corners as commutators? To be honest there is really only ONE LL corner 3-cycle i do not instantly (with practice) see easily as a commutator. It is the following:

R'FU2F'RFR'U2RF'.

Oh well, for bld it would be different ;-) One would just memorise -0+, --- or whatever for the orientations !!

- Per

 

[Lucas Garron]

What's wrong with doing/seeing corners as commutators? To be honest there is really only ONE LL corner 3-cycle i do not instantly (with practice) see easily as a commutator. It is the following:

R'FU2F'RFR'U2RF'.

Oh well, for bld it would be different ;-) One would just memorise -0+, --- or whatever for the orientations !!

- Per

Come on! Real men memo and women memo stickers!

Also, seen this? x (RUR'U')(RUR'U')(RUR'U') B' (RUR'U')(RUR'U')(RUR'U') B x'
(You can write it as a comm if you want...)

 

[mrCage]

What's wrong with doing/seeing corners as commutators? To be honest there is really only ONE LL corner 3-cycle i do not instantly (with practice) see easily as a commutator. It is the following:

R'FU2F'RFR'U2RF'.

Oh well, for bld it would be different ;-) One would just memorise -0+, --- or whatever for the orientations !!

- Per

Come on! Real men memo and women memo stickers!

Also, seen this? x (RUR'U')(RUR'U')(RUR'U') B' (RUR'U')(RUR'U')(RUR'U') B x'
(You can write it as a comm if you want...)

I think yours is too long To me it would at least be more beautiful to write it as x ([R,U]*3 B' [U,R]*3 B) x' - since [U,R]*3 = [R,U]*3

This one is nicer : [R U F2 D2 F2 U' R', U']. On the other hand yours will generalise to megaminx :rolleyes:
"Wrap-around" sequences do not translate from cube to MM

- Per

 

[Lucas Garron]

What's wrong with doing/seeing corners as commutators? To be honest there is really only ONE LL corner 3-cycle i do not instantly (with practice) see easily as a commutator. It is the following:

R'FU2F'RFR'U2RF'.

Oh well, for bld it would be different ;-) One would just memorise -0+, --- or whatever for the orientations !!

- Per

Come on! Real men memo and women memo stickers!

Also, seen this? x (RUR'U')(RUR'U')(RUR'U') B' (RUR'U')(RUR'U')(RUR'U') B x'
(You can write it as a comm if you want...)

I think yours is too long To me it would at least be more beautiful to write it as x ([R,U]*3 B' [U,R]*3 B) x' - since [U,R]*3 = [R,U]*3

This one is nicer : [R U F2 D2 F2 U' R', U']. On the other hand yours will generalise to megaminx :rolleyes:

- Per

Oho! But that is not all!
x [[R, U]*3, B'] x'
That's 3 moves of notation (and don't you try to count cube rotations!).
Also x [[R, U]*3, F'] x', etc., and the most compact (F can be B):

[[R,U]3,F]

(Ha! Beat ten characters that will evaluate here!

 

[mrCage]

Hi

I'm not quoting - in order to save the environment
And yes that notation is the most compact, unless i invent something that is not yet supported anyway :rolleyes:
I still don't like the algorithm - for solving purposes ...

- Per

 

[cmhardw]

haha ok this discussion is getting a little silly ;-)

Also, maybe you both do this too, but I just visualize the double conjugate alg R' F U2 F' R F R' U2 R F' as really this commutator [F' R F R', U2] only with a cyclic shift done such that you start on the 3rd to last move. The commutator [F' R F R', U2] I think is a pretty straight forward "direct insert" case into the U layer.

Also, for the record, I do actually use the alg R' F U2 F' R F R' U2 R F' and rotations of this alg onto other faces like F' D L2 D' F D F' L2 F D' in my actual blindfold solving for corners. I didn't discover this alg on my own Daniel Beyer showed it to me, so again many thanks to him for helping improve my blindsolving.

Also, I'm about 25% done optimizing my corner commutators for all cube sizes. Daniel Beyer has already optimized all corner cases for his solving, but I'm being stubborn and trying to do the work on my own - and using his help when I can't find an alg I like for a case ;-)

So now when I see a letter pair, I don't just visualize the corner sticker cycles but I also think of the alg. I literally consider the letter pair word as the name for that alg. So to me R' F U2 F' R F R' U2 R F' isn't just any alg, it's RN or "Ron", as in van Bruchem. ;-)

Chris

 

[mrCage]

Hi
I thought it was a "one-pass" job to find optimal algs with CubeExp as it can process several "scrambles" as a "single input". Doing them one by one is tedious. Doing it all manually the hardcore way should be fun. But since qtm is not a FM metric i will not do it :rolleyes:

- Per

PS! Chris, did you also work out all the 1x1x1 cases ? Hehe ... What's cube size gotta do with the corner perms anyway ??

 

[cmhardw]

PS! Chris, did you also work out all the 1x1x1 cases ? Hehe ... What's cube size gotta do with the corner perms anyway ??

haha Per I guess that's a good point. I do handle parity in the corner permutation in a different way for 2x2, 3x3, 4x4, and 5x5, but other than that everything I do is exactly the same for corners and I just wasn't thinking ;-)

I do have to admit though, I really do like this commutator for 1x1x1:
x y z y' x' z'

;-)

Chris

 

[Johannes91]

[[R,U]3,F]

(Ha! Beat ten characters that will evaluate here!

Ten, what a rookie! I could do it in, let's say two characters, if I wanted.

 

[Mike Hughey]

Johannes, I think you have a slight advantage in that particular competition.

 

[cuBerBruce]

What's wrong with doing/seeing corners as commutators? To be honest there is really only ONE LL corner 3-cycle i do not instantly (with practice) see easily as a commutator. It is the following:

R'FU2F'RFR'U2RF'.

Oh well, for bld it would be different ;-) One would just memorise -0+, --- or whatever for the orientations !!

- Per

Come on! Real men memo and women memo stickers!

Also, seen this? x (RUR'U')(RUR'U')(RUR'U') B' (RUR'U')(RUR'U')(RUR'U') B x'
(You can write it as a comm if you want...)

I think yours is too long To me it would at least be more beautiful to write it as x ([R,U]*3 B' [U,R]*3 B) x' - since [U,R]*3 = [R,U]*3

This one is nicer : [R U F2 D2 F2 U' R', U']. On the other hand yours will generalise to megaminx :rolleyes:

- Per

Oho! But that is not all!
x [[R, U]*3, B'] x'
That's 3 moves of notation (and don't you try to count cube rotations!).
Also x [[R, U]*3, F'] x', etc., and the most compact (F can be B):

[[R,U]3,F]

(Ha! Beat ten characters that will evaluate here!

I will also note that all these will also evaluate on my Windows CubeApp that I mentioned on another forum.

See:
http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/39811
and
http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/39812

 

[dbeyer]

Several things Per:
A) I did all 376 cases by hand from one buffer
B) Learning the big "376" is really just learning 5-6 different cases and being able to reflect them across the cube.
C) From the information that Stephan gave:
There are actually 6 cases that are 12 HTM (the 6 16-QTM cases)

Quote:
8 qtm: 90 cases
10 qtm: 108 cases
12 qtm: 156 cases
14 qtm: 18 cases
16 qtm: 6 cases
---
8-qtm is obviously 8 htm
9-HTM has a HTM cancelation, but there is still 12 or 14 quarter moves.
10-qtm ranges from 8-10 htm D' L'U2L D' L'U2L D2, There are some more optimal HTM cases that lead to worse QTM cases, no 10-HTM case is 12 HTM (at least not the data that I've gathered, which wasn't looking for optimal QTM) I would take an 8-HTM case with 10-QTM over a 10-HTM with 10-QTM anyday.
12 qtm ranges from 8-12 htm

16 qtm: 6 cases this is actually the 6 cases that are 12 HTM
R2 U F2 U' F2 U' R2 U F2 U F2 U' (12f) //BV - Angry Beavers
U F2 U' F2 U' R2 U F2 U F2 U' R2 (12f) //VB - Volley Ball
B2 U' L2 U L2 U B2 U' L2 U' L2 U (12f) //BW - Bowser
U' L2 U L2 U B2 U' L2 U' L2 U B2 (12f) //WB - Webslinger
R2 D B2 D' B2 U' B2 U B2 U R2 U' (12f) //VW - Buggy
U R2 U' B2 U' B2 U B2 D B2 D' R2 (12f) //WV - Wyvern
There are the algs for me personally. Cycling from the URB

 

[dbeyer]

None the less, it was really a big research project. Due to my dedication to it, I lost sight of cubing, and then really lost the desire to test it and fully apply it. I was able to use what I had learned from it at the VA open, and I was able to pass my knowlege onto Chris the night before the competition and to a handful of others. Since I've learned of commutators, I've sworn by them. They are the way to win in execution.

 

[mrCage]

[[R,U]3,F]

(Ha! Beat ten characters that will evaluate here!

Ten, what a rookie! I could do it in, let's say two characters, if I wanted.

Hi Johannes !!

Give me evidence. Give me the link with the mere 2 character doing that alg. It's not hard to make it more compact than 10 chars, the problem would be to have it shown in that applet with that notation

-Per

 

[Johannes91]

I could do it in, let's say two characters, if I wanted.

Give me evidence.

#\

It's not hard to make it more compact than 10 chars, the problem would be to have it shown in that applet with that notation

It's not an applet. I hate Java.

 

[dbeyer]

To have compact notation leads to a larger library of variables which must be used. Seriously, what you wrote was quite ridiculous Johannes. Perhaps you should tell us what that actually means. Without using other notations.

 

[Johannes91]

Perhaps you should tell us what that actually means.

Nothing, it's quite arbitrary. The point was just that I could beat Lucas' ten characters because it's my site. You guys have no sense of humor.

[[R,U]3,F]

(Ha! Beat ten characters that will evaluate here!