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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 12:27 am 
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bmenrigh wrote:
I used very close to the same technique that you propose and Julian used. I was expecting the puzzle to take me an hour but the setups were brutal. I really struggled to use setups that would put the edges in place with the correct orientation. I'm sure you'll do great but I think the puzzle is deceptively hard.
Yes the setups for the edges were a little odd but orientation was by far the easiest part, they are far enough away (at least with my commutator) that you just twist the individual edge if it is wrong, i now have 470 moves to cycle 55 triangles with a (3,1) alg i have never used before... i actually don't know how this will turn out with these stupid setup moves.
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Two hours and another 340 moves later and i beat Michaels record by 130 moves, turns out i wasn't being optimistic with my guess of beating it by 100 :D
Julian wrote:
That's the way I solved it. I thought I did a good job until Michael solved it 110 moves faster, and it is humbling that you reckon you could beat even Michael's solve by a good margin! I found the setups tricky with this one.
I'm not going to lie, set up moves on this thing are horrendous, far harder than i expected, but it's very nice to know i wasn't being over confident for my move count estimate.

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Last edited by Elwyn on Sat Oct 16, 2010 2:35 am, edited 1 time in total.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 2:34 am 
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bmenrigh wrote:
I was expecting the puzzle to take me an hour but the setups were brutal. I really struggled to use setups that would put the edges in place with the correct orientation. I'm sure you'll do great but I think the puzzle is deceptively hard.
Brandon, I absolutely agree with what you're saying. My last 3-cycle, I had to use about 15 set-up moves. And I assumed that it would be a relatively quick solve, but yet I spent over 3 hours solving it. That surprised me a lot. As for my move count: I could easily have gotten it sub-2000 if I weren't so sloppy. I basically just threw away moves by not undoing unnecessary moves.

Elwyn, good job on getting so far using so few moves! I'll be excited to see what you're final move count will be.

EDIT: you just edited your post with your completed solve, haha. Good job on beating the record!

Second edit: I didn't have any trouble with the orientation of the edges: I did exactly what Elwyn did here and it worked fine. The issue was the other set-up moves. Maybe because I insisted on not writing any of them down and also because they were hard in general. But it made me feel better that I'm not the only one who experienced that.


Last edited by Katja on Sun Oct 17, 2010 4:30 am, edited 1 time in total.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 5:14 pm 
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I decided to give 1.1.19 a second try tonight, and just as last time: I got stuck with the same situation I didn't manage to solve before:
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Skjermbilde 2010-10-17 kl. 00.08.24.png
Skjermbilde 2010-10-17 kl. 00.08.24.png [ 268.64 KiB | Viewed 7005 times ]
This time though, I was more optimistic about finding a solution to the problem. Until I accidentally hit the "go back one page" button! Arg, now I've lost my motivation to try for a third time. I'm only 4 puzzles away from getting my name on the leader board, so I though 4 challenging puzzles would be most fitting. Maybe this just has to wait. This puzzle really hates me, I'm sure :lol:


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 5:16 pm 
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Gelatinbrain: I find your "rolling spheres cubes" beautifull. :) The spheres have 12 orientations. That is new and it is a real challange to solve the puzzle. There is one thing I dont understand. Maybe you can help me. How do the spheres change orientation, when I do one move. Is it like they would roll physically on the outline of a circle? :?: I tried to get the ball at the up-right-fore location into other orientations and found that I can only get 11 other orientations. 6 cases I reached by one move, 2 cases I reached by 3 moves and 3 cases I reached by 4 moves. With reversing the moves with an other ball it is possible to solve the puzzle all but one ball, and never need more than 6 moves for one ball. For the last ball I use 8, 13 or 15 moves. Some cases of two remaining balls I can solve at once. When solving the ball before the last ball it might be possible, to get the last ball into a good case or solve it even. But two balls make a lot of possibilities. I also wonder, if an U6 move for instance can be counted as one move. In the half turn metric a double-move is counted as one move, because it performs in time nearly as a single move. Now can you say that about a move that is repeated 6 times? The answer is clearly not.
I have read all the recent posts about this puzzle from Brandon and Julian. I guess they are on the same level than me.
Stefan.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 6:04 pm 
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Katten wrote:
I decided to give 1.1.19 a second try tonight, and just as last time: I got stuck with the same situation I didn't manage to solve before: [...snip...] This time though, I was more optimistic about finding a solution to the problem. Until I accidentally hit the "go back one page" button! Arg, now I've lost my motivation to try for a third time. I'm only 4 puzzles away from getting my name on the leader board, so I though 4 challenging puzzles would be most fitting. Maybe this just has to wait. This puzzle really hates me, I'm sure :lol:
It hurts to put a bunch of time into a puzzle and then lose the progress.

As was said before, that isn't any sort of parity case and just requires you to swap two identically colored triangles in a three-cycle. Here is a 16-move fix for that case:

/* Setup first triangle */
F2,
/* Setup second triangle */
[H'2&2,K',D2],
/* Perform (3,1) commutator */
[F&2,A',F'&2],K&2,[F&2,A,F'&2],K'&2,
/* Undo second triangle setups */
[D'2,K,H2&2],
/* Undo first triangle setups */
F'2


The heart of the routine is the 3-cycle [F&2,A',F'&2],K&2,[F&2,A,F'&2],K'&2. The setups just move the right pieces into place so that we get orange 1-> orange2 -> red1 -> orange1.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 7:12 pm 
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Julian wrote:
Stefan Schwalbe wrote:
The spheres have 12 orientations. That is new and it is a real challange to solve the puzzle. There is one thing I dont understand. Maybe you can help me. How do the spheres change orientation, when I do one move. Is it like they would roll physically on the outline of a circle? :?:
[...] With each 90 degree move, each ball first rotates 90 degrees as if being "pushed" by the ball "behind" it (about to take its position); then each ball rotates 180 degrees in the same axis as the move. So when you make a U move, each sphere finishes with y2: UFR moves to UFL and spins x'y2; UFL moves to UBL and spins z'y2; UBL moves to UBR and spins xy2; and UBR moves to UFR and spins zy2.
I have been imagining a gearing mechanism on the spheres where there is a floating point above the center of the face I'm about to turn and 4 axles descend from that point down through the diagonal of each for the 4 spheres about the face. Then I can look at a face and see how each of the spheres will turn without thought but mechanical intuition.

Stefan, you said you have a 8-move sequence to solve for the last sphere? I've only been able to find a 10 move sequence. I got a special case of Julian's 14 move routine down to 13 moves. I'm working on screenshots for these cases for a post I hope to do tomorrow.

Stefan's low move counts motivated me to try harder. It hasn't hit the scoreboard yet but I got a 17 move solve 8-) .

Edit: fixed to properly quote Julian.

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Last edited by Brandon Enright on Sat Oct 16, 2010 7:21 pm, edited 2 times in total.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 7:16 pm 
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Edit: Brandon, I immediately deleted my post because I realized it had an important mistake in it and I wanted to avoid confusing anyone with incorrect info. My corrected response is below...

Stefan Schwalbe wrote:
The spheres have 12 orientations. That is new and it is a real challange to solve the puzzle. There is one thing I dont understand. Maybe you can help me. How do the spheres change orientation, when I do one move. Is it like they would roll physically on the outline of a circle? :?:
Hi Stefan. I noticed yesterday that you have joined the cube-spheres party! Very low move counts from you. :solved: If it helps, here is the way I see the way the spheres rotate:

With each 90 degree move, each ball first rotates 90 degrees as if being "pushed" by the ball "behind" it (about to take its position); then each ball rotates 180 degrees in the same axis as the move. So when you make a U move, each sphere finishes with y2: UFR moves to UFL and spins x'y2; UFL moves to UBL and spins z'y2; UBL moves to UBR and spins xy2; and UBR moves to UFR and spins zy2. That's why each ball can only be in one of 12 orientations wherever it is, because it has always rotated through an even number of quarter moves if it is in the same Skewb corner orbital that it started, and it has always rotated through an odd number of quarter moves if it is in the different Skewb orbital.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 16, 2010 7:38 pm 
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3.11.1 state counting attempt 2:

There are 8 spheres divided into 2 groups of 4. Spheres in each group are indistinguishable but the orientation of a sphere reveals which group it is in. Therefore, there are 8! / (4! * 4!) ways to order the spheres.

Each sphere can be in 12 different orientations independently of all other spheres. This allows for 12^8 possible orientations.

The overall orientation of the puzzle doesn't matter which reduces the number of possibilities by 24.

That results in (8! * 12^8) / (4! * 4! * 24) == 1,254,113,280 possible positions.


The reason the orientation of the puzzle is a factor of 24 rather than 12 is that you could solve the puzzle into a state where all of the even spheres and odd spheres have swapped position. You wouldn't know this and each sphere would have a quarter-turn in the same axis.

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Last edited by Brandon Enright on Mon Oct 18, 2010 12:23 am, edited 1 time in total.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 4:36 am 
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bmenrigh wrote:
The heart of the routine is the 3-cycle [F&2,A',F'&2],K&2,[F&2,A,F'&2],K'&2. The setups just move the right pieces into place so that we get orange 1-> orange2 -> red1 -> orange1.
Brandon, that's such an easy fix! I cannot believe I didn't figure this out by myself :oops: I was really frying my brain with a fix for it last night and this crossed my mind, but I didn't find the right way to execute it. Anyways, thanks a lot for sharing; tonight this puzzle is going down! :twisted:


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 12:40 pm 
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bmenrigh wrote:
1.11.1 state counting attempt 2:
Psst... it's 3.11.1. :P

bmenrigh wrote:
That results in (8! * 12^8) / (4! * 4! * 24) == 1,254,113,280 possible positions.

The reason the orientation of the puzzle is a factor of 24 rather than 12 is that you could solve the puzzle into a state where all of the even spheres and odd spheres have swapped position. You wouldn't know this and each sphere would have a quarter-turn in the same axis.
:idea: Of course! Thank you; now I have seen the light. This is the closest answer yet. It is a slight undercount though, because some scrambles look the same from different orientations. For example, if you spin FUR, FDR, BUL, BDL all y2, then the puzzle looks exactly the same from 2 different orientations, so you would need to count this state twice before dividing by 24. This is a pesky situation that doesn't come up often -- identical pieces with distinct orientations. I don't know how many rotationally symmetric states there are with this puzzle. It could well be a trivial number compared to the scale of 1.25 billion, and not enough to push the exact result up to 1.26 billion to 3 sig figs.

P.S. 17 moves?! 8-)


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 1:54 pm 
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Stefan Schwalbe wrote:
There is one thing I dont understand. Maybe you can help me. How do the spheres change orientation, when I do one move. Is it like they would roll physically on the outline of a circle?

The amazing thing is you solved it nevertheless,and even broke the record. :lol:
Maybe the animation is not smooth enough or too fast to follow the movement.
The balls rotate simulatniously around itself and a common center. Just like the earth turning around the sun or the teacups in an attraction park.
The axis of self-rotation is the line connecting the
center of each ball and the face center of the circumscribing cube.
The balls spin 120º while turning 90º around the common axis so that they settle down in their original orientation after 3 rounds.
In fact the resulting orientation is as same as Julian explained.

Katja, thank you for help links. Be careful to do not click "go back". The area outside the gray rectangle is the browser's territory. The applet cannot show a warning for clicks outside its own area.

Check new puzzles:
3.1.27 ~ 3.1.30 & 3.2.11(TomZ's "Compy Skewb" compatible, I think...)

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 2:08 pm 
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gelatinbrain wrote:
Be careful to do not click "go back". The area outside the gray rectangle is the browser's territory. The applet cannot show a warning for clicks outside its own area.
Yes, I blame only myself for that happening. I was really tired and didn't pay attention to where the pointer was before I pressed the button and then it was too late.
gelatinbrain wrote:
3.2.11(TomZ's "Compy Skewb" compatible, I think...)
I'd say yes, except for one thing: when you do a Dino Cube rotation, the corners won't move. Check out Tom's video.

As for the other new puzzles: they look like hard puzzles!


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 3:09 pm 
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On my Compy Skewb, there are tips that are fixed to the inner skewb. When you make a skewb rotation they rotate with it, but they're stationary when you do a compy/dino move. The tips show the orientation of the internal skewb corner, making the puzzle yet a little harder.

Maybe you could include the puzzle's name under it? Or have you stopped doing that? I remember that for instance, the megaminx was properly labeled (the nxnxn cubes still bear labels). If that's not the case, a link would be really, really nice :)

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 3:40 pm 
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Thank you for explaining the 3.11.1 spheres behavior to me. Now I understand. :D
I admire the new puzzles (3.1.27 ~ 3.1.30 ). Another new idea. The circle pieces turn 45 deg while the other pieces turn 90 deg. Dont know how to solve it. Maybe I find some time for it.


Last edited by Stef-n on Sun Oct 24, 2010 2:53 pm, edited 1 time in total.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 17, 2010 5:32 pm 
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TomZ wrote:
On my Compy Skewb, there are tips that are fixed to the inner skewb. When you make a skewb rotation they rotate with it, but they're stationary when you do a compy/dino move. The tips show the orientation of the internal skewb corner, making the puzzle yet a little harder.

Maybe you could include the puzzle's name under it? Or have you stopped doing that? I remember that for instance, the megaminx was properly labeled (the nxnxn cubes still bear labels). If that's not the case, a link would be really, really nice :)

So, tips are not trivial as I thought. Then this one
should come close.
Image

I stripped names of puzzles for two reasons
First, I thought it's better to avoid using possible trademarks or something of that sort. No complaint so far,just for a precaution.
Second reason is that personally I don't like those arbitrary names like brabrabra-minx.
It's just a question of taste, but I think someone should elaborate a mathematically correct naming convention applicable to all possible twisty puzzles.

If you want links to your site, then PM me. Maybe not everyone appreciates such an initiative. So I will do only for demand.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 18, 2010 12:51 am 
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Julian wrote:
bmenrigh wrote:
1.11.1 state counting attempt 2:
Psst... it's 3.11.1. :P
Damn, thanks for pointing that out. My brain has an aggressive auto-correct for these sorts of things. I would have never caught that. I never see mistakes like if/of, or/on/of, if/it an/on, etc. In general a single replacement of a letter of a transposition of letters gets me every time. The other day I even misspelled Elwyn as Elywn ... twice! :oops: Believe it or not, I actually proofread my posts 2-3 times to try to reduce these sorts of mistakes and I still fail. It's really a wonder I can solve any twisty puzzles at all!

Julian wrote:
bmenrigh wrote:
That results in (8! * 12^8) / (4! * 4! * 24) == 1,254,113,280 possible positions.

The reason the orientation of the puzzle is a factor of 24 rather than 12 is that you could solve the puzzle into a state where all of the even spheres and odd spheres have swapped position. You wouldn't know this and each sphere would have a quarter-turn in the same axis.
:idea: Of course! Thank you; now I have seen the light. This is the closest answer yet. It is a slight undercount though, because some scrambles look the same from different orientations. For example, if you spin FUR, FDR, BUL, BDL all y2, then the puzzle looks exactly the same from 2 different orientations, so you would need to count this state twice before dividing by 24. This is a pesky situation that doesn't come up often -- identical pieces with distinct orientations. I don't know how many rotationally symmetric states there are with this puzzle. It could well be a trivial number compared to the scale of 1.25 billion, and not enough to push the exact result up to 1.26 billion to 3 sig figs.
Okay I'm pretty sure I see what you're saying and this seems like a hard problem to tackle. I think we're actually over-counting though, not under-counting. I'm not positive on this but here's my argument for why:

Take any scramble that looks identical from two different orientations. The reason we still need to divide by 24 on that scramble is that there are still 24 different orientations that we could make that scramble. By shuffling the pieces around we could create 24 identical scrambles that we need to cancel to 1.

The trouble with the scramble is that if it is indistinguishable from two different orientations then that means there will be two "different" scrambles that will result in the same visual scramble. I think my calculation counts those twice but we need to cancel one and only count it once.

I suppose I could be thinking about this the totally wrong way though...

My gut tells me that there are only a few hundred to a thousand different patterns like this? I'm not sure.

Julian wrote:
P.S. 17 moves?! 8-)
Yeah I was really happy with it too. I should just fess up though and say it was mostly luck. I partially solved about 50 scrambles, getting to the last 1-4 spheres. If it took me more than 12 moves to get there I just re-scrambled.

Eventually I got lucky and hit on a scramble where two adjacent spheres were already in the same orientation. turning another face a few times placed a third sphere in the correct orientation right next to them. I then played with a simple 3-move sequence that put the 4th sphere on the face in place. I solved half the puzzle in 4 moves. Then I was even luckier on the top -- three spheres were already in the same orientation and a few quarter turns re-oriented them the same way as the bottom four. I got to the last sphere which had twists in two different axes in 5 moves. I used a 13-move version of your 14-move fix to solve it. I worked it out so that the first move of the fix nicely canceled with the 5th move of the solve. 17 moves of pure luck. It only took about 50 tries.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 18, 2010 6:42 am 
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bmenrigh wrote:
It's really a wonder I can solve any twisty puzzles at all!
I'm the same way with proofreading my own posts, so no worries!

bmenrigh wrote:
Julian wrote:
bmenrigh wrote:
That results in (8! * 12^8) / (4! * 4! * 24) == 1,254,113,280 possible positions.

The reason the orientation of the puzzle is a factor of 24 rather than 12 is that you could solve the puzzle into a state where all of the even spheres and odd spheres have swapped position. You wouldn't know this and each sphere would have a quarter-turn in the same axis.
:idea: Of course! Thank you; now I have seen the light. This is the closest answer yet. It is a slight undercount though, because some scrambles look the same from different orientations. For example, if you spin FUR, FDR, BUL, BDL all y2, then the puzzle looks exactly the same from 2 different orientations, so you would need to count this state twice before dividing by 24. This is a pesky situation that doesn't come up often -- identical pieces with distinct orientations. I don't know how many rotationally symmetric states there are with this puzzle. It could well be a trivial number compared to the scale of 1.25 billion, and not enough to push the exact result up to 1.26 billion to 3 sig figs.
Okay I'm pretty sure I see what you're saying and this seems like a hard problem to tackle. I think we're actually over-counting though, not under-counting.
Here's an illustration of why I say it's an undercount. I always like to take extreme cases for illustrative purposes so here's a very extreme case... let's say that a cube puzzle has just two overall states: black and white. When you make a move the whole of the outside turns from black to white, or from white to black. If we take the 2 states and divide by 24, we get 1/12, which is clearly incorrect. The all-black state looks the same from 24 orientations so we must count it 24 times, and the same with the white. Then 48/24 = 2, resulting in the correct answer. The same logic applies but obviously in a less extreme way with this and other puzzles. The example I gave above, where four corners are twisted y2, is only counted 12 times in the part of the calculation before we divide by 24. Our count of possible states is effectively paired/fused by a y2 rotation of the whole puzzle when we are oriented to see the four spheres spun y2 from solved. I find it quite confusing to think and write about, but I remember that Burnside's Lemma is relevant to the logic.

Julian wrote:
My gut tells me that there are only a few hundred to a thousand different patterns like this? I'm not sure.
I agree, I don't think there are that many.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 18, 2010 7:52 am 
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Julian wrote:
Julian wrote:
My gut tells me that there are only a few hundred to a thousand different patterns like this? I'm not sure.

I agree, I don't think there are that many.
Agreeing with yourself :)

Sorry for the wasted post i just thought it was funny after the whole proofreading comment.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 18, 2010 6:43 pm 
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Elwyn wrote:
Sorry for the wasted post i just thought it was funny after the whole proofreading comment.
:lol: Not a wasted post in my opinion. Okay, I've quadruple checked that I'm quoting you and not me before clicking Submit...


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 18, 2010 8:51 pm 
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On discussing the available states of 3.11.1:

Julian wrote:
Here's an illustration of why I say it's an undercount. I always like to take extreme cases for illustrative purposes so here's a very extreme case... let's say that a cube puzzle has just two overall states: black and white. When you make a move the whole of the outside turns from black to white, or from white to black. If we take the 2 states and divide by 24, we get 1/12, which is clearly incorrect. The all-black state looks the same from 24 orientations so we must count it 24 times, and the same with the white. Then 48/24 = 2, resulting in the correct answer. The same logic applies but obviously in a less extreme way with this and other puzzles. The example I gave above, where four corners are twisted y2, is only counted 12 times in the part of the calculation before we divide by 24. Our count of possible states is effectively paired/fused by a y2 rotation of the whole puzzle when we are oriented to see the four spheres spun y2 from solved. I find it quite confusing to think and write about, but I remember that Burnside's Lemma is relevant to the logic.

This IS quite confusing to write and talk about! This puzzle has consumed my every thought for a few days now. I'm not sure what face best describes all the thought and effort I've put into this problem but :D, :?, :x, :oops:, :evil:, and :roll: all seem sadly appropriate. This post is long but I hope it is worth the read!

I like your extreme toy example and I agree we need one. Unfortunately I don't think your black/white cube is up to the task. In your example you say "if you make a move the whole of the outside turns from black to white". This is fine but we haven't really defined moves that are equivalent to 3.11.1. The reason we have to divide by 24 on 3.11.1 is that we can assemble the same puzzle in 24 different orientations -- namely (8! / (4! * 4!)) * (12^8). This term overestimates by 24 so we have to account for that. In your puzzle the number of ways you can arrange it is ((8! / 8!) * (2^1) == 2) because every piece is indistinguishable and the orientation of every piece is determined by the orientation of just 1 piece. There is no over-count and no factor of 24 to divide by. So instead of counting the all black state 24 times and the all white state 24 times and getting (24 + 24) / 24 == 2 I say there is no re-arrangement phase and therefore no over-count by a factor of 24.

But, I think there is an extreme toy example that is illustrative. Suppose we had the 3.11.1 puzzle but 7 of the 8 pieces were pure white. Only one sphere had the 6 colors on it. We can see the position and orientation of our one sphere and the state of the rest of the spheres doesn't matter/isn't visible.

Given a single state, it is easy to see how the puzzle can be re-oriented into 24 different states. Therefore in any calculation we do that allows us to arrange pieces we have to account for this extra factor of 24.

So to count this puzzle, the number of permutations available is (8 choose 1 == 8) or by another method, (8! / 7! == 8).

The number of orientations available for that sphere is 12. (hold on, I know there is a mistake here...)

So that give us 8 * 12 / 24 == 4 total unique states.

We can count them: {no rotation, x2, [x y], [x -y]}. It's easy to see that re-orienting the puzzle gives us the other rotations.

But wait, what about the orientations that are 1 or 3 twists away? The other 7 spheres don't matter so if we put an even sphere in an odd slot there should be another 12 orientations available. The calculation must have a mistake.

The mistake is in the colored sphere choosing phase. Is our 1 colored sphere an even sphere or an odd sphere? There should be 24 orientations possible for 1 sphere, not 12. The other 12 are available if we chose the other type of sphere to be the only non-white sphere. That's just (2 choose 1 == 2).

That gives us 2 * 8 * 12 / 24 == 8 total unique states.

They are (If I've counted right): {none, x, x2, [x y], [x -y], [x2 y], [x2 -y], -x}. You should be able to re-orient the puzzle to get all other possible redundant states.

I'm pretty sure this toy example illustrates an issue with my previous calculation. I said we have to divide by 24 because you could swap all of the even and odd spheres and it would look like the puzzle was rotated a quarter turn. We didn't account for actually swapping the even and odd spheres though. If all of the even spheres stayed even then there are only 12 distinct orientations available and we would have to divide by 12.

Thus the calculation should be:
(2 * (8! / (4! * 4!)) * 12^8) / 24) == 2,508,226,560

But this still doesn't tell us if we under or over counted the issue of the one orientation of the puzzle looking identical from two different orientations.

To figure that out I think we need to look at a slightly bigger toy example. Lets add one additional colored sphere. So now the puzzle has 2 colored spheres and the other 6 are pure white.

First using intuition:

Doing a calculation that ignores re-orientation of the puzzle, we can affix one sphere by the above calculation to one spot. That's 8 states. Then the other sphere can be in the other 7 spots on the puzzle and in any orientation. That's 8 * 7 * 24. But, if we re-orient the puzzle we can always choose a view where the two spheres appear to have swapped places which causes us to over-count by a factor of 2 that we must correct for. That results in (8 * 7 * 24) / 2 == 672. This may be confusing but recall that because we have affixed one of the spheres positions but it is in a sort of "quantum superposition" where it counts as 8 states at once.

Now, using math:
If we break it up piece-meal then the sphere choosing phase can result in {[odd odd], [odd even], [even odd], [even even]}

Odd-Odd and Even-Even:
(8! / (6! * 2!) * 12^2) / 24 == 168 (for each).

Odd-Even and Even-Odd:
((8! / 6!) * 12^2) / 24 == 336 (for both).

168 + 168 + 336 = 672


So with all that being said, I think the 3.11.1 calculation should be:
(2 * (8! / (4! * 4!)) * 12^8) / 24) == 2,508,226,560

I'm not sure if that is an over count but I'm almost certain it isn't an under count. More thought and rebuttal is needed :lol: Perhaps we've beaten this topic to death? I hope not :mrgreen:

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Tue Oct 19, 2010 7:06 am 
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bmenrigh wrote:
On discussing the available states of 3.11.1:

<snip>

Perhaps we've beaten this topic to death? I hope not :mrgreen:
Multiplying by 2 and then dividing by 24 goes back to my original instinct that we must divide by 12, but we have done far more than come full circle, because the discussion has helped us understand more about the odd/even spheres.

Regarding symmetries and undercounting, you are right that my black or white puzzle was not a good example. Let's look again at the case where FUR, FDR, BUL, and BDL are all twisted y2 from their solved state. This can only happen when all of the spheres are in their correct odd/even orbitals -- one of the 8! / (4! * 4!) possibilities -- and this state of those 4 spheres twisted y2 from an orientation is 1 of the 12^8.

But those 4 spheres are twisted y2 from the point of view of two orientations, because the state looks identical when we reorient the puzzle by y2. Hence the undercounting problem. If we are to divide by 12 (or double the possibilities by counting all odd/even inversions and then divide by 24) we must make sure we count each state once in every orientation it is seen. We assume that the state has been counted from both orientations, but it has not. This state, in its 12 different even orientations, comes up just 6 times in the 12^8 (this one plus the ones corresponding to x2, xy, xz, x'y, and x'z reorientations of the puzzle).

All states with 2-fold rotational symmetry must be counted twice, those with 4-fold rotational symmetry must be counted four times, etc., then we can divide by 12 and get the correct result.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Tue Oct 19, 2010 12:37 pm 
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Julian wrote:
[...snip...]But those 4 spheres are twisted y2 from the point of view of two orientations, because the state looks identical when we reorient the puzzle by y2. Hence the undercounting problem. If we are to divide by 12 (or double the possibilities by counting all odd/even inversions and then divide by 24) we must make sure we count each state once in every orientation it is seen. We assume that the state has been counted from both orientations, but it has not. This state, in its 12 different even orientations, comes up just 6 times in the 12^8 (this one plus the ones corresponding to x2, xy, xz, x'y, and x'z reorientations of the puzzle).

All states with 2-fold rotational symmetry must be counted twice, those with 4-fold rotational symmetry must be counted four times, etc., then we can divide by 12 and get the correct result.

:idea: You're right! I kept thinking that this case would get counted a second time later in the sphere permutation calculation. That is, I thought the puzzle would be assembled into the same state a different way and we'd count it again. I see now though that the calculation won't count it again. We do have an undercount. Another way to look at it is that we over-canceled.

Compared to 2.5 billion states the number of over-canceled symmetric states has to be tiny. I'm sure there is a way to calculate them though. I don't have any experience with such things and I can't find any puzzle on Jaap's page that has this problem to see how he did it.

I'm really glad we have different ways of thinking about things or this problem would have gone totally unnoticed by me. With this puzzle Gelatinbrain has lived up to his namesake and turned my brain to mush :oops: .

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Tue Oct 19, 2010 7:19 pm 
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Solved 3.2.11

Attachment:
File comment: Solved
Screen shot 2010-10-20 at 1.09.43 PM.png
Screen shot 2010-10-20 at 1.09.43 PM.png [ 74.88 KiB | Viewed 5899 times ]


I solved it Corners and Centers first similar to a skewb.
I then solved the Center corners(squares next to centers) with a simple 4 move algorithm.
Then the Centers next to the edges with an 8 move algorithm.
Lastly the Edges with a 30 move algorithm.

I got a certificate. I'm not sure how to submit the certificate, so I'll just post it for now, so Gelatinbrain can use it to get the solve information for highscores:


9d910535ba9c0a3de217dae8765ec6bc
263cf5de4a7aebf40b2d26b24c7986f2
0db3467a8cfd16de2c6e98ef31331aca
34e619cd32a7584fb09d623bc49668c6


Edit: I also solved 3.2.12 in a similar way.

Here is the certificate:


c8fe6adb54b12779a6935ebd2373ebe9
73814865f1ce5fe51a3f34e719cd32a7
5845b094622fc49a680cd09d6489c89e
613dc27689ec135ea1bc436e91dd2393

Edit2: I also solved 3.2.9 today. Thin parts first, then corners, and then corner centers.

Attachment:
File comment: Done
Screen shot 2010-10-20 at 4.09.53 PM.png
Screen shot 2010-10-20 at 4.09.53 PM.png [ 50.77 KiB | Viewed 5890 times ]


And the certificate:


239abd2108879f0bf930e04f1c86950d
dcf9178e8016bd2c676cce30e51acb34
e613cd3ba74c4fa09d6b3bcc97442fd1
9b6437c89e613dc27689ec135ea3bc11

Edit3: Also solved 3.2.10

Certificate:


2930a99c133fa976a930405fc4bc246e
f4d24778ecf465d926474c7886f20db9
46798ce016c72c7298dd31f61a3534e7
19cd32a7584fb09d623bc497682ed03b


-Mark- :)

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Wed Oct 20, 2010 11:01 am 
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Skjermbilde 2010-10-20 kl. 11.01.52.png [ 94.76 KiB | Viewed 5873 times ]
Easier than I thought! These where my steps:

1.) Solve centers, non-pure
2.) Solve wide triangles, non-pure
3.) Solve thin triangles, pure
4.) Solve like Megaminx

Placing the thin triangle pieces on this one was really easy, due to the not-at-all difficult set-up moves. I wish placing the thin triangles always where that easy :lol:

As for something else: anyone know how to calculate the odds of not getting a parity case when pairing up edges on puzzles like 1.1.18, 1.1.20 etc?


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Wed Oct 20, 2010 11:29 am 
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Katten wrote:
[...snip...]Placing the thin triangle pieces on this one was really easy, due to the not-at-all difficult set-up moves. I wish placing the thin triangles always where that easy :lol:
Good job, welcome to the 1.1.15 over 2000 moves club, I was getting lonely :lol:

Katten wrote:
As for something else: anyone know how to calculate the odds of not getting a parity case when pairing up edges on puzzles like 1.1.18, 1.1.20 etc?
Getting a parity case in edge pairs like on 1.1.18 or 1.1.20 or the 4x4x4 is really a question of the permutation parity of how the individual edge pieces were paired.

When you are pairing for edge pieces to reduce them to edge groups, you have to assemble the groups into an even permutation. When you have two groups swapped, the groups are in an odd permutation and can't be solved. By breaking up the groups into their constituent pieces and re-forming the groups into an even permutation you can then solve the groups.

It's 50/50 -- assemble into an odd permutation and you get a parity, assemble into an even permutation and you don't.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Wed Oct 20, 2010 2:12 pm 
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bmenrigh wrote:
Good job, welcome to the 1.1.15 over 2000 moves club, I was getting lonely :lol:
Thanks, you can always count on me :lol: Speaking of: I'd like to welcome myself to the 1.2.12 over 1000 moves club! I'll blame this on the fact that I juggled this solve with babysitting my very attention-seeking younger siblings :lol: I'll resolve this some later time. For now I'm more focused on the actual solves than how well I do compared to others. At least now I'm only one puzzle away from appearing on the leader board :D

EDIT: Looks like I'm already on it :D


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Thu Oct 21, 2010 2:56 pm 
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Gelatinbrain, can you change my name on the rankings page? I dont want to continue with that old slogan. :roll: If possible, replace it with "Agamemnon" please. "Agamemnon" would be ok, for a friend called me Agamemnon once.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Thu Oct 21, 2010 3:56 pm 
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ERRR... This is not the page to write this on (again) but my java is not running the applet. I have checked on the java.com page and on a page called java tester. Both show that my browser is supporting the applet and that it is working. But when I start the applet I get the loading sign (windows loading sign in firefox and the big blue loading circle with the java symbol in the middle in internet explorer) and after it is done loading I see nothing.
Can anyone tell me what I might have to do? I haven't used the applet in a long time so I can't tell since when java isn't working. :cry:
Shall I send the report from the java console to their support? Will they answer mails of that kind?

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Thu Oct 21, 2010 4:40 pm 
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Hm I have an update on my problem:
So I have played some java games and they all work. And when I download the applet from the page where it says "To download this applet click here!(about 200 KB)" I can open the files. However there is a warning (translated by me from german to english):

Java found application components that could present a security threat.
Name: com.sun.opengl.util.JOGLApplet Launcher

It asks me if it shall block the potentially unsave components. I hit no and the applet works. I don't even know if it would also have worked if I hadn't checked "no". Because now that I did it once, the question won't be asked anymore. Still the applet doesn't work from the page itself (only opening the files on my computer works.) Also, there are some newer puzzles in the applet that have not been added to those downloadable files yet so I can't play with those :(

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Thu Oct 21, 2010 5:13 pm 
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alaskajoe wrote:
Hm I have an update on my problem:
So I have played some java games and they all work. And when I download the applet from the page where it says "To download this applet click here!(about 200 KB)" I can open the files. However there is a warning (translated by me from german to english):

Java found application components that could present a security threat.
Name: com.sun.opengl.util.JOGLApplet Launcher

It asks me if it shall block the potentially unsave components. I hit no and the applet works. I don't even know if it would also have worked if I hadn't checked "no". Because now that I did it once, the question won't be asked anymore. Still the applet doesn't work from the page itself (only opening the files on my computer works.) Also, there are some newer puzzles in the applet that have not been added to those downloadable files yet so I can't play with those :(

This is already a classic problem. The reason is not yet known.
The only solution so far is to change the security setting.
For details:
viewtopic.php?f=8&t=7830&start=2067

For applet troubles, please PM me rather than posting here.
This thread became too big. I don't know if it good or bad (though personally it makes me happy :) ).

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Fri Oct 22, 2010 7:51 am 
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If I can contribute something to the 3.11.1 discussion.

Here is my number of possible configurations: 7 x 5 x 12^7 == 1,254,113,280.
Here is the proof:

First compute the number of possible permutations.

You have 8 identical balls like in the puzzle, but because they can only reach 12 of 24 orientations, you have 4 balls of one sort, and 4 balls of the other sort. I paint them virtually black and white for the permutation calculation.
You have 8 identical locations in the puzzle, but they are split into 2 groups of 4 locations, let's say 4 A-locations and 4 B-locations.
Place the first black ball into 1 of the 8 locations.
If it is an A-location, all A locations are from now on black, and all B locations are from now on white. Vice versa if it is a B-location.
Because it was the first ball we multiply only 1 instead of 4 or 8.
Place the second black ball into one of 7 locations. Multiply 7.
Place the third black ball into one of 6 locations. Multiply 6.
Place the last black ball into one of 5 locations. Multiply 5.
You can exchange the last 3 black balls, so divide 6.
You have 4 white balls left, because there is no difference, multiply 1.
The result is 35 permutations.

Compute the orientations:

The first ball gives the orientation of the whole puzzle. Multiply 1.
The other 7 balls can be twisted 12 ways. Multiply 12^7.
That gives 35,831,808 orientation- configurations.
Multiply permutations and orientations: 35x12^7.

It's the same result than Brandons earlier result. But a different proof.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Fri Oct 22, 2010 3:48 pm 
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Stefan Schwalbe wrote:
[...snip...]If it is an A-location, all A locations are from now on black, and all B locations are from now on white. Vice versa if it is a B-location.
Because it was the first ball we multiply only 1 instead of 4 or 8.

Hi Stefan, I want to read and digest your method some more before I provide any thoughtful comments. Regarding the placement of the first ball though, you have two options, pick a white or a black ball. Or to look at it another way, you have a white ball and you have two spots to place it, an A spot or a B spot.

Doesn't this mean the calculation could count 2 at this phase rather than 1, 4, or 8?

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Fri Oct 22, 2010 3:59 pm 
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3.3.7 (Little Chop / 24 Cube):

I'm at a conference today where half the talks are quite slow and boring so I thought it would be fun to program the Little Chop. Unlike the Pentulitmate, programming it was smooth.

I have found two different (7,1) pure three-cycles 8-)
[UF, UR, RB, UB, RB, UR, UF], [FR], [UF, UR, RB, UB, RB, UR, UF], [FR]
[UB, UL, RB, UF, RB, UL, UB], [UR], [UB, UL, RB, UF, RB, UL, UB], [UR]


I haven't worked on finding non-commutator sequences yet.

This is still a terribly hard puzzle but now we know something better than (10,1).

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Fri Oct 22, 2010 4:27 pm 
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bmenrigh wrote:
Stefan Schwalbe wrote:
[...snip...]If it is an A-location, all A locations are from now on black, and all B locations are from now on white. Vice versa if it is a B-location.
Because it was the first ball we multiply only 1 instead of 4 or 8.

Hi Stefan, I want to read and digest your method some more before I provide any thoughtful comments. Regarding the placement of the first ball though, you have two options, pick a white or a black ball. Or to look at it another way, you have a white ball and you have two spots to place it, an A spot or a B spot.

Doesn't this mean the calculation could count 2 at this phase rather than 1, 4, or 8?


I'm starting to wobble. I'm not sure. After you have placed the first ball, the first ball makes right or wrong for the permutation of all the other 7 balls. The first ball is already solved in permutation. So I multiply only 1 for it's permutation.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Fri Oct 22, 2010 4:52 pm 
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Brandon, I have an even simpler 3-cycle for the little chop:

(UR, RF, LU,
RF,
UR, RF, LU,
RF, ) x2

Which is just a (3,1) done twice. It will result in this 3-cycle:
Attachment:
Skjermbilde 2010-10-22 kl. 23.50.14.png
Skjermbilde 2010-10-22 kl. 23.50.14.png [ 28.98 KiB | Viewed 5683 times ]
At least I think this is a commutator that's not too hard to discover intuitively. Still it's of the same length as yours, so I'll look forward to seeing what else your program comes up with :D

EDIT:
Attachment:
Skjermbilde 2010-10-23 kl. 01.21.35.png
Skjermbilde 2010-10-23 kl. 01.21.35.png [ 56.74 KiB | Viewed 5668 times ]
It worked out fine, but man this puzzle is a whole lot harder than it looks! Basically what I did was that I started out with making two opposite faces, and then tried to get as many correct colors on the correct faces as possible without using the commutator. That didn't take long, I'll assure you, and then I hit some serious trouble with set-ups to get the remaining cycles done.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Fri Oct 22, 2010 7:01 pm 
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Katten wrote:
Brandon, I have an even simpler 3-cycle for the little chop:

(UR, RF, LU,
RF,
UR, RF, LU,
RF, ) x2

Which is just a (3,1) done twice. It will result in this 3-cycle:
hahaha nice find!
And nice solve, your method is a lot like fusions which was discussed a LONG time ago in this thread

As a side not your 3 cycle also works on 4.3.3 (octahedral little chop)

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 12:23 am 
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Katten wrote:
Brandon, I have an even simpler 3-cycle for the little chop:

(UR, RF, LU,
RF,
UR, RF, LU,
RF, ) x2

Which is just a (3,1) done twice. It will result in this 3-cycle: [...image snipped...] At least I think this is a commutator that's not too hard to discover intuitively. Still it's of the same length as yours, so I'll look forward to seeing what else your program comes up with :D
Whoa :shock:, great find! You shouldn't sit on such great routines like this. This isn't actually a commutator. It's just [UR, RF, LU, RF]x4 which seems like an unusual construction for a 3-cycle to me. I'd expect a 2-2 swap or two 3-cycles. This is great. I'll save the details for another post but you may be surprised to learn that 16 moves is the shortest length 3-cycle on the puzzle -- both for commutated routines as well any other free-form routine like this one.

Katten wrote:
EDIT: [...image snipped...] It worked out fine, but man this puzzle is a whole lot harder than it looks! Basically what I did was that I started out with making two opposite faces, and then tried to get as many correct colors on the correct faces as possible without using the commutator. That didn't take long, I'll assure you, and then I hit some serious trouble with set-ups to get the remaining cycles done.
Indeed the setups on this puzzle are horrible. I tried some months ago to solve the little-chop using a 2-2 swap I found. I got down to the last 3-cycle and tried setups for more than an hour before giving up. Congratulations on solving one of the hardest twisty puzzles ever conceived. In my opinion the little chop has the highest "difficulty density" -- that is how hard it is for its size / piece count.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 1:23 am 
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3.3.7 / Little Chop / 24-Cube

The Little Chop is one of the hardest puzzles on Gelatinbrain's site and amazingly, the shortest pure 3-cycles are all at least 16 moves.

There are quite a few different unique triangle patterns and setup moves on the puzzle are quite hard so making more patterns by hand is difficult. 16 move routines can only make 2 different patterns.

For the sake of labeling the different patterns I'll use tri(X, Y, Z) where X, Y, and Z are each the square of the length of one leg of the triangle formed by the 3-cycle. The numbers are only meaningful when related to each other.

These are the shortest commutators for each pattern.

tri(4, 2, 2):
(8,6); [FR, UL, FR, UL, RB, UL, RB, UL], [RB, FR, UL, RB, FR, UL], [UL, RB, UL, RB, UL, FR, UL, FR], [UL, FR, RB, UL, FR, RB]

tri(6, 2, 2):
(11,3); [UR, FR, UF, UL, FR, RB, UB, RB, FR, UF, RB], [FR, UF, FR], [RB, UF, FR, RB, UB, RB, FR, UL, UF, FR, UR], [FR, UF, FR]

tri(6, 6, 4):
(10,3); [FR, UB, FR, UR, UF, UB, RB, FR, UF, UR], [FR, RB, FR], [UR, UF, FR, RB, UB, UF, UR, FR, UB, FR], [FR, RB, FR]

tri(6, 6, 6):
(9,3); [FR, UB, RB, FR, UR, FR, RB, UB, FR], [UF, RB, UF], [FR, UB, RB, FR, UR, FR, RB, UB, FR], [UF, RB, UF]

tri(8, 6, 2):
(11,3); [FR, UB, UL, UF, RB, UL, RB, UF, UL, UB, FR], [UB, RB, UB], [FR, UB, UL, UF, RB, UL, RB, UF, UL, UB, FR], [UB, RB, UB]

tri(10, 4, 2):
(9,5); [FR, UF, RB, UF, UL, UF, RB, UF, FR], [UR, UL, FR, UL, UR], [FR, UF, RB, UF, UL, UF, RB, UF, FR], [UR, UL, FR, UL, UR]

tri(10, 8, 6):
(10,3); [FR, UF, UR, UL, FR, UB, UL, UF, FR, UL], [FR, UR, FR], [UL, FR, UF, UL, UB, FR, UL, UR, UF, FR], [FR, UR, FR]

tri(12, 8, 4):
(10,3); [UB, FR, UB, UF, FR, RB, FR, UB, RB, UF], [UL, UR, UL], [UF, RB, UB, FR, RB, FR, UF, UB, FR, UB], [UL, UR, UL]

tri(12, 10, 6):
(7,6); [FR, UB, FR, UL, UF, UR, UF], [RB, UL, UB, UL, RB, UB], [UF, UR, UF, UL, FR, UB, FR], [UB, RB, UL, UB, UL, RB]

tri(14, 6, 4):
(10,3); [FR, RB, UL, FR, UL, UR, RB, FR, UR, RB], [FR, UR, FR], [RB, UR, FR, RB, UR, UL, FR, UL, RB, FR], [FR, UR, FR]

tri(14, 6, 6):
(9,3); [UB, FR, UL, FR, UB, FR, UL, FR, UB], [UL, FR, UL], [UB, FR, UL, FR, UB, FR, UL, FR, UB], [UL, FR, UL]

tri(14, 8, 2):
(11,3); [FR, RB, UB, RB, UL, UB, UF, UB, UL, FR, UF], [FR, UR, FR], [UF, FR, UL, UB, UF, UB, UL, RB, UB, RB, FR], [FR, UR, FR]

tri(14, 10, 8):
(10,3); [FR, RB, UB, RB, UF, RB, UB, RB, FR, UF], [FR, UF, FR], [UF, FR, RB, UB, RB, UF, RB, UB, RB, FR], [FR, UF, FR]

tri(14, 12, 2):
(12,1); [FR, UB, FR, RB, UR, UF, UR, UL, UF, UB, FR, UR], [RB], [UR, FR, UB, UF, UL, UR, UF, UR, RB, FR, UB, FR], [RB]

tri(14, 12, 10):
(7,1); [FR, UF, FR, UB, FR, UF, FR], [UB], [FR, UF, FR, UB, FR, UF, FR], [UB]

tri(14, 14, 4):
(8,5); [FR, UB, UL, RB, UB, RB, UL, FR], [UB, FR, RB, FR, UF], [FR, UL, RB, UB, RB, UL, UB, FR], [UF, FR, RB, FR, UB]

tri(14, 14, 6):
(10,3); [FR, RB, UB, RB, FR, UF, FR, UB, FR, UF], [FR, UB, FR], [UF, FR, UB, FR, UF, FR, RB, UB, RB, FR], [FR, UB, FR]

tri(14, 14, 14):
(10,3); [FR, UB, UF, FR, UR, UB, RB, FR, UB, UR], [UL, UB, UL], [UR, UB, FR, RB, UB, UR, FR, UF, UB, FR], [UL, UB, UL]

tri(16, 6, 6):
(10,3); [FR, RB, UB, FR, UL, UR, UL, FR, UB, FR], [UL, UR, UL], [FR, UB, FR, UL, UR, UL, FR, UB, RB, FR], [UL, UR, UL]

tri(16, 8, 8):
(11,3); [FR, UB, RB, FR, UR, UF, UB, RB, FR, UF, UR], [FR, RB, FR], [UR, UF, FR, RB, UB, UF, UR, FR, RB, UB, FR], [FR, RB, FR]

tri(16, 10, 2):
(12,1); [UB, FR, RB, FR, UF, UR, UB, UF, RB, UL, RB, UF], [UR], [UF, RB, UL, RB, UF, UB, UR, UF, FR, RB, FR, UB], [UR]

tri(16, 12, 12):
(7,1); [FR, UB, FR, UF, FR, UB, FR], [UF], [FR, UB, FR, UF, FR, UB, FR], [UF]

tri(16, 14, 6):
(10,3); [UB, FR, RB, UF, RB, UL, UR, UF, UL, FR], [UF, RB, UF], [FR, UL, UF, UR, UL, RB, UF, RB, FR, UB], [UF, RB, UF]

tri(16, 14, 14):
(6,3); [FR, RB, UF, FR, UF, RB], [UF, FR, UF], [RB, UF, FR, UF, RB, FR], [UF, FR, UF]

tri(18, 10, 4):
(10,4); [FR, UB, RB, FR, UR, UF, RB, UB, UF, UR], [UF, UL, UF, FR], [UR, UF, UB, RB, UF, UR, FR, RB, UB, FR], [FR, UF, UL, UF]

tri(18, 10, 6):
(10,3); [UB, UF, FR, UL, UB, RB, UB, UL, FR, UF], [UL, UB, UL], [UF, FR, UL, UB, RB, UB, UL, FR, UF, UB], [UL, UB, UL]

tri(18, 12, 6):
(10,1); [FR, UB, UF, UR, FR, UB, UF, FR, UR, UB], [RB], [UB, UR, FR, UF, UB, FR, UR, UF, UB, FR], [RB]

tri(18, 14, 2):
(13,1); [FR, RB, UB, UL, UR, FR, UB, UF, UB, RB, UR, UL, UF], [RB], [UF, UL, UR, RB, UB, UF, UB, FR, UR, UL, UB, RB, FR], [RB]

tri(18, 14, 8):
(10,1); [FR, UF, FR, RB, UB, RB, UF, RB, UB, RB], [UF], [RB, UB, RB, UF, RB, UB, RB, FR, UF, FR], [UF]

tri(18, 14, 10):
(8,1); [FR, UF, UB, FR, UF, FR, UB, FR], [UF], [FR, UB, FR, UF, FR, UB, UF, FR], [UF]

tri(18, 14, 12):
(12,1); [FR, RB, UL, RB, UF, UB, RB, UL, UR, UF, FR, UR], [UF], [UR, FR, UF, UR, UL, RB, UB, UF, RB, UL, RB, FR], [UF]

tri(18, 16, 2):
(9,5); [FR, UL, UB, FR, RB, FR, UB, UL, FR], [UF, FR, UR, FR, UF], [FR, UL, UB, FR, RB, FR, UB, UL, FR], [UF, FR, UR, FR, UF]

tri(18, 16, 10):
(11,1); [FR, UF, RB, UR, UB, FR, UL, UF, RB, UB, UL], [UR], [UL, UB, RB, UF, UL, FR, UB, UR, RB, UF, FR], [UR]

tri(18, 18, 4):
(10,1); [FR, UF, FR, RB, UL, UR, UL, FR, UB, RB], [UR], [RB, UB, FR, UL, UR, UL, RB, FR, UF, FR], [UR]

tri(18, 18, 6):
(10,1); [FR, RB, UR, FR, RB, UR, UL, FR, UL, RB], [UR], [RB, UL, FR, UL, UR, RB, FR, UR, RB, FR], [UR]

tri(20, 10, 10):
(11,3); [FR, UB, UR, UF, UR, UB, UR, RB, UR, FR, UL], [FR, RB, FR], [UL, FR, UR, RB, UR, UB, UR, UF, UR, UB, FR], [FR, RB, FR]

tri(20, 12, 8):
(11,1); [FR, UF, FR, UL, UB, UF, UB, UL, RB, UB, RB], [UR], [RB, UB, RB, UL, UB, UF, UB, UL, FR, UF, FR], [UR]

tri(20, 14, 6):
(10,3); [FR, UB, FR, RB, UR, UB, FR, RB, UB, UR], [FR, UF, FR], [UR, UB, RB, FR, UB, UR, RB, FR, UB, FR], [FR, UF, FR]

tri(20, 16, 4):
(9,5); [FR, UB, FR, UR, UF, UB, RB, FR, UF], [UL, FR, RB, FR, UL], [UF, FR, RB, UB, UF, UR, FR, UB, FR], [UL, FR, RB, FR, UL]

tri(20, 18, 2):
(12,1); [FR, UB, UR, UL, UF, FR, RB, FR, UB, UL, UF, RB], [UR], [RB, UF, UL, UB, FR, RB, FR, UF, UL, UR, UB, FR], [UR]


I'm not sure if that's all the possible patterns or not but it's all of them that can be reached using a commutator with less than 30 moves. I really hope there isn't a bug in my program because cataloging those took forever!

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Last edited by Brandon Enright on Tue Jan 18, 2011 9:56 pm, edited 7 times in total.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 2:34 am 
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I'm surprised to know that it's not actually a commutator :shock: I've know it for some time, but I wasn't aware that it would work for the little chop. I first came up with it when I tried solving puzzles like 5.2.1 or the Helicopter cube etc. But take 5.2.1 as an example. If you perform the sequence once, you'll have isolated 3 pieces and when you perform it the second time, you'll preserve these pieces and at the same time restore the others, resulting in this:
Attachment:
Skjermbilde 2010-10-23 kl. 09.14.15.png
Skjermbilde 2010-10-23 kl. 09.14.15.png [ 31.97 KiB | Viewed 5632 times ]
Now, this works exactly the same on the Helicopter cube. Also I know it to work for all 5.2.x puzzles plus 1.4.2. Also now that Elwyn pointed it out, it also works for 4.3.3 :shock: I had no idea that this would be so universal. It was just the simplest thing that came to mind when I tried to solve half turn puzzles, and I think 5.2.1 was the one I solved first.
bmenrigh wrote:
I'll save the details for another post but you may be surprised to learn that 16 moves is the shortest length 3-cycle on the puzzle -- both for commutated routines as well any other free-form routine like this one.
I can't wait! This puzzle really surprised me. And I can't believe I came up with a free-form routine without knowing it! Really, 16 is the shortest? This right here is proof that this puzzle is not by any means as easy as it may look.
bmenrigh wrote:
Indeed the setups on this puzzle are horrible. I tried some months ago to solve the little-chop using a 2-2 swap I found. I got down to the last 3-cycle and tried setups for more than an hour before giving up. Congratulations on solving one of the hardest twisty puzzles ever conceived. In my opinion the little chop has the highest "difficulty density" -- that is how hard it is for its size / piece count.
I have to tell you, even with a proper 3-cycle this was very very hard. But I'm certain that you can solve it with one of the cycles you've already found using your program. I think I'll try a different approach next time I solve it. I'm wondering if it would be better to start off with making 3 adjacent faces instead of two opposite. And maybe with your program we'll know the optimal routine for it and can work on the best possible method to solve it with.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 8:09 am 
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Amazing routine, Katja! I suspected that a 16 move pure cycle would be possible for the Little Chop, but I did not expect it to be a commutator found by a human! Your algo can be viewed as a (7,1) commutator, if we reverse the order:

RF, LU, RF, UR, RF, LU, RF,
UR,
RF, LU, RF, UR, RF, LU, RF,
UR

Here is a diagram I made of the position after the first 3 moves to help me understand how the main 7 move conjugate sequence works:
Attachment:
's Little Chop algo works.jpg
's Little Chop algo works.jpg [ 35.87 KiB | Viewed 5623 times ]

I have drawn blue lines around the half we are interested in, the one centered on DL. Of the 12 pieces in the half, the 5 pieces that have stayed in the same position are colored gray, the 4 pieces that have swapped places in pairs are shown in black, and the other 3 are shown normally. The purple lines show how those 3 pieces have been swapped. Obviously if we undo the first 3 moves we just get back to the solved position. But what if we do UR first, and then undo the first 3 moves? The gray and black pieces will still end up in their solved places, the red and white pieces marked S in the other half will end up back in their former places but swapped, and the green piece marked with a dot will be replaced by the yellow piece marked with a dot.

So after 7 moves we have a half with a single swapped piece and a pair that has swapped places. So we can turn the other half, undo the 7 moves, turn the other half again, and we have a pure cycle in 16 moves. This clever trick -- not minding about a non-symmetrical additional piece swap -- only works with deep cut puzzles.

I love that this algo originated from an edge-turning tetrahedral puzzle! :D

I believe that Brandon's (7,1) commutators are this algo and its mirror image.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 9:17 am 
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Katten wrote:
Now, this works exactly the same on the Helicopter cube. Also I know it to work for all 5.2.x puzzles plus 1.4.2. Also now that Elwyn pointed it out, it also works for 4.3.3 :shock: I had no idea that this would be so universal. It was just the simplest thing that came to mind when I tried to solve half turn puzzles, and I think 5.2.1 was the one I solved first.

It's like we're all on a search for beautiful diamonds and this forum is us sitting around, discussing our strategies. Short routines on the Little Chop are like mythical blue diamonds that we've scaled Everest and scoured the deserts looking for -- then you walk up and say "oh what, you guys mean these little things?" as you pull a handful of them out of your pocket. :lol:

You should share diamonds when you find them so that the rest of us can marvel at their beauty. Routines from my program are like cheep knock-offs and make poor substitutes.

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Prior to using my real name I posted under the account named bmenrigh.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 1:43 pm 
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bmenrigh wrote:
You should share diamonds when you find them so that the rest of us can marvel at their beauty. Routines from my program are like cheep knock-offs and make poor substitutes.
Hahaha, I love the metaphors! I shared it as soon as I found it though! I decided to give the little chop some twists after reading your post yesterday. And this wild thought hit me: what if the routine that I created for 5.2.1, which also works for the helicopter cube would actually work for this as well? And it did! :shock: I really stumbled upon something great here, didn't I?
Julian wrote:
Amazing routine, Katja! I suspected that a 16 move pure cycle would be possible for the Little Chop, but I did not expect it to be a commutator found by a human!
Wow, I guess this confirms what I said above! Also I loved your illustrative and technical outline of how it actually works. And this just further confirms the fact that this is one complex puzzle. But I'm not really sure that I know what a conjugate is?

Also, I will be looking out for other puzzles this routine might work for. As my intention when I came up with it were for the edge-turning tetrahedral only. So far it's worked for these puzzles:

5.2.1 - 5.2.8, 1.4.1, 1.4.2, 1.4.8, 1.4.9, 3.3.1, 3.3.7, 3.3.9, 4.3.3

EDIT: all of this makes me draw the following conclusion: I was horribly mistaken when I said that this was probably the most intuitive commutator to find for the little chop. Also, I called the routine "simple", but now I realize that it's not :lol:


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sat Oct 23, 2010 11:35 pm 
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I just did a 105 move little chop solve but i don't understand how... i am very bad at getting the first face and can't actually get the second face intuitively, in fact i have no intuition whatsoever on this puzzle, every setup move was difficult. I think i am going to have to practice this one a lot before i actually feel confident solving it.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 24, 2010 3:18 am 
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Elwyn wrote:
in fact i have no intuition whatsoever on this puzzle, every setup move was difficult. I think i am going to have to practice this one a lot before i actually feel confident solving it.
This is what makes you 105 move solve even more impressive! Did you use the same method as I did? And I also feel that I don't fully understand this puzzle yet. I'm not even sure why I managed to solve it :lol: But I got better at the end of the solve, so hopefully with some practice I'll be able to get rid of my 900 move solve.

Btw, Elwyn: do you think you'll be able to beat the 50 move record for this with practice?


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Sun Oct 24, 2010 7:56 am 
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Katten wrote:
Did you use the same method as I did? And I also feel that I don't fully understand this puzzle yet. I'm not even sure why I managed to solve it :lol: But I got better at the end of the solve, so hopefully with some practice I'll be able to get rid of my 900 move solve.

Btw, Elwyn: do you think you'll be able to beat the 50 move record for this with practice?
well i suppose i did use a similar method, one face then 3 pieces on the other at which point i saw getting the last piece wouldn't be worth it because i had a fair few solved pieces on the other 4 faces then just cycles.

I doubt i could beat it without a lot of practice mixed with a lot of luck.

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 25, 2010 6:44 am 
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Katten wrote:
I'm not really sure that I know what a conjugate is?
A commutator is an algorithm like p q p' q' and a conjugate is an algorithm like p q p'. With the conjugate you don't undo the q sequence at the end.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 25, 2010 7:01 am 
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Elwyn wrote:
I just did a 105 move little chop solve but i don't understand how... i am very bad at getting the first face and can't actually get the second face intuitively, in fact i have no intuition whatsoever on this puzzle, every setup move was difficult. I think i am going to have to practice this one a lot before i actually feel confident solving it.
I also struggle to solve the first face, and I have never succeeded with making a second face intuitively. For me, one of the funniest moments in the history of this thread was when fusion said something casually like, "First you solve one face, then you solve the opposite face, then..." And everyone else was responding along the lines of, "Whoa, you've lost me at 'solve the opposite face'. How do I do that?!" This evening I'll do a search and see if the link to his guide still works.

Little Chop setups are really tough, but for a whole new level of evilness try playing around with the petal-shaped circle pieces of 3.3.10 and 3.3.11! I found the setups so difficult for 3.3.7 and 4.3.3 that a while ago I wrote a table of all the possibilities (by my count, 64 perms not including reflective symmetries, 93 perms including). All of them are reachable in a maximum of 6 setup moves from the perm of my (10,1) cycle. I suspect the same is true with Katja's (7,1) cycle.


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 25, 2010 8:03 am 
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I have had a bit more practice and i think i have the hang of getting the first face well and have gotten the second face once...
Julian wrote:
This evening I'll do a search and see if the link to his guide still works.

Little Chop setups are really tough, but for a whole new level of evilness try playing around with the petal-shaped circle pieces of 3.3.10 and 3.3.11! I found the setups so difficult for 3.3.7 and 4.3.3 that a while ago I wrote a table of all the possibilities (by my count, 64 perms not including reflective symmetries, 93 perms including). All of them are reachable in a maximum of 6 setup moves from the perm of my (10,1) cycle. I suspect the same is true with Katja's (7,1) cycle.
The link still works viewtopic.php?p=109603#p109603 , i went back and looked at a bit of it after i struggled through the solve, (it comes as lots of documents because it was apparently too big for one) i haven't learnt anything from it yet though except the pseudo 2-2 swap UR UF UB UF UR which is meant to be used once you have the top and bottom faces to get some more pieces on the last 4. I say pseudo because in actual fact it is far from a 2-2 swap it just appears to be on a solved puzzle

Alright i just downloaded it all and got it into one doc, I think i might have an actual read through it tomorrow. I put it into one document and attached it, i see no reason for fusion to be upset with this as i am just making his guide available to people in the same thread he posted it. Is there a reason he thought it was too big for one doc?

I will PM him and take it down if there is a problem.

Also i'm amazed at the 6 setups for every cycle. This is the bit that will take practice to be able to see them quickly (or see them at all would be nice for now).

I just did another solve, i used the guide for the second face but i have gotten better at setups on my own which is good, i wasn't being careful with moves and got a 150 move solve.

Just got the second face intuitively again... i hope that means i'm getting better not lucky.

Alright i have now done a few solves, it's still hard but it's also 1am, i'll read the guide and have more practice tomorrow, this is a fun and challenging puzzles :)


Attachments:
Little Chop.doc [3.25 MiB]
Downloaded 88 times

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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Mon Oct 25, 2010 10:11 am 
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Attachment:
Skjermbilde 2010-10-25 kl. 16.55.33.png
Skjermbilde 2010-10-25 kl. 16.55.33.png [ 55.93 KiB | Viewed 5503 times ]
Well, that was hard! Making one and a half face intuitively went relatively smooth, but after that I had to go over to the (7,1) cycle. After I had gotten 3 full faces, I got completely stuck for about 30 minutes before I was able to find the right set-up moves which allowed me to continue. Surprisingly, the last cycle I did, only required one easy set-up. Unlike the rest, which were quite horrible. But I got pretty good at discovering them easily and quickly towards the end, so with practice I think I'll be able to understand this puzzle much better.

As for 3.3.7: I also struggle with making the second face intuitively, but I did manage to do it... once. A little practice is definitely needed!
Julian wrote:
for a whole new level of evilness try playing around with the petal-shaped circle pieces of 3.3.10 and 3.3.11!
I noticed them yesterday, actually, and gave them some twists. And my jaw literally dropped to the floor; as if the normal little chop wasn't hard enough :lol: I will definitely set it as a high priority goal of mine to solve them. So far I'm stumped on how though.

EDIT: Just did my first re-solve of the little chop and submitted a 275 moves, 16 minutes solve :D I'm getting better at the set-ups already!


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 Post subject: Re: Gelatin Brain's Applet Solutions Discussion Thread
PostPosted: Thu Oct 28, 2010 2:45 am 
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So yesterday I came up with all the missing commutators for 1.1.39 and I actually managed to find pure (3,1)'s for most of them. However, my pure (not (3,1)) cycle for the thin triangles kinda backfired a little on me. Here's the pieces it cycles:
Attachment:
Skjermbilde 2010-10-27 kl. 18.47.43.png
Skjermbilde 2010-10-27 kl. 18.47.43.png [ 60.76 KiB | Viewed 5445 times ]
It only works for the pieces in those exact spots and as far as I experienced, getting the other pieces into those spots turned out not to work at all. This surprised me a lot as I didn't think this would occur. I'd like your thoughts on this; is there anything extremely obvious that I'm not seeing here?

I also put in a 1.1.8 solve yesterday. I did not use reduction to Megaminx, but I'm looking forward to trying that some time. Also, I've been putting this one off for a long time, as I've read that others had trouble with it. However, cycling the wide triangles does not have to be done using a pure routine. Meaning I've known how to solve it for some time now.


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