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Scramble lengths for big cubes.

blade740

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Mockskin and I were discussing bigger cubes (6x6x6, or "6c3" and up) on IRC today, and he had the idea that scrambles could be made more optimal on the 4c3 and 5c3 by using multiple-layer turns instead of single-layer slice turns. I decided to calculate the number of modifications for each piece of each puzzle for scramble of turn length t.


USING SINGLE LAYER TURNS:

3c3
average mods per corner (t/2)
average mods per edge (t/3)

4c3
average mods per corner (t/4)
average mods per edge (t/4)
average mods per center (t/4)

5c3
avg mods per corner (t/4)
avg mods per outer edge (t/4)
avg mods per center edge (t/6)
avg mods per T center (t/6)
avg mods per X center (t/4)

Assuming the standard scramble lengths, these scrambles allow for an average piece complexity of 10 modifications.

USING THICK TURNS:

4c3
average mods per corner (t/2)
average mods per edge (5t/12)
average mods per center (t/3)

5c3
avg mods per corner (t/2)
avg mods per outer edge (5t/12)
avg mods per center edge (t/3)
avg mods per T center (t/4)
avg mods per X center (t/3)

With thick layer scrambles, we can reduce the MINIMUM, not even the average, complexity per piece, while putting scramble lengths for the 4c3 and 5c3 at 30 and 40 turns, respectively.

Anyway, I started to work out the scramble length required for the same complexity on the 6c3.

I labeled the pieces like so:

[1][2][3][3][2][1]
[2][4][5][5][4][2]
[3][5][6][6][5][3]
[3][5][6][6][5][3]
[2][4][5][5][4][2]
[1][2][3][3][2][1]

Now, we found another problem. Should the ratio of face turns to slice turns be even (2:1:1), or should the ratio of face turns:second slice turns:third slice turns be even(1:1:1)?
Here are the average complexities with a 1:1:1 ratio:
[1]: (t/6)
[2]: (t/6)
[3]: (t/6)
[4]: (t/6)
[5]: (t/12)
[6]: (t/6)

And here are the complexities with a 2:1:1 ratio:

[1]: (t/4)
[2]: (5t/24)
[3]: (5t/24)
[4]: (t/6)
[5]: (t/6)
[6]: (t/6)

This means that we'd need a minimum of 60 single layer turns to scramble a 6c3 either way (if you allow for less complexity on the 5 pieces). The difference is, you get more complexity on the edges and corners and on the [5] pieces.


Here's what you get with multiple layer turns. This time, it will take 35 turns to scramble all pieces except [5] to a complexity of 10.
[1]: (t/2)
[2]: (7t/18)
[3]: (7t/96)
[4]: (4t/9)
[5]: (5t/36)
[6]: (5t/16)






Sorry if I bored you with that. Anyway, that's how we spend our nights in IRC. Mock an I also came up with a scalable notation for bigger cubes than 5c3 (as well as the term 5c3, meaning a cube of order five and dimension three)

We're very bored.
 

Johannes91

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Why do you define the quality of a scramble by counting those "complexities of pieces"?? The depth of the position seems much more logical to me. 40 moves to scramble a 5x5 can't be enough, I'm sure there are tons of positions that are farther than 40 moves from solved. You could just rotate the cube (xy'z2x2yzy'x2y'zx' etc.) and that would move every single piece many times, but it doesn't mean that the cube is scrambled well.

Originally posted by blade740+Jan 25 2007, 08:40 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td>QUOTE (blade740 @ Jan 25 2007, 08:40 AM)</td></tr><tr><td id='QUOTE'>Here's what you get with multiple layer turns. This time, it will take 35 turns to scramble all pieces except [5] to a complexity of 10.[/b]

Ummh, are you saying that 35 moves is enough to scramble a 6x6? There's seriously something wrong with your logic.

Originally posted by [email protected] 25 2007, 09:38 AM
Now here is a question for the 3c3: Obviously thick turns and face turns are the same, but what would result in a better scramble: thick turns or single layer turns?
You answered your own question. They are obviously the same, it doesn't matter how you write the moves.

<!--QuoteBegin-blade740
@Jan 25 2007, 02:55 PM
thick turns lower the turns needed to 20, instead of the original 25.[/quote]
:S:S:S
Every scramble can be written using only UDFBRL (single layer turns) or only udfbrl (double layer turns). It doesn't matter, the moves are still the same. It can't change the quality of the scramble.
 

blade740

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We're counting complexity just to see if it matters. I'm not saying that this is a better way of doing it. This post was actually more for speculation on how to determine scramble length for the 6c6.

You're right the 3c3. I was forgetting that since you can do cube rotations for free, the extra 4 pieces you move on a double layer turn don't actually move with respect to the fixed centers. That's why I didn't originally put it in my first post. None of the other scrambles assume going past the center in multi-layer turns.

About the 6c6, 35 turns will NOT solve it to where every piece has moved an average of 10 times. The pieces labeled [5] will take 720 turns to average that. Nobody wants to scramble 720 turns.
 

AvGalen

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Originally posted by AvGalen+Jan 25 2007, 09:38 AM--></div><table border='0' align='center' width='95%' cellpadding='3' cellspacing='1'><tr><td>QUOTE (AvGalen @ Jan 25 2007, 09:38 AM)</td></tr><tr><td id='QUOTE'>Now here is a question for the 3c3: Obviously thick turns and face turns are the same, but what would result in a better scramble: thick turns or single layer turns?[/b]

<!--QuoteBegin-Johannes91
@Jan 25 2007, 06:55 PM
You answered your own question. They are obviously the same, it doesn't matter how you write the moves.[/quote]

No I did not. A layer is not the same as a face! Single layer turns on a 3x3x3 could also be slice moves.
 
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