# The Naturally Desired Scramble State



## Dane man (Jun 16, 2014)

I think we all at one point or another have been messing around with the cube and been looking to scramble it in such a way that no single color is aligned with itself anywhere on the cube (without using algorithms). We turn and turn trying to break the pairs that continue to show up as we turn it, until eventually, in rare instances, we finally manage to scramble a cube in such a way as to separate all colors from their own kind. One example of this kind of scramble is known as the super flop. And I just achieved one recently, and it got me thinking... 

What is the minimum number of moves required to achieve one? (Answered)
How many of these states exist on the 3x3? 

This is either a mathematical problem or one that requires computer power to be thrown at it as well. 

But what do you guys think? Can it be found?


_Hey look, hundredth post._


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## Coolster01 (Jun 16, 2014)

6 moves: 
R2 L2 U2 D2 F2 B2


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## XTowncuber (Jun 16, 2014)

Dane man said:


> What is the minimum number of moves required to achieve one?



6 

R2 L2 F2 B2 U2 D2


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## kcl (Jun 16, 2014)

3 

M2 E2 S2


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## Rocky0701 (Jun 16, 2014)

kclejeune said:


> 3
> 
> M2 E2 S2


Those count as two each for FMC though, so six.


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## Ninja Storm (Jun 16, 2014)

Is there a position without either a color touching itself or the opposite color?


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## Coolster01 (Jun 16, 2014)

Ninja Storm said:


> Is there a position without either a color touching itself or the opposite color?



Super flip, super flip + checkerboard.


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## Dane man (Jun 16, 2014)

Coolster01 said:


> 6 moves:
> R2 L2 U2 D2 F2 B2


There goes the second question. How about the first one?


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## Lucas Garron (Jun 16, 2014)

Dane man said:


> But what do you guys think? Can it be found?



I think yes. If not for 3x3x3, then for puzzles which are nearly as complex. Very clever dynamic programming.
Maybe we can interest Tom in this. 

In any case, "naturally desired" isn't really a good term for this. If you want to discuss it a lot, I suggest something accurate and objective like "color-separating".



Dane man said:


> There goes the second question. How about the first one?



No, it doesn't? I don't see a proof in this thread that 5 moves are not enough.


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## Tempus (Jun 16, 2014)

Lucas Garron said:


> No, it doesn't? I don't see a proof in this thread that 5 moves are not enough.


On a solved cube, there are 72 adjacent pairs of like-colored stickers, 12 per side, each of which must be broken up. When you turn one outer layer, you're only cutting through 12 of these, and so one such move cannot break up more than 12. 72/12=6, ergo 6 turns is a lower bound. Since we already have an example of a 6-turn solution, 6 is also the upper bound. Ergo, the number is 6. Q.E.D.


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## DeeDubb (Jun 16, 2014)

I was wondering if it was possible to generate something that looks like a random scramble (and follows whatever rules are required for regulation scrambling), but ends up solving the cube (without just looking like half a scramble and then undoing it).


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## Lucas Garron (Jun 16, 2014)

DeeDubb said:


> I was wondering if it was possible to generate something that looks like a random scramble (and follows whatever rules are required for regulation scrambling), but ends up solving the cube (without just looking like half a scramble and then undoing it).



There are no rules for what the scramble "looks like", only for what it does (produce a random state).

However, you can make it a goal to produce an identity alg that looks like the reversed output of a two-phase solver.
I don't know if anyone's done that, but I did something similar for an April Fools' prank back when the scramble was just random moves: R' U F R2 D B' R2 L2 F B2 D2 F' B R2 B L2 F' B U2 F R D' F' R2 U'


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## DeeDubb (Jun 16, 2014)

Lucas Garron said:


> There are no rules for what the scramble "looks like", only for what it does (produce a random state).
> 
> However, you can make it a goal to produce an identity alg that looks like the reversed output of a two-phase solver.
> I don't know if anyone's done that, but I did something similar for an April Fools' prank back when the scramble was just random moves: R' U F R2 D B' R2 L2 F B2 D2 F' B R2 B L2 F' B U2 F R D' F' R2 U'



Wow, yeah, that's what I was looking for. Actually, it would be funny if on Aprils Fool's, QQtimer was pumping out random scrambles that just solve the cube. 

It seems like it would be a fun challenge to program something to generate random scrambles that just solve the cube.


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## guysensei1 (Jun 16, 2014)

DeeDubb said:


> Wow, yeah, that's what I was looking for. Actually, it would be funny if on Aprils Fool's, QQtimer was pumping out random scrambles that just solve the cube.
> 
> It seems like it would be a fun challenge to program something to generate random scrambles that just solve the cube.



cubeexplorer doesn't handle solved cubes well. It insists that 8 moves is 'optimal'...

EDIT:
F' U D' L2 U F2 R2 U' D B2 R2 U' F' (13f)

it's short but meh.


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## Rocky0701 (Jun 16, 2014)

DeeDubb said:


> Wow, yeah, that's what I was looking for. Actually, it would be funny if on Aprils Fool's, QQtimer was pumping out random scrambles that just solve the cube.
> 
> It seems like it would be a fun challenge to program something to generate random scrambles that just solve the cube.


Yeah, but then there would be a ton of noobs going around being like: "One time, I got a complete whole cube skip, so my PB is 0" or something stupid like that


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## megaminxwin (Jun 16, 2014)

I wonder if there are any positions where no colours touch, even diagonally? I'd try to find out but I am tired and lazy.


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## whauk (Jun 16, 2014)

Sth like M E M' E M' S ?
http://alg.cubing.net/?setup=M_E_M-_E_M-_S
Don't know whether it's optimal. I just messed around with slice-moves.


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## Dane man (Jun 16, 2014)

DeeDubb said:


> I was wondering if it was possible to generate something that looks like a random scramble (and follows whatever rules are required for regulation scrambling), but ends up solving the cube (without just looking like half a scramble and then undoing it).


You could use cube explorer to generate two different solutions for a scramble, make the reverse of one of them, then put them together. (Ex: L' B2 R' B' U' D B' L' B2 R' L R B D' U B L R)



megaminxwin said:


> I wonder if there are any positions where no colours touch, even diagonally? I'd try to find out but I am tired and lazy.


See whauk's scramble.


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## 10461394944000 (Jun 16, 2014)

whauk said:


> Sth like M E M' E M' S ?
> http://alg.cubing.net/?setup=M_E_M-_E_M-_S
> Don't know whether it's optimal. I just messed around with slice-moves.



S E2 M'


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## rokicki (Jun 16, 2014)

Interesting challenge; I think it can be done as Lucas says with dynamic programming.
I'm not sure it can be done just with pencil and paper.

What if we extend it to colors not touching diagonally---even across a cube edge?
(So if the upper sticker on the up-front-right corner is yellow, the right sticker on
the up-right edge cannot be yellow either).


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