## Rubik's Cube Conjecture, true or not?

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### Rubik's Cube Conjecture, true or not?

I was moving a solved rubik's cube U and R like for 5 minutes, and then the cube became solved again! I conjectured that if you keep on moving a cube to a certain sequence of moves, the cube will go back to its original position.

I wrote a Java applet to test more complex sequences, and all of them worked so far.
The applet is at http://davidpan.co.cc/Rubiks_Cube_Conjecture/

Does anyone have any ideas on this? It kind of makes sense if you think about it, but I'm wondering how you can rigorously prove it.
function

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### Re: Rubik's Cube Conjecture, true or not?

It's true but I don't know how to rigorously prove it.  If you repeat an algorithm in the same way over and over it the cube will go back to its original postion.

Concerning the U R alg: If you count the quater turns as you do it, every 6 quarter turns the 'urb' corner is in the correct postion, every 14 quarter turns all the edges are in the correct position and every 30 quarter turns the corners are in the right postion.  If you take the LCD of 1/6, 1/14, 1/30 you'll have 1/210 invert that and you'll get 210, so it's 210 quarter turns or face turns (what ever way you count them.)   the cube will go back to its original position. At 70 quarter turns the pieces are in the right places but the 6 corners are 240 degrees out of phase  counterclockwise spin - whatever you call it. At 35 quarter turns [17(U R) U] there's a neat sysmetrical pattern.  At 15 quarter turns [7(U R) U] the 6 corners are upsidedown, which is useful for the 4x4 cube or corners first method.

Blue

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### Re: Rubik's Cube Conjecture, true or not?

Thanks for your explanation, it's really interesting... U R does take 210 turns!
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Joined: Wed May 19, 2010 11:35 am

### Re: Rubik's Cube Conjecture, true or not?

actually, it is true! proof here: http://dalpi.co.cc/rubiks/
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Posts: 3
Joined: Wed May 19, 2010 11:35 am