Rigorous Proof of Laws

[PeterNewton]

All right Stefan, I understand that stuff about parity now. Took some time, but I get it, thanks. But like you and Herbert said, only the impossible is proven, so it is only an upper bound. How would I go about proving the possible, the lower bound? Only then can one say that since lower bound = upper bound, this is the exact fraction of all conceivable states that are possible to reach through legal moves.

 

[Matt S]

The lower bound is proven by construction. Essentially, you demonstrate a method of generating any cube state that isn't explicitly disallowed by the Laws.

I think the most straightforward way is to prove that T-Perm (or any other that swaps two edges and two corners), a two edge flipper, a two corner twister, and congugates are sufficient to generate the positions. Actually, T-Perm and congugates alone are sufficient to do the orienting also, but that's more complicated to prove.

 

[Stefan]

Yeah, show a way to actually construct (or equivalently, solve) the possible states. Again have a look at this:
http://arcsecond.wordpress.com/2008/12/06/answer-parity-of-permutations/
Note how part one proves that any permutation is possible by describing a way to do it.

 

[PeterNewton]

Hmmm ok I was able to do this for the 3x3 without too much difficulty, since I am familiar with the 3x3 laws. But my friend and I are doing a project on the general formulas for the number of combinations of any odd nxn cube and any even nxn cube (one for odd, one for even).
Unfortunatly, neither of us are very good with big cubes and so, while we understand orbits and such, we do not know the laws for a general odd or even cube. Do you mind listing them or linking them?
The upper bounds should not be too difficult to derive then using methods similar to the 3x3. But then what would the construction algorithms be for deriving the lower bounds? I'm guessing they constitute of things like 'there is an alg to transpositon (transepose?) two wing edges in orbit in an odd cube' and obviously a fair bit of additional logic to show that it can be extended to any two wings in orbit. What kind of these type exist in general for odd and/or even cubes?

 

[mrCage]

Me and commutators are even

Per

 

[PeterNewton]

Me and commutators are even

Per

Huhhh? I think you forgot to finish typing or something.

 

[PeterNewton]

All right. Forget what I said earlier. There are only 4 things that I need now to complete the proof.

I need confirmation that:
1) There is an algorithm that can switch two center pieces in the same orbit without disturbing the rest of the cubies, even or odd. Any pair case will do since the argument can then be extended to all possible pairs.
2) There is an algorithm that can switch two wing edges in the same orbit in an odd cube, or two edges in the same orbit in an even cube, without disturbing the rest of the cubies. Again, any pair case will do.

Proof that:
1) There is only one possible orientation of any center piece on even or odd cubes, other than central centers, for each location it can occupy in its orbit.
2) There is only one possible orientation of any non-central edge on odd cubes and any edge on even cubes for each location it can occupy in its orbit.

For the two proofs, I know that you can talk about how Verdes' general nxnxn design does not allow it because of the internal structure, but I will use that only as a last resort, since it is not very elegant.

EDIT: maybe I should explicitly say that some help for the above 4 would be nice...