Unreachable positions

[Jaspar]

How do we know that there are unreachable positions on a Rubik's cube?
How do we know what they are?

I'm interested in a proof or proof sketch that:
(i) there are positions on a Rubik's Cube that cannot be reached by legal moves.
(ii) there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge ect.

 

[Cride5]

Every quarter turn on a Rubik's cube permutes the pieces on that layer, such that there is an odd-parity permutation of edges and an odd-parity permutation of corners. Together they create an overall even-parity permutation of the pieces on that face. Because all legal moves on a Rubik's cube can be broken down into a sequence of quarter turns, it is impossible for the cube to enter a permutation with odd parity if only legal moves are used.

So for:
(i) A single swap of two edges, or any odd-parity permutation is a state which cannot be reached by legal moves.
(ii) A swap of a single pair is an odd-parity permutation, and is impossible to achieve with quarter-turns according to the above.

EDIT: More information on this wiki page.

 

[Stefan]

http://www.ryanheise.com/cube/cube_laws.html

 

[Jaspar]

So you only need to analyze one quarter turn. I was expecting something a lot more complicated.

 

[qqwref]

Yeah, since any move sequence is just a bunch of quarter turns after each other, looking at one quarter turn is enough to figure out what kinds of positions are unreachable.

It's a bit trickier to prove that every position you think is reachable IS, though.

 

[M4rQu5]

If it can be reached with legal moves, atleast it is undoable with the inverted moves.

 

[meichenl]

I gave a lecture on why you can't swap two cubies for the Berkeley cubing class earlier this year. You can get it here:

http://dl.dropbox.com/u/1377494/parityLecture.wmv

Sound isn't great - my photographer decided to sit right next to the camera and cube - and it's more rushed than I'd like because I was trying to fit it all in half an hour. The basics of the entire thing are there, though.