To expand on what Maarten said I have included a proof about the 4x4x4 cube below. Also, Maarten :fp I did not see your post before I started typing my own, so I apologize for the repeat parts.
Despite quite a thorough search, I haven't been able to fine a proper explanation for OLL parity. As far as I can see (by looking at the cube), OLL parity is when only two dedges are swapped. The reason this maifests its self as an orientation problem is that when a pair of dedges are swapped in place, then they must also be flipped too because the internal mechanics of the pieces wouldn't allow otherwise.
Mostly correct. "Parity" is the state of a permutation of pieces, however we misuse this term to describe the state when the parity is odd (i.e. the permutation is in an odd permutation state). The proper term we should use is "wings have odd parity" or something like this. However, we incorrectly say "wings have parity", or "the cube has parity". In fact, each group of pieces on the cube
always have parity, because pieces are always in either an even or odd permutation.
Your assertion that "odd parity" in the wings asserts itself as two wings swapped is the only part I truly, and completely, disagree with. I'll give you another example of odd parity. From a solved cube do the turn
r. You cube now has OLL parity, by the true definition of what OLL parity is. So as not to be a complete stickler for terminology, I will add to your definition of OLL parity the additional: "When the wing edges are in an odd parity state when the cube is being solved with the reduction method, and centers are solved and edge groups are built, then this manifests itself as an impossible edge orientation case, considering the rules of a 3x3x3 cube."
My question is: How can only two dedges be swapped, and nothing else. Its counter to my intuition on how twisty puzzles operate. Does OLL parity possibly mean that two centre pieces are also swapped? If anyone could shed light on this I'd be very interested. I'd like to be able to solve OLL parity intuitively, but I've no idea how to at the moment
This is my favorite question in cubing theory, and it's the reason why I love the 4x4x4 more than any other n x n x n cube. Let's for a second not only consider the 4x4x4 cube, but the 4x4x4 supercube. On the 4x4x4 supercube it is indeed possible to switch any two wing edges, and
affect no other part of the cube at all! Not only is that true, but the 4x4x4 supercube is the only n x n x n supercube where you can swap any two wing edges in the same orbit without affecting any other piece on the cube. For all other cube sizes that have edge pieces, swapping only two of those edges
must have an effect on some other pieces of the cube!
Here is the proof that it is possible to swap two wings on a 4x4x4 and not affect any other pieces on the cube:
When turning an inner layer of a 4x4x4 supercube by one quarter turn you are affecting four wing edges, and you are 4-cycling them (toggling the permutation parity, and yes I mean parity in the correct sense of the word here). You are also creating two 4-cycles of centers that are in the same orbit. This net effect is an even permutation, so the parity of the centers remains the same. From here you can use commutators and 3 cycles to return all the centers to their correct location (all even parity positions are achievable using 3 cycles) and you can do 1 commutator on the wing edges to solve 2 of the now unsolved pieces back to their original location. This leaves 2 wing edges swapped, and everything else in its original location.
Proof for why this does not work on any other cube size:
When turning an inner layer slice by one quarter turn on a cube larger than 4x4x4 there are now two types of center pieces. There are the "x" centers, and the "oblique centers" as I call them (you can rename them how you like, but I define my terms in the next sentence). Each inner slice contains exactly 8 "x-centers" from the same orbit, and only from one orbit of "x-centers". Each inner slice contains exactly 4, and no more, "oblique centers" from the same orbit, for 1 or more different orbits. I could provide you with the formula for how many orbits exactly, but I just want you to get the basic idea.
Here's how this affects the pieces when doing a single quarter turn of an inner layer. When turning the inner layer the wing edges are 4-cycled (toggling the parity). The "x-center" orbit that is affected has had two 4-cycles of pieces created which is a net even permutation. Parity of this orbit is not affected. Now, all the remaining center pieces on this slice are "oblique" centers, and each orbit contained on this slice has only 4 of its pieces on that slice. By doing the inner slice quarter turn you create a 4-cycle of pieces in every single one of these orbits, thus toggling the parity of the permutation in every single one of those orbits.
Therefore the 4x4x4 supercube is the only supercube where the parity of the wing edge permutation is completely independent of the parity of all other orbits on the cube.
Hope this helps, and yes I agree that the 4x4x4 is the most beautiful cube from a theory standpoint for exactly this same reason.
Chris
