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What causes Actual parities, is a 2-cycle of some type of pieces, an example is OLL parity, which is a 2cycle of wing edges.
There are also vitrual parities, which are cases that are called parity, but are not, such as PLL parity, which is a 2+2 cycle, but if the case was on a 3x3, it would be a parity.

As for solving them, Once you know the algs, its easy

There are many types of parity. On a 4x4 All parity is basically that the cube has been reduced from an even layered cube to an odd layered cube in a way it couldn't have been scramble (as an odd layered cube). I don't own a void cube, but i believe that to avoid parity you must know the color scheme (correct me if i'm wrong)

What causes Actual parities, is a 2-cycle of some type of pieces, an example is OLL parity, which is a 2cycle of wing edges.
There are also vitrual parities, which are cases that are called parity, but are not, such as PLL parity, which is a 2+2 cycle, but if the case was on a 3x3, it would be a parity.

As for solving them, Once you know the algs, its easy

You can't cycle 2 pieces (unless it's a 2x2) on a nxnxn cube. On a 4x4 it isn't about cycling 2 pieces, it's about placing the center pieces in the wrong order. For example: The OLL-Parity algorithm (All L and R moves are inner slices only) R2 B2
U2 L U2 R' U2 R U2 F2 R F2 L' B2 R2) Flips one edge (two pieces) and Rotates the top center by 180 degrees. Obviously you don't notice this when the center is all the same color, but Do a Bw move and then do the algortihm, you'll notice it.

You can't cycle 2 pieces (unless it's a 2x2) on a nxnxn cube. On a 4x4 it isn't about cycling 2 pieces, it's about placing the center pieces in the wrong order.

Your first sentence is not completely correct. You are missing a non-trivial special case where you can perform a pure 2-cycle on an nxnxn cube. I won't tell you which n, though. I'll let you figure it out

Your second sentence is not true :/

As to the OP:

"Fixing" parity is done by simply turning a quarter turn on a slice that contains the piece that is in an odd permutation such as to create an odd permutation. As an example turning U would not fix wing parity (it does two 4-cycles of wings), but turning u would fix wing parity (it does a 4-cycle of wings). So in that sense fixing any parity comes down to performing only 1 quarter turn, so they are all very easy to fix. Of course I know that you mean to fix them in a speedsolving sense, without destroying the progress you've already made on the puzzle while solving it after reduction. In that sense I'd say that I think void cube parity is the most involved for me. I don't have any cool memorized algs for that one and I just do an M or M' slice turn and solve the DF and DB cross edges again Roux methods and intuition (I don't solve the void cube often).

You can't cycle 2 pieces (unless it's a 2x2) on a nxnxn cube. On a 4x4 it isn't about cycling 2 pieces, it's about placing the center pieces in the wrong order. For example: The OLL-Parity algorithm (All L and R moves are inner slices only) R2 B2
U2 L U2 R' U2 R U2 F2 R F2 L' B2 R2) Flips one edge (two pieces) and Rotates the top center by 180 degrees. Obviously you don't notice this when the center is all the same color, but Do a Bw move and then do the algortihm, you'll notice it.

I know, but since the rotation is a 180 on the centre, this is a 2+2 cycle of centres, which can be solved purely using comms and doesn't change the parity of other pieces. The parity of edges and centres isn't connected in the sense that one determines the other.

Your first sentence is not completely correct. You are missing a non-trivial special case where you can perform a pure 2-cycle on an nxnxn cube. I won't tell you which n, though. I'll let you figure it out

I only have a 4x4 and void, but the permutation parity on the void is hard for me because the alg I know fixes the permutation, but messes up the orientation.

I only have a 4x4 and void, but the permutation parity on the void is hard for me because the alg I know fixes the permutation, but messes up the orientation.

None of my cubes have ended up with parity except after they've been scrambled and I tried to solve them. I'm pretty sure that's what causes parity to appear, trying to solve after a scramble.

I'd vote for the 6x6x6, but I don't know know what the void parity problem is. Maybe it's worse.

You know there isn't just 1 parity alg for square-1 right? If are referring to the adj parity alg then i can understand but there are many other algs that fix parity on the puzzle while affecting other edges and they are quite a bit easier. Just sayin'

My biggest parity problem with a 4x4 is looking ahead. When I get to OLL parity I have two a algs. I like the longer 1-up, 2-up, 3down thing but it will add PLL parity if it isn't there or cancel out PLL if it is. The standard speedsolving alg doesn't do this. If I could see to the next few steps I could avoid one type of parity by choosing the right alg to apply.