I'm going to be posting some of my longer posts on speedsolving.com here. I think I've written some very useful things down there, but it's a kinda big forum (and I've made a lot of posts!) so it's easy for stuff to get lost. The plan is to copy some of them here, with a link to the original post (for those who want to view the thread). Here goes...
From http://www.speedsolving.com/forum/showthread.php?p=852756:
In speedcubing there are two things we think of as parity: "reduction
parity" and mathematical parity. Some things we call parity fall into
both categories, but others only fall into one of them, and you will
often see people disagree on whether certain things count as parity,
because they disagree that both of these two definitions are valid.
Reduction parity occurs when you try to reduce the puzzle so it can be
solved by a constrained set of moves, putting it into some subset of the
positions. However, you can often reach a position which seems like it
is in your subset, but which is actually not, and to solve the puzzle
you have to briefly go outside your constrained set of moves to bring
the puzzle back into the subset you want. Typically the number of
positions you can encounter is some small multiple of the number of
positions you expect. The obvious example is PLL parity in 4x4x4: all
the centers and edges are properly paired, so you expect to be able to
finish the puzzle with only outer layer turns, but this isn't quite
possible. OLL parity falls under this definition too (so the reduced
4x4x4 has four times as many positions as you would expect). Square-1
parity also falls under this definition - your constrained set of moves
are any moves that keep the puzzle in cubeshape. BLD parity is not of
this type (the solver has not reduced the puzzle).
Mathematical parity is based on the idea of the mathematical definition
of an odd permutation. Basically, at least one orbit of pieces has an
odd permutation, and thus cannot be solved with just 3-cycles. You see
this type of parity crop up in blindfolded solves, because blindfolded
solvers attempt to solve most or all of the puzzle with 3-cycles and
thus an odd permutation is very noticeable. OLL parity on 4x4x4 and
Square-1 parity can also be thought of as this way, as they originate
from some kind of 2-cycle. PLL parity on 4x4x4 is not of this type (it
can be solved with 3-cycles).
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