Hi guys. I think a few things need clearing up.
- You can get parity on face-turning cube puzzles because each face has 4 sides. What happens is that doing a quarter turn of a face (or slice) does 4-cycles of pieces, and if there are an odd number of 4-cycles the parity is changed (since a 4-cycle is an odd permutation). On a face-turning dodecahedron puzzle, you can NEVER have parity, because each face has 5 sides. That is, when you turn a face or slice you are only creating 5-cycles, and a 5-cycle is an even permutation so you will never get into a parity state (a 5-cycle is an even permutation, and to get parity you need to get to an odd permutation state).
- The 5x5 DOES have parity. I actually find it really hard to believe that there are people who don't realize this. The difference between the 4x4 OLL parity and the 5x5 parity is only that on the 5x5 you can tell you have parity at the end of reduction, and so you can fix it then, although you don't have to. On the 4x4 you can't tell until you get up to the LL because it looks like the edge groups is paired up correctly. But these are *both* parity situations and in fact can be - and typically are - solved by the same algorithm.
- I think everyone agrees that a 4x4 is basically a combination of a 2x2 and 3x3. What people don't agree on is what defines a 2x2 (and even cubes in general). Is it the lack of fixed centers? The fact that it only has corners? The fact that it is deep cut (each layer is half the puzzle)? Each of these three definitions has a dodecahedral equivalent - the most common ones are, respectively, the Pyraminx Crystal, the Impossiball/Kilominx, and the Pentultimate. I personally follow the last definition, because the reason it is named 2x2 is that on each axis there are only 2 layers, and this happens ONLY when the puzzle is deep cut. Similarly, a 4x4 has 4 layers on each axis, and the only way this can happen is when one of the layers (per axis) is deep cut. So for me the 2x2 version of a Megaminx is actually a Pentultimate! Of course, the 4x4 version would then be a puzzle with both Megaminx and Pentultimate cuts. I guess this would be solved in a similar way to the 3x3 + Skewb puzzles on gelatinbrain.
