# There Is No True OLL Parity above Three Dimensions

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### Raymond Zhao

Dec 18, 2022, 8:28:52 PM12/18/22
to hypercubing
Hi y'all,

This thread is a continuation of a question I asked in 2021, which was inspired by Rty's two posts on 4^4 3c parity in 2020 (here and here). I speculated that Rty's situation, which I interpreted as a 2-swap of 3c pieces, was not actually possible on the 4^4; turns on the 4^4 result in even permutations, whereas a 2-swap is an odd permutation. However, I didn't have a way to easily prove that property for higher-dimensional 4^n puzzles.

Recently, I remembered Joel, who appears in the linked threads above, mentioning that there might be a thread on higher-dimensional parity in the message archive, but I never bothered to check until now. I found one thread there from 2009 on higher-dimensional parity, and the last message in that thread from David Smith not only confirmed my conjecture, but provided a generalization of it. David didn't provide long-winded proofs for his claims, but his helpful explanations gave me motivation to write an article elaborating on the topic.

And so, I present my article, There Is No True OLL Parity above Three Dimensions. The article contains proof sketches of some of David's propositions. It also contains some screenshots of what I call pseudo-OLL parity, which is a situation that looks like OLL parity on a higher-dimensional even-layered cube, but is not caused by an odd permutation.

Kind regards,
Raymond

### Joel Karlsson

Dec 19, 2022, 6:55:05 AM12/19/22
to Raymond Zhao, hypercubing
Hi Raymond,

Nice write-up! In your proof sketch of Thm 2 it was not clear to me that you also consider "deep slices", although the proof works for those as well.

One can perhaps simplify the proof slightly. Instead of splitting a face (or deeper slice) into a 2x2x2 grid of eight octants, you can just split it in two through the middle in the plane of rotation (this is possible as long as the slice has at least dim=3, so for n at least 4). By pairwise matching up 4-cycles in the two halves you see that no odd permutations are possible (a 90 degree turn always splits into disjoint 4-cycles).

Consider the case of an arbitrary (odd or even) number of layers in dimension at least 4. Similar to your proof for the 3^n case, it immediate follows, by splitting in two, that only pieces that intersect the midplane of the slice can undergo odd permutations. Now, by splitting this midplane into quadrants you see (by D_4 (Dihedral_4) symmetry, which again requires dim at least 4) that odd permutations are only possible of the pieces intersecting the axes separating the quadrants or the "diagonals". In this way, you can see what's possible for e.g. a 5^4.

Best regards,
Joel

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