Relationship of Duplicate Positions and Non-Trivial Identities
Submitted by Jerry Bryan on Thu, 01/29/2009 - 08:36.
(Reconstructed from the Drupal archives. Much thanks to Mark for all the work he does in supporting this site.)
This message addresses both the Non-Trivial Identities thread and the Generalizing Dan Hoey's Syllables thread. I thought that I had a good handle on the relationship between Non-Trivial Identities and Duplicate Positions, but I find a confusing discrepancy.
Consider the following four positions.
w = F R' F' R U F' w' = F U' R' F R F' x = U F' L' U L U' x' = U L' U' L F U' y = U' R U R' F' U y' = U' F R U' R' U z = F' U L F' L' F z' = F' L F L' U' F
It is the case that w=x=y=z, what I call a duplicate position. This duplicate position is obviously related to the list of 1440 non-trivial identities from the Non-Trivial Identity Thread. Therefore, I was thinking that (for example) we would find wx', wy', and wz' in the list of 1440 non-trivial identities. But we don't, or at least not exactly. We find wx' and wy', but not wz'. Why not?
Well, we can write wz' as F R' F' R U F' F' L F L' U' F, which is not in the list. But F R' F' R U F F L F L' U' F is in the list. Clearly, in some sense these two identities are the same. They differ only in that the first one has F'F' in the middle and the second one has FF in the middle. And F'F' and FF are just two different names for the same permutation. On the other hand, the F'F' version corresponds directly to wz' and the FF version does not.
mdlazreg's very nice analysis of the 1440 identities did make note of the fact that there are half-way positions in the 1440 identities such that 4 of them are equal, as exemplified by w=x=y=z. And the final numbers in mdlazreg's analysis seem correct. Yet I'm bothered that wx' and wy' are in the list but wz' is not. Is there a simple explanation? (By the way, the same discrepancy, if indeed it's a discrepancy, occurs for each of the 24 unique half-way positions that have four different optimal processes. That is, each optimal process is paired in the list with the inverse of two of the other three optimal processes, but not with the third of the other three optimal processes.)