**Computational proof**

Every bound since Thistlethwaite's 52 for this puzzle depended on exhaustive search. Earlier, higher bounds were based on the exhaustive search of smaller groups and factor spaces. Some might consider it a feature of Rubik's cube, that the distance of a position cannot be easily estimated by some structural property.

Personally, I was holding out hope for finding a distance-21 position. Now that would have been very nice.

I've been trying to simplify, analyze, and solve this problem using non-exhaustive methods for years, and have come up all but empty. There are some small observations (for instance, some small number of the H-cosets have distance 18) but, in the end, very little "simple structure" to the group that I could figure out how to exploit. For many of these puzzles, I fear that's the nature of the beast.