Thirty-One QTM Moves Suffice

As of last Tuesday, with 1200 QTM cosets solved, all at a distance of 25 except two at 24, I've been able to show that 31 moves suffices in the QTM to solve every cube position.

Surprised by the dearth of 26s (none found out of more than 20 trillion positions solved), I decided to take a cue from Radu (and others) and explore the symmetrical positions. In the past few days I've run about 500 cosets with 16-way, 8-way, and 4-way symmetry and have not found any distance-26 positions other than the one that is known.

This is in contrast to the face turn metric, where distance-20 positions pop up much more frequently.

Once I finish all 16-way, 8-way, and 4-way cosets, I plan to explore enough additional cosets to show that 30 moves suffice (I am doing this on my single fast machine so progress will be much slower than it was with the face turn metric, where Sony Pictures Imageworks provided huge computational resources for a short while.)