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Thirty QTM Moves Suffice

Submitted by rokicki on Thu, 02/19/2009 - 00:17.
With 10,114 cosets solved in the quarter turn metric, I have shown
that 30 or fewer quarter turns suffice for every Rubik's cube
position. Every coset was shown to have a bound of 25 or less,
except the single coset containing the known distance-26 position.

I also solved every coset exhibiting 4-way, 8-way, and 16-way
symmetry, and each of these also were found to have a bound of
25 or less. Thus, if there is an additional distance-26 or
greater position, it must have symmetry of only 2, 3, or 6, or
no symmetry at all. I believe, based on this, that it is likely
that on other distance-26 positions exist.

This effort has required in total, so far, 19 CPU days on a
i7 920 and 31 CPU days on a Q6600.

I believe most QTM cosets actually have a worst-case distance of
24 or less; I will be investigating this by solving 25 random
QTM cosets fully, if possible.
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Slight correction

Submitted by rokicki on Thu, 02/19/2009 - 12:45.
When I say every coset with 4-way, 8-way, or 16-way symmetry has a bound of 25 or less, I mean every coset *except* the one containing the known distance-26 position (that is, the coset with the four middle edges in the middle, corners in the solved orientation, but all edges flipped).

It looks like totally solving the 25 sample cosets will take some significant time. When run with a phase one depth through 21, they take about six hours each, but typically end with hundreds of possible distance-24 positions, each of which must be examined to determine if it is really a distance-24 or just a recalcitrant distance-22 position. Yet, running with a phase one depth through 22 will take about 2 1/2 days per coset on my fast machine.
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