# 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.

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.