i've also done some searching for short maneuvers for superflip,
although not to the extent that dik has. i was never really
satisfied with my efforts to exploit its symmetry and centrality.
however, i've recently had some new thoughts about this which
I have indeed considered this, but have not yet come to a conclusion.
suppose that there is a minimal sequence for superflip which
contains a half-turn. then, by applying R' U2 to superflip,
we've moved 3q (or 2f ) closer to start.
I do not know whether this is clear for all readers. My reasoning
was similar but the conclusion different, but someway equivalent:
If the minimal sequence contains a half-turn, we may just as well
assume that that half turn is the last, and F2. I do not know
whether the proof has been shown on this list, but it is simple.
Suppose M is a minimal sequence, and Z is some random sequence,
in that case Z M Z' is also superflip. Take Z the maximal
sequence at the end consisting of quarter-turns only, we end with
a sequence of equal length terminating with a half-turn.
Because of symmetry we may just as well consider it to be F2.
otherwise, every minimal sequence contains only 90 degree turns.
then either R' U' gets us 2q (or 2f ) closer to start,
or R' U gets us 2q (or 2f ) closer to start. (and probably
it would be nice to reduce this latter case to only one of R' U'
or R' U . can anyone do this?
This needs more than simple symmetry. There are 12*8 different
endings, and we have 48 symmetries (24 by rotation * 2 by inversion).
Leaving 2 cases. I considered this, but have not yet come to
conclusions. On the other hand I do not yet know what to conclude
from M M' = I for every superflip sequence.
when searching for superflip in the face turn metric, it's
sufficient to search through depth 17 in stage 1!
suppose we have a 19f sequence for superflip. then, by considering
parity, some turn must be a half-turn. now we may suppose (as above)
that the last two face turns are U R2 , which is in stage 2!
Yes, I had seen that. One of the major reasons I was not amused when
the system crashed doing depth 17 in stage 1! I will restart the
program doing depth 17, but I will first redo the counting so that
counts larger than 2^32 are correct.