[next] [prev] [up] Date: Wed, 12 Jul 95 10:31:10 -0400
[next] [prev] [up] From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
[next] [prev] [up] Subject: Re: Partial Results, Edges only (with Face Centers), Qturns
On 07/10/95 at 16:20:29 Jerry Bryan said:

The local maximum unique up to M-conjugacy is the 6-H position
(see Symmetry and Local Maxima). I do not yet have a process
for the 6-H, but I should be able to have one soon.

I can now give one way to do it as (UD)(RR)(LL)(UD). It seems too
simple once you chase it down. Note that this is *not* the 6-H
position when you apply the process to the whole cube; you have
to omit the corners for this process to yield the 6-H. Nonetheless,
the pattern is a nice one when the process is applied to the whole
cube, one that looks familiar, although I cannot place it. It is
*almost* what I described as the "interesting" part of three of the
10q local maxima on the whole cube, but the "interesting" part of
the 10q local maxima is (U'D')(RR)(LL)(UD) instead.

It might be noted that the length of this position
is also 8q on an edges-only-without-centers cube (see my note
of 8 Dec 1993 22:41:38). I did not actually provide a process
for the without-centers case, but the same process
works for the 6-H edges-only with or without centers for this position.
Such is not always true. I have talked about it before, but many
minimal processes for without-center cubes induce an invisible
rotation which becomes visible when the Face centers are included.

This is probably as good a time as any to correct an old error, pointed
out to me by Dan Hoey. The length of a position without centers is
the minimum taken over C of the length of the same position with
centers -- that is, the minimum of the respective lengths of the
same position rotated 24 different ways. For searches without centers
I store representatives of the form Y=Repr{m'Xmc}. At one point,
I said |Y|=min{|Yc|}. This is certainly not true. Y is just one
of the {Yc}, and it is totally arbitrary which one it is. The difficulty
is really a notational one. It is the length of Y without centers
which is min{|Yc|}, not the length of Y itself (with centers). But
I don't have a good way to say "Y without centers" or especially to
say "length of Y without centers".

But in any case, the most interesting cases to me are the ones where
the length without centers matches the length with centers, so that the
minimal process for the without centers case does not induce an
invisible rotation. The position at hand is such a case.

Finally, the position is in the anti-slice group (i.e.,
(UD)(RL)(RL)(UD)), so the position is a local maximum in the
anti-slice edges only group with a length 4a.

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax
837 Chestnut Ridge Road                              BRYAN@WVNVM
Morgantown, WV 26505                                 BRYAN@WVNVM.WVNET.EDU

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