From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 19 22:57:36 1998
Return-Path:
Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP
id WAA17267; Wed, 19 Aug 1998 22:57:36 -0400 (EDT)
Precedence: bulk
Errors-To: cube-lovers-errors@mc.lcs.mit.edu
Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Aug 19 17:10:15 1998
Date: Wed, 19 Aug 1998 17:09:54 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: minimal maneuvers for X symmetric positions
In-Reply-To: <199808182028.QAA24353@euclid.math.brown.edu>
To: michael reid
Cc: cube-lovers@ai.mit.edu
Message-Id:
On Tue, 18 Aug 1998 16:28:28 -0400 michael reid
wrote:
> X is the subgroup of the cube symmetry group which preserves
> the U-D axis. there are 128 positions which have X symmetry:
>
> the UR edge can go in any of the 8 positions UR, RU, DR,
> RD, UL, LU, DL, LD; this determines the location of the
> edges UB, UL, UF, DR, DB, DL, DF.
>
> the FR edge can go in any of the 4 positions FR, RF, BL, LB;
> this determines the location of the edges FL, BR, BL.
>
> the UFR corner can go in any of the 4 positions UFR, UBL,
> DRF, DLB; this determines the location of all the corners.
>
> any combination of these is possible, which gives 128 positions.
> 4 of the positions have more symmetry, namely M symmetry.
> (these positions are start, superflip, pons asinorum, and
> pons asinorum composed with superflip.)
>
> minimal maneuvers for the other positions are:
>
> 1. F2 R L' D2 F2 D2 R' L F2 D2 (16q*, 10f*)
>
> 2. U F' B' R2 U' D' F' B U D R2 F B D' (16q*, 14f)
> U F2 U2 F2 R L F2 U2 F2 U2 R' L' U (13f*, 20q)
>
I don't think Mike has said so explicitly, but he appears to
have adopted a very useful convention from Herbert Kociemba's
Cube Explorer 1.5. To wit, Cube Explorer 1.5 flags the length
of a maneuver with an asterisk when the length has been shown to
be minimal. Cube Explorer 1.5 operates only in face turns, so it
omits the q or f designation of units.
But for example, Cube Explorer 1.5 might show the length of a
cube upon which it is operating as (13) meaning 13f, then later
in the search show the length as (12), and still later show the
length as (12)* to show that 12 face moves have been proven to
be minimal. The only difference between Mike's style and Cube
Explorer's style is that Cube Explorer puts the asterisk outside
the parentheses.
I loaded Mike's E-mail into Cube Explorer to take a quick look
at the X symmetric positions. Many of them are familiar to
readers of this list, and all of them are quite pretty.
(Loading Mike's E-mail into Cube Explorer "just worked". I
didn't have to edit it at all to remove extraneous text. Cube
Explorer's maneuver reader seems to have a remarkable ability to
extract maneuvers in BFUDLR notation which are imbedded in other
extraneous text.) If you have Cube Explorer 1.5 (and you
should!), I would encourage you similarly to load Mike's X
symmetric patterns into it and take a look.
The patterns look as expected for patterns which preserve the
U-D axis. The U and D faces are the same pattern. The F, R, B,
and L faces are the same pattern and may be described as
being in the same orientation with respect to rotations of the
square.
For positions #1 through #62, the U and D faces may be described
as being symmetric with respect to the symmetries of the square.
They range from being solid, to having one dot, to being a +, to
being an X, etc. All are glyphs.
Positions #63 through #124 are essentially the first 62
positions composed with superflip. I had never noticed it, and
I don't *think* it has been described on the list, but for every
symmetry group, half of the corresponding positions can be
described as "basic" positions and the other half can be
described as the basic positions composed with superflip. That
is, if Symm(x)=K, then Symm(xf)=Symm(fx)=K, where x is any
position and f is the superflip.
----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us