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