From BRYAN@wvnvm.wvnet.edu Tue Feb 14 13:11:30 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA10450; Tue, 14 Feb 95 13:11:30 EST Message-Id: <9502141811.AA10450@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 7478; Tue, 14 Feb 95 13:10:42 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 9940; Tue, 14 Feb 1995 13:10:42 -0500 X-Acknowledge-To: Date: Tue, 14 Feb 1995 13:10:41 EST From: "Jerry Bryan" To: "der Mouse" , "Cube Lovers List" Subject: Re: Re: superflip in quarter turn metric In-Reply-To: Message of 02/13/95 at 18:30:32 from , mouse@collatz.mcrcim.mcgill.edu Ok, let's try, try again. I was incorrect in my response to der Mouse, and Martin Schoenert's correction is appreciated. The original issue was as follows: suppose you have created a data base for N levels of God's algorithm, beginning with Start at the root of a tree. With quarter-turns as generators, there is 1 position at level zero, 12 positions at level one, 114 positions at level two, etc. Now, suppose you want to create a data base for N levels of God's algorithm, starting with X at the root of the tree. Can you simply compose your first tree with X on an element by element basis in order to obtain the X-rooted data base? (You do have to be careful about pre-multiplying vs. post-multiplying as Martin indicated!) I stated essentially that the superflip was fairly unique in that you could compose the superflip with the Start-rooted data base in order to obtain the superflip-rooted data base, but that for X-rooted data bases in general you would have to perform a complete search starting with X. der Mouse (correctly) noted that the same procedure would work for any position X as for the superflip. I (incorrectly) took exception with der Mouse, citing my fallacious distance argument. However, the way my data bases work, I still think that the superflip is fairly unique in its ability to be composed with a Start-rooted data base. der Mouse was on the right track in his first post when he questioned whether the issue was the fact that the data bases only store representative elements of M-conjugacy classes. I responded that the storage of representative elements was not the issue, but in fact it is. For example, when storing only representative elements of M-conjugacy classes, consider an F-rooted data base. Strictly speaking, we would have to speak of a Repr{m'Fm}-rooted data base, because F might not be the representative element of {m'Fm} -- it could be any of the twelve elements of Q. When storing only representative elements for a Start-rooted data base, there is 1 element at level zero, 1 element at level one, and 5 elements at level two. For a Repr{m'Fm}-rooted data base, there is 1 element at level zero, and six (!) elements at level one -- namely those same elements that are at level zero and level two of the Start-rooted data base. Hence, we cannot take the Start-rooted data base and pre-multiply each element by Repr{m'Fm} to obtain a Repr{m'Fm}-rooted data base. And in general, we cannot take the Start-rooted data base and pre-multiply each element by an arbitrary Repr{m'Xm} to obtain a Repr{m'Xm}-rooted data base. But for the superflip E, we *can* take the Start-rooted data base and pre-multiply each element by Repr{m'Em} to obtain a Repr{m'Em}-rooted data base. In fact, note that |{m'Em}|=1, so we must have Repr{m'Em}=E. However, note that for each Y in the Start-rooted data base, it is not sufficient to calculate (EY). Rather, we must calculate Repr{m'(EY)m}. That is, we know by definition that we have Y=Repr{m'Ym}, but we do not necessarily have (EY)=Repr{m'(EY)m}. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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