From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 22 19:05:00 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id TAA18829; Tue, 22 Sep 1998 19:05:00 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Sat, 19 Sep 1998 09:13:58 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Two-face and three-face subgroups In-Reply-To: <199809151621.MAA19286@Twig.Rodents.Montreal.QC.CA> To: der Mouse Cc: Cube Lovers Message-Id: On Tue, 15 Sep 1998 12:21:58 -0400 (EDT) der Mouse wrote: > I've been playing with the two-face subgroup [%] of the 3-Cube and got > to wondering - how much work has been done on the two-face and > three-face subgroups? Certainly the two-face subgroup "feels" like a > much smaller object than even the 2-Cube (though perhaps more tedious > for human solution), perhaps about the size of the Pyraminx. > The subgroup has been explored fairly thoroughly. For example, look in the archives 8/31/1994 for a summary of the first complete God's Algorithm search of this particular subgroup. There are a number of articles in the archives thereafter. has been searched in both the quarter turn metric and the face turn metric, and local maxima have been investigated as a part of the search. has a very small branching factor and a corresponding large diameter of 25 in the quarter turn metric, at least I think it's a large diameter for such a small group. Until Mike Reid recently showed that the diameter of G in the quarter turn metric was at least 26, the diameter of was the largest known for the 3x3x3 cube or any of its subgroups. Frey and Singmaster's book discusses both two face and three face subgroups, among other things giving their sizes. To my knowledge, no God's Algorithm searches have been performed for the three face subgroups. We have ||=73483200, so is slightly smaller than the corners group at 88179840. The 2x2x2 is 24 times smaller than the corners group, at 3674160. However, I am of the school of thought that tends not to equate the size of the group (or search space, for problems that are not actually groups) with the difficulty of the problem. ---------------------- Jerry Bryan jbryan@pstcc.cc.tn.us