From ronnie@cisco.com Thu Aug 5 19:55:48 1993 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23537; Thu, 5 Aug 93 19:55:48 EDT Received: from localhost.cisco.com by lager.cisco.com with SMTP id AA23583 (5.67a/IDA-1.5 for ); Thu, 5 Aug 1993 16:55:37 -0700 Message-Id: <199308052355.AA23583@lager.cisco.com> To: acw@bronze.lcs.mit.edu (Allan C. Wechsler) Cc: Dik.Winter@cwi.nl, cube-lovers@life.ai.mit.edu Subject: Re: Diameter of cube group? In-Reply-To: Your message of "Thu, 05 Aug 1993 17:02:45 EDT." <9308052102.AA22841@bronze.lcs.mit.edu> Date: Thu, 05 Aug 1993 16:55:36 -0700 From: "Ronnie B. Kon" Disclaimer: this sounds more authoritative than is intended--I really don't know what I'm talking about. It couldn't be very pointy. From the most distant configuration, there are 6 positions immediately before it. There are 6^2 two steps away, 6^3 three steps, etc. (well, 6^2 - 1 and 6^3 - ?) actually. This is necessarily so, as if any of the configurations reachable with two twists (for example) are closer in than (max - 2) steps then you could reach the maximum configuration by getting there and then doing the two steps. This gives me the feeling that Monte Carlo is fairly valid. (How's that for rigor?) Ronnie > I wonder about the validity of your Monte Carlo analysis. It seems > to be based on an intuition about how fast the number of configurations > falls off with the distance from SOLVED. I share the intuition, but > I'm not sure I can rigorize it, and that makes me cautious. > > What prevents a group from having a "pointy tail", that is, a "corridor" > of elements at increasing distances from the identity? In fact, does > the number of elements as a function of distance have to be unimodal? > Could this function have a "waist"? Intuitively, this sounds > impossible, but I am wondering what constraints on such functions are known. >