From acw@bronze.lcs.mit.edu Thu Aug 5 17:03:17 1993 Return-Path: Received: from bronze.lcs.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16999; Thu, 5 Aug 93 17:03:17 EDT Received: by bronze.lcs.mit.edu id AA22841; Thu, 5 Aug 93 17:02:45 EDT Date: Thu, 5 Aug 93 17:02:45 EDT From: acw@bronze.lcs.mit.edu (Allan C. Wechsler) Message-Id: <9308052102.AA22841@bronze.lcs.mit.edu> To: Dik.Winter@cwi.nl Cc: cube-lovers@life.ai.mit.edu In-Reply-To: Dik.Winter@cwi.nl's message of Tue, 3 Aug 93 03:10:22 +0200 <9308030110.AA23300.dik@boring.cwi.nl> Subject: Diameter of cube group? 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.