From @mitvma.mit.edu:DWR2560@TAMZEUS.BITNET Fri Aug 20 09:47:26 1993 Return-Path: <@mitvma.mit.edu:DWR2560@TAMZEUS.BITNET> Received: from mitvma.mit.edu by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA19046; Fri, 20 Aug 93 09:47:26 EDT Message-Id: <9308201347.AA19046@life.ai.mit.edu> Received: from MITVMA.MIT.EDU by mitvma.mit.edu (IBM VM SMTP V2R2) with BSMTP id 3102; Fri, 20 Aug 93 06:56:16 EDT Received: from TAMZEUS.BITNET (DWR2560) by MITVMA.MIT.EDU (Mailer R2.10 ptf000) with BSMTP id 4997; Fri, 20 Aug 93 06:56:15 EDT Date: Fri, 20 Aug 93 05:56 CST From: Subject: pointy tails To: cube-lovers@life.ai.mit.edu X-Original-To: cube-lovers@life.ai.mit.edu, DWR2560 Allan C. Wechsler writes: > 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. > >Very good. This is a necessary insight, regardless of the exact >numerical details. (For example, you mean 12, not 6.) But the >possible flaw is that there might be more than one maximally distant >state; if their sets of neighbors overlap viciously enough, this >effect could make the tail pointier. You can make valence-12 graphs [deletia] All this misses the point (so to speak) which is that 12^N is _exceedingly_ pointy for our purposes. If one samples only 1000 positions out of ~10E19, then one could very well miss a 12^N tail of length 14 moves! The estimate of 22 as an upper limit relies on the intuition that the distribution is MUCH blunter than this. Dave Ring dwr2560@zeus.tamu.edu