From cube-lovers-errors@mc.lcs.mit.edu Thu Apr 23 11:51:20 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA04842; Thu, 23 Apr 1998 11:51:19 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Apr 23 11:42:59 1998 From: whuang@ugcs.caltech.edu (Wei-Hwa Huang) Message-Id: <199804231547.IAA09346@gluttony.ugcs.caltech.edu> Subject: Re: Hamiltonian circuits on the cube To: jbryan@pstcc.cc.tn.us (Jerry Bryan) Date: Wed, 22 Apr 1998 16:58:41 -0700 (PDT) Cc: cube-lovers@ai.mit.edu In-Reply-To: <9804231425.AA10935@sun28.aic.nrl.navy.mil> from "Dan Hoey" at Apr 23, 98 10:25:30 am Reply-To: whuang@ugcs.caltech.edu Jerry Bryan typed something like this in a previous message: > It makes me wonder of you actually saw Dan's unfolding in your mind's > eye, as it were, as you worked out your solution. Or another way to put > it, did you work out your solution in 2-D or in 3-D? It also makes me > wonder if there is any other unfolding that would lead as naturally to a > Hamiltonian circuit. I tend to think not, but I could well be wrong. > Actually, I didn't visualize any unfolding at all, so I guess I did it in 3-D. Here's approximately the line of reasoning that led to my solution. As Dr. Singmaster notes, there is only one way to draw a Hamiltonian on a 1x1x1 cube where all the faces are identical, and that is with a right angle on each face. Naturally one's first impulse is to find a path that enters each 3x3 face in one place and exits in another -- and these two ends must be on edges 90-degree apart. One quickly sees that the two exits must be on edge cubies, since if any were on corner cubies there would be a parity problem between "inner corners" and "outer corners." But if they were edge cubies, then no Hamiltonian path exists (as the inner corner must join to the ends already). However, another extension is the "three parallel paths" pattern: put this on each face: A B C | | | | | +-D | +----E +-------F This leads to three paths on the cube, where the center one is the traditional 1x1x1 Hamiltonian. If this can be rearranged to a solution, we must try to reconnect the ends so that there is some "interaction" between the three paths. C must connect to D, but we can connect A to B instead -- and this leads to a solution, which surprised me when I visualized it on a 3-d cube. (I most definitely find visualizing in 3-D easier than visualizing the links in an unfolded cube.) -- Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/ --------------------------------------------------------------------------- Smoking cigarettes are bad for you, so smoking cigarettes is bad for you.