From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 22 11:53:02 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA00421; Wed, 22 Apr 1998 11:53:02 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Apr 22 11:43:05 1998 Date: Wed, 22 Apr 98 11:42:49 EDT Message-Id: <9804221542.AA10123@sun28.aic.nrl.navy.mil> From: Dan Hoey To: whuang@ugcs.caltech.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: <6hfr60$lfq@gap.cco.caltech.edu> Subject: Re: Hamiltonian circuits on the cube whuang@ugcs.caltech.edu (Wei-Hwa Huang) writes: > I was pretty surprised to come up with this within ten minutes of reading > the question: Wow, I'm impressed. I thought I'd have to write a program to find them, and here's a nice symmetric solution. The symmetry is more visible in a different unfolding: +-@-+-@-+-@-+---+---+---+ | @@@@@ @@|@@@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@@|@@ @@@@@ | +---+---+---+-@-+-@-+-@-+---+---+---+ | @@@@@ @@|@@@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@@|@@ @@@@@ | +---+---+---+-@-+-@-+-@-+---+---+---+ | @@@@@ @@|@@@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@@|@@ @@@@@ | +---+---+---+-@-+-@-+-@-+ It shouldn't be that hard to solve a cube with these markings--there are only two different kinds of corner cubies, three kinds of edge cubies, and the face centers need only be oriented mod 180 degrees. Working from one of the symmetric corners, it's not hard to see that this is the only continuous solution. I've noticed a minor modification to your pattern that also admits an isoglyphic Hamiltonian path: +-@-+-@-+-@-+-@-+---+---+ |@@ @@@@@ | @@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@ | @@@@@ @@| +---+---+-@-+-@-+-@-+-@-+-@-+---+---+ |@@ @@@@@ | @@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@ | @@@@@ @@| +---+---+-@-+-@-+-@-+-@-+-@-+---+---+ |@@ @@@@@ | @@@@@@@@@ | + + + + + + @ + | @@@@@@@@@@|@@@@@@@@@@ | + @ + + + + + + | @@@@@@@@@ | @@@@@ @@| +---+---+-@-+-@-+-@-+-@-+ Anyone who's working on an exhaustive search to see if there are any others, send me e-mail before I hack again! Dan Hoey@AIC.NRL.Navy.Mil