From cube-lovers-errors@mc.lcs.mit.edu Thu Apr 23 20:24:58 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id UAA05974; Thu, 23 Apr 1998 20:24:58 -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 20:22:45 1998 Date: Thu, 23 Apr 98 20:21:11 EDT Message-Id: <9804240021.AA11374@sun28.aic.nrl.navy.mil> From: Dan Hoey To: cube-lovers@ai.mit.edu Cc: whuang@ugcs.caltech.edu Subject: Re: Hamiltonian circuits on the cube I wrote: "...send me e-mail before I hack again!" Too late. The only chiral Hamiltonian isopaths are the two we've already seen, and: +---+---+-@-+---+-@-+---+ | @@@@@ @@|@@@@@@ @@| + @ + @ + + + + @ + | @ @@@@@@|@@@@@@ @ | + @ + + + + @ + @ + |@@ @@@@@@|@@ @@@@@ | +---+---+-@-+---+-@-+---+-@-+---+---+ | @@@@@ @@|@@@@@@ @@| + @ + @ + + + + @ + | @ @@@@@@|@@@@@@ @ | + @ + + + + @ + @ + |@@ @@@@@@|@@ @@@@@ | +---+---+-@-+---+-@-+---+-@-+---+---+ | @@@@@ @@|@@@@@@ @@| + @ + @ + + + + @ + | @ @@@@@@|@@@@@@ @ | + @ + + + + @ + @ + |@@ @@@@@@|@@ @@@@@ | +---+-@-+---+-@-+---+---+ I actually generated all the continuous chiral isopaths, and the following is the other extreme--the only one with nine disjoint paths. Yet one of the paths goes through one third of the facelets. +-@-+-@-+---+-@-+-@-+-@-+ |@@ @@@@@@|@@ @ @ | + + + + + @ + @ + |@@ @@@@@@|@@@@@@ @@| + @ + @ + + + + + | @ @ @@|@@@@@@ @@| +-@-+-@-+---+-@-+-@-+-@-+---+-@-+-@-+ |@@ @@@@@@|@@ @ @ | + + + + + @ + @ + |@@ @@@@@@|@@@@@@ @@| + @ + @ + + + + + | @ @ @@|@@@@@@ @@| +-@-+-@-+---+-@-+-@-+-@-+---+-@-+-@-+ |@@ @@@@@@|@@ @ @ | + + + + + @ + @ + |@@ @@@@@@|@@@@@@ @@| + @ + @ + + + + + | @ @ @@|@@@@@@ @@| +-@-+-@-+-@-+---+-@-+-@-+ Dan Hoey@AIC.NRL.Navy.Mil