From ncramer@bbn.com Wed May 15 22:52:53 1991 Return-Path: Received: from LABS-N.BBN.COM by life.ai.mit.edu (4.1/AI-4.10) id AA21390; Wed, 15 May 91 22:52:53 EDT Message-Id: <9105160252.AA21390@life.ai.mit.edu> Date: Wed, 15 May 91 22:09:23 EDT From: Nichael Cramer To: Ronnie Kon Cc: dn1l+@andrew.cmu.edu, ncramer@bbn.com, Cube-Lovers@life.ai.mit.edu Subject: Re: ARGGHHH!! [was: 5by cubes] Ronnie Kon writes: >I write: >>The state of the cube is not: >> >>X|O|X|X|X X|A|X|C|X >>X|X|X|X|X X|X|X|X|X >>X|X|X|X|X But rather: X|X|X|X|X >>X|X|X|X|X X|X|X|X|X >>X|X|X|O|X X|X|X|B|X >> >>Where cubie "C" just "looks" like it's in the right place. >> >>You need an operator that rotates A->B->C->A. [...] >> >>This will very likely leave an inconvenient number of edges flipped. For >>the answer to _this_ problem, see my last post. ;) > >I think you must be wrong here (but would love to be proved wrong--I'm no >mathematician so group theory is very much beyond me). > > [Proofs deleted.] Hi. I think we're in complete agreement, at least up to here. (I particularly enjoyed your "proof by hardware ;). I didn't mean to imply that the A->B->C->A operator preserved flipped-ness of the Non-Central-Edge[NCE] Cubies. Moreover, I was being imprecise where I said "a NCE cubie is simply flipped"; rather "the cubie *appears* as if it were in the right place (i.e. judged by its colors) and flipped". As you point out, *really* means that it is in the slot of its "twin". To recap more succinctly, what I was proposing was a rather pedestrian, two-step solution to the original problem. Starting from the initial state in FIG1 (where the cube is completely "solved" except that the cubies marked "O" are swapped. Also they are swapped in such a way that the visible face is all a single color). FIG1: X|O|X|X|X FIG2: X|Q|X|Q|X X|X|X|X|X X|X|X|X|X X|X|X|X|X A->B->C->A gives: X|X|X|X|X X|X|X|X|X X|X|X|X|X X|X|X|O|X X|X|X|X|X STEP1] If we then apply the A->B->C->A operator, we end up with the state in FIG2, where the cube is completely "solved" except that the cubies marked "Q" "appear" to be "simply" flipped. STEP2] We can then solve this problem, which (imo) is easier. For example see the method that I described in an earlier post; this involves turning the non-central plane (containing the flipped cubie) through a quarter turn. Of course, now that I say it, it seems that the correct course would be to *start* with the quarter turn of the non-central plane. This would leave five NCE cubies out of place, but the cube would be in the right orbit.