From BRYAN@wvnvm.wvnet.edu Mon Oct 23 16:38:39 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA29047; Mon, 23 Oct 95 16:38:39 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R3) with BSMTP id 8685; Mon, 23 Oct 95 16:38:12 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7909; Mon, 23 Oct 1995 16:38:13 -0400 Message-Id: Date: Mon, 23 Oct 1995 16:38:12 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: pull out the corner? In-Reply-To: Message of 10/23/95 at 20:42:30 from hazard@niksula.hut.fi On 10/23/95 at 20:42:30 Mikko Haapanen said: >I have a question (yes, again). This subject may be discussed here before, >but i don't understand set theory or other high math, so i ask: >If i had a 3x3x3 cube and i pull out a corner piece. I turn it and push >back. Now the cube cannot be solved. I think the cube is now 'on the other >orbit'. If i pull now an edge piece and flip it, the cube is again on some >other orbit. >Only one of those orbits are legal. How many different illegal orbits there are ? In the terms you are using, there are 12 orbits. Of these, 1 is "legal" (contains Start), and 11 are "illegal" (do not contain Start). There is a factor of 3 from twisting the corners. Pull out a corner piece. There are 3 ways to put it back in. You can put it back in the way it came out, you can twist it right, or you can twist it left. There is a factor of 2 from flipping the edges. Pull out an edge piece. There are 2 ways to put it back in, flipped or unflipped. There is a factor of 2 from parity. The edges can be said to be in even parity or in odd parity, and the corners can be said to be in even parity or odd parity. Normally, the corners and edges are in the same parity. A quarter turn changes the parity both for the edges and for the corners. But pull out 2 edges pieces (or 2 corner pieces). Put them back where they came from, and their parity remains the same. Exchange them, and their parity changes. We therefore have 12=3x2x2. However (and draw a deep breath), for every expert there is an equal and opposite expert. This use of the term "orbit" agrees with some experts. However, other experts would say that the corners form an orbit, that the edges form an orbit, and that the face centers form an orbit. I don't know which use of the term orbit is correct (perhaps both are in the proper context). But in any case, if you take a cube apart, there are 12 disjoint sets of positions that you choose from when you put the cube back together. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU