From cube-lovers-errors@mc.lcs.mit.edu Fri Jan 16 16:29:01 1998 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id QAA14189; Fri, 16 Jan 1998 16:28:59 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Jan 16 12:59:32 1998 Date: Fri, 16 Jan 1998 17:58:10 +0000 (GMT) From: Jonathan Tuliani To: Cube-Lovers Subject: MEGAMINX Message-Id: The following is based on an email I sent recently to Kurt Endl. He says that he does not have time to work on this at present, and, with a thesis to write, nor do I! I expect that someone has heard of this and the answer is known--I am new to this discussion group. Otherwise, hopefully someone will find it sufficiently interesting to think it through. I was delighted to be given a MEGAMINX this Christmas, together with Kurt Endl's instruction booklet. I resolved to attempt the puzzle without looking in the booklet, at least at first. My approach was similar--I built a solution in layers, starting at the bottom (which I shall call the north pole, consistent with the notation in the latter part of the book) and proceeding, layer by later towards the south pole. I was successful in my efforts until I reached the south cap, at which point I became stuck. My problem was to position and orient the south pole edges. Try as I might, the best I could do was to reach a position where two south pole edges needed to be exchanged. And try as I might, I could not find a way to do this. After a week, I gave up and turned to the instructions. I was delighted to see that their approach was similar to mine, and fascinated by the simple moves L_{**}, L^{**}, R_{**} and R^{**} used. My methods were, of course, far less elegant. I was able to start at section 8, `Setting the South Pole edges'. The procedure for setting the edges affects the southern equatorial corners, which you then arrange later. My layer-by-layer approach had, of course, already set these corners correctly before attempting the south pole edges (and indeed before setting the southern equatorial edges). Perhaps this was the key? Having the southern equatorial corners set should not affect the validity of the book's method, which should work with any arrangement of these corners. But following the instructions, I was unable to position the last two south pole edges correctly. The statement ``The remaining two South Pole edges will be correctly placed again at the same time'' on page 21, section 8 of the instructions must be incorrect--here after all was a counterexample! My last two south pole edges needed to be exchanged. I ignored the problem for the time being. I oriented the two edges correctly, with them still in the wrong positions. I was then able to complete the MEGAMINX, positioning and orienting the southern equatorial and south pole corners as in the instructions. The result was a complete MEGAMINX, except that it appeared that two little triangular stickers, each on the border of the southern cap, had been exchanged. After some thought, I have found a way out of this problem. I believe that this is a detail that may be required for solution in some circumstances that is not in this instruction booklet. I will try to describe what I think went wrong. The twelve faces of the MEGAMINX are coloured using only six colours, with opposite faces bearing the same colour. Thus, each edge piece has an `identical twin' on the opposite side of the completed MEGAMINX. When solving the MEGAMINX from a totally jumbled position, these twins are indistinguishable and may therefore be assigned either to their own original position, or to their twin's position at random. In this sense the solution of MEGAMINX is not unique. (Strictly, the solution may still be unique but we have not proved conclusively that it is so, and have demonstrated reason to believe that it may not be.) Now, suppose we return to the two edges I wished to exchange. Imagine them as south pole edges also in adjacent faces. Turn the MEGAMINX so that the two faces concerned are in the left and right positions (using your terminology). We need to exchange these edges, but the problem we have is that every sequence of moves seems to rotate 3 edges cyclicly, rather than exchange a pair! For example, the book uses L_{**} or R_{**} to bring one of the two edges concerned down into the front southern equatorial edge (``...the edge we have misused so brutally...''), and the same moves to move it back up again. But these moves, together with any I can find, cycle 3 edges. How can moves cycling 3 edges be used to exchange just 2 edges? I was stuck. The solution comes from the `identical twins' I talked about before. Suppose one of the two edges I'm interested in is, say, yellow/blue, and the other yellow/orange, so the southern pole is yellow. Tucked away on the opposite side of MEGAMINX is *another* yellow/blue edge. By a simple sequence of moves, this may be brought into the postion of the front southern equatorial edge. Now consider cycling these three edges. As two of the edges are identical, cycing these three edges *looks* like a swapping of just two edges! Now we return to the other side of MEGAMINX the twin of the piece that was originaly there. (Some fixing of the MEGAMINX is required to repair the damage caused by bringing an edge from the opposite side of the MEGAMINX to the front and sending it's twin back, but this isn't too hard.) Now, having apparently `swapped' two adjacent edges, we can proceed as per the instructions and complete the MEGAMINX. This I have done. Have any other people encountered this problem, or was I just extremely unlucky? The question arising is, of course, just how many solutions of MEGAMINX are there? There are 10 of these `identical twin' edge pairs. I reckon swapping just one twin pair is not possible in a complete solution, but that swapping any even number of twins may be (unproven), and so there are 512 solutions, each of which would be distinct if 12 different colours had been used on the original puzzle. Does anyone fancy having a stab at this conjecture? Jonathan Tuliani Mathematics Department Royal Holloway, University of London Egham Surrey TW20 0EX U.K. jont@dcs.rhbnc.ac.uk