From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 11:46:24 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id LAA10792; Fri, 14 Nov 1997 11:46:23 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Nov 14 10:27:59 1997 Date: Fri, 14 Nov 1997 15:25:06 GMT From: David Singmaster To: tenie1@juno.com Cc: cube-lovers@ai.mit.edu Message-Id: <009BD499.F2FD74E5.202@ice.sbu.ac.uk> Subject: RE: 6x6x6 cube design First, regarding the 6^3 and 7^3. As noted, when you get to these sizes, the connection of the corners while turning becomes problematic. For the 6^3, the overlap is about 15% of the edge length of the cubie, probably too small to be practicable. One can imagine some clever mechanism to hold onto the corners, but it would be tricky and I've never seen one clearly described. However, if you think about it, there's no reason for all the levels of the cube to be the same size. That it, the parallel cutting planes of the entire cube do not have to be equally spaced. One can thus have the corner cubies be very large with much smaller centre cubies. The edge cubies will be cuboids, rather than cubes. Using this idea, one can make arbitrarily large cubes, but the interior pieces become impossible to manipulate. Now let me try my hand at describing three versions of the 2^3. I'll start with the simplest which was sent to me from Japan about 1980. This had a steel sphere in the middle and each cubie had a magnet in it. Although the sphere and the cubies were carefully machined, when one moved it quickly, a piece would catch against another piece and lift off and then fall off. Not very successful. The second version was patented by Ishige in Japan about 1977? and several versions were made. I received a batch of seven with different colouring patterns made by a German sports firm - three or four had broken just in the post! This version has a central sphere and six of what I call 'umbrellas' sticking out toward each face centre. Each of the pieces has a notch around the part that rest against the inner sphere. The umbrellas catch into these notches. One can also think of the cubies as having their own umbrellas, but of triangular form and concave. This is the same mechanism used in the Impossiball. The third version is the most common and is shown in Rubik's Hungarian patent, but is hard to interpret as I've never had the text translated. Basically, his 2^3 is a 3^3 with the edge and centre pieces concealed. I gather from earlier messages that there were several versions of this, but I only recall one, but I only ever took a few apart. At the very centre was a cube. On each face was a square rod extending almost to the face center. The ends of these had a + groove. Between the rods were pieces in the form of a quadrant with a groove on the outer, curved, edge. When all these pieces are in place, each of the midplanes of the cube is seen to contain a circle with a groove on its outer edge. The corner pieces are basically hollow, but each interior face is a layer ending in a quarter-circular curve, which fits as a tongue into the groove just mentioned. Where two of these meet, at the interior edge of the piece, a section is cut away to allow the piece to slide past the projections of the end of the square rods. In theory, one might be able to avoid the quadrant pieces, but I think they give the structure stability. A more serious problem is that the inner, concealed, pieces can get out of synch with the visible pieces, The early patent of Gustafson left gaps so one could see the inner pieces and move them. The method used by Rubik and in some similar puzzles is to fix one corner piece to the inner structure by some method. Rubik's 2^3 did this by making some of the rod ends solid rather than grooved (or perhaps they were fixed to the central cube so they couldn't rotate). One could also not notch one of the corner pieces. Whatever one does, it must have the effect of preventing one corner from moving in relation to the inner structure. I seem to recall that the 4^3 uses this idea also. Don't know how much this helps, but that's the best I can do off-hand. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk