, where Q is the set of quarter-turns Q={F,B,U,D,L,R,F',B',U',D',L',R'}. Elements of Q move the corners and edges, but Q is the identity on the centers. C, on the other hand, is generally considered to move the centers. Hence, the group generated asis a supergroup of G, and there are elements of the supergroup which are not in G. (This supergroup, by the way, is not The Supergroup. The Supergroup is generated by Q alone, but with orientations of the (otherwise fixed) centers considered.) Therefore, our first order of business is to make C into a sub-group of G. We observe that since the elements of Q are the identity on the centers, the primary function of the centers is to provide a frame of reference. But we can provide a frame of reference without the centers actually being there. For example, consider the group GC consisting of cube centers and corners. You can model this group by removing the edge labels from a physical cube. Establish the cube at Start and perform RL'. The corners will be rotated forward, and will be positioned properly with respect to each other, but the cube is clearly not solved. You can tell that the cube is not at Start because the corners are not aligned properly with the centers. Now, do the same thing except remove both the edge and center labels. If you perform RL' at Start, the cube "looks" solved but rotated forward. However, we can adopt the convention that the cube is solved only if the Up color is Up, the Front color is Front, etc. With this convention in place, RL' is clearly seen not to be solved; it is two moves from Start. The convention provides the fixed frame of reference. Furthermore, RL' (which is in GC) is equal to an element of C, and indeed all elements of C are in GC, as are all elements of the form Xc or cX for c in C and X in GC. Hence, we have=. Similar comments apply to GE, the group of edges and centers, except that processes composed from elements of Q to accomplish rotations in C are not quite so short in GE as they are in GC. G, the full 3x3x3 cube group consisting of corners, edges, and centers is a bit more difficult. The problem is that if X is in G, then objects of the form Xc or cX are in G only if c is even. Twelve elements of C are even and twelve are odd. Indeed, C[even] is a sub-group of C, but C[odd] is not. We will deal with this situation (as circumstances require) in two different ways. One is simply to restrict ourselves to C[even] when dealing with G. The other is to define a new group we will call GS. In our model for G in which the centers are implied by a frame of reference convention rather than by actual physical centers, we can easily add slice moves to the standard face moves. If the centers were physically present, then the slice moves would move the centers, but without the physical centers there is no problem. If S is the set of slice moves, then GS is generated as. GS is essentially G with parity restrictions removed. Hence we observe that |G|=|GC|*|GE|/2, |GS|=|GC|*|GE|, and |GS|=|G|*2. Also, if X is in G or in GS, then elements of the form cX or Xc are in GS for all c in C. In those occasions where we are willing to think of GS rather than G, we can use C rather than C[even]. At this point, we can say that GS/C, G/C[even], GC/C, and GE/C are cosets of C in GS, C[even] in G, C in GC, and C in GE, respectively. To be a little more conformant with standard coset notation, we will write cube elements as lower case letters for the remainder of this note, and hence for a particular cube x a coset of C is denoted as Cx={y: y=cx} or xC={y: y=xc}. Now, we propose a group operator for the cosets: Cx Cy = C(xy) and xC yC = (xy)C. Showing that we have a group is easy. I originally included a proof in this note, but there is a proof in Chapter 8 of Frey and Singmaster's _Handbook of Cubik Math_. Hence, I will defer to their proof instead. According to Frey and Singmaster, G/C is called the factor group of C in G, or the quotient group of G by C. Of most significance to us right now is the fact that the identity of the factor group is Ci or iC, where i is the identity of G. But Ci or iC is just C. Hence, the identity of the factor group is C. This justifies our identification of G/C with a centerless cube. In English, it means that we can rotate a centerless cube in space without changing anything. I think this would comply with most people's intuitive sense of what it means for a cube to be centerless. Finally, as to whether this model retains the "symmetrical nature of the problem", I will have to leave that as an open question, depending on precisely what we mean by "symmetrical". It seems to me that this model does a better job of being "symmetrical" than a model which includes only seven corner cubies or only eleven edge cubies, but maybe not. What does "symmetrical" mean when it comes to centerless cubes? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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 If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow?