Date: 2 August 1981 03:43-EDT From: Alan Bawden Subject: Administrivia and an assault on God's number. To: CUBE-LOVERS at MIT-MC First the administrivia. Starting with this message mail to Cube-Lovers is no longer automatically redistributed to everyone on the list. This was done because the "please add me to this list" type of message is now almost as frequent as messages discussing topics of general interest. Now the "editors" (Dave Plummer and myself) will have a chance to catch these boo-boos. Hopefull this change will be completely invisable to the rest of you except that the mail headers will contain our names as the senders, and the turn-around time will be a little slower. If this really offends anyone then we can put it back the way it was, but lets try it this way for a while. You should continue to send your messages to Cube-Lovers at MIT-MC. Now, on to God's number! As you may recall, a somewhat complicated counting argument sets a lower bound on God's number (the worst case counting of the best possible algorithm) at 21 Quarter twists. Dan Hoey's message of January 9 1981 (the second message in a series about the "Supergroup") contains an excellent summary of that argument, so I won't repeat any of it here. That argument takes into account certain trivial identities such as FFFF=I and FB=BF in order to reduce the amount by which the counting overestimates the number of configurations a certain number of twists away from "solved". The same argument ignoring the identities only leads to a lower bound of 19. It is thus natural to expect that taking even more identities into account would lead to an even higher lower bound. Well, the next smallest identities are those of the form FR'F'RUF'U'FRU'R'U=I. It is known that there are none smaller that aren't a consequence of those "trivial" identities mentioned above (See Hoey's message of 22 March 1981), although there might be others of the same length. What happens if we take these additional no-ops into account? The conceptual problems in applying these new identities to the counting have had me stumped for quite some time now, but last week I finally figured a way that would cover at least some (maybe even all, I haven't worked on a proof of that yet) of the consequences. Well, all right, I'm keeping you in suspence, what did I learn? Nothing. The lower bound still stands at 21 (similarly the Supergroup lower bound still stands at 25). Even after taking FR'F'RUF'U'FRU'R'U=I and his many friends into consideration it seems that the numbers that fall out of the counting scheme (and it is amazingly complicated!) are only slightly smaller than those we already knew. The relevant numbers are: Size of the cube group: 43252003274489856000 Under old counting: positions 20 Q's or less away from start: 39812499178877773072 positions 21 Q's or less away from start: 373188814849923987472 Under new counting: positions 20 Q's or less away from start: 39726356237445007600 positions 21 Q's or less away from start: 372326146413193718032 As you can see the numbers are depressingly close. This seems to shut the door on any further improvements of this kind to this argument. It is hard to imagine that the effects of any other identities (remember they have to be at least 12 Q's long) could be signifigantly greater than the effect here. (Of course, if we knew ALL of them... but then we would understand the group completely!) It is of course possible that some deeper property, deeper than just the knowledge of one identity, could improve this style of counting argument. It is of course also possible that I have screwed up somewhere. I sould really let some of the rest of you into the details of this thing. As you can guess, I am not very excited at the idea of having to explain the details of the argument to you all. The proof is complicated and kludgey, and I at least am convinced that it leads nowhere. People who are interested in the gory details can contact me and we can work something out.