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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.