Date: 30 May 1982 16:30-EDT
From: Alan Bawden <ALAN at MIT-MC>
Subject: God's number
To: CUBE-LOVERS at MIT-MC

OK, it's been some time since I pointed out where I keep archives and
things...  Old cube-lovers mail is archived in the following places: 

MC:ALAN;CUBE MAIL0	;oldest mail in forward order
MC:ALAN;CUBE MAIL1	;next oldest mail in forward order
MC:ALAN;CUBE MAIL2	;more of same
MC:ALAN;CUBE MAIL3	;still more of same
MC:ALAN;CUBE MAIL	;recent mail in reverse order

(Files can be FTP'd from MIT-MC without an account.)

In addition, I have the following two excerpts from the archives sitting on my
directory since they contain some of the more asked-for material:

MC:ALAN;CUBE 4X4X4	;Contains some pre-release speculations on the 4x4x4
			;cube.  Some are out of date, but it contains the only
			;analysis this list has seen of the 4x4x4 group, I
			;believe.
MC:ALAN;CUBE S&LM	;While most of the speculation about the diameter of
			;the 3x3x3 group is scattered randomly through the
			;archives, this file contains the single message with
			;the highest content.  Hoey and Saxe's message on
			;Symmetry and Local Maxima.

To briefly remind you all of ALL that we know about the diameter of the 3x3x3
group (refered to as "God's number" in many of our discussions):

We know that God's number is greater or equal to 21 quarter twists.
 (See Hoey's message of January 9 1981: "The Supergroup -- Part 2 ..." in MAIL1
 for a good explanation of this, as well as some other interesting bounds.)
We know that God's number is greater or equal to 18 half twists.
 (See Singmaster.)
We know that God's number is less than or equal to 52 half twists.
 (See Singmaster again, this is Thistlethwaite's algorithm of several years 
 ago.  I'll bet it's been improved upon by now.  There is a persistent rumor
 that he was trying for 41.)

We have never bothered to figure out an upper bound on God's number in quarter
twists ("Q"s).  It must be less than 104 Qs because of the half twist result,
but we could probably do better than that if we took the trouble to understand
Thistlethwaite's algorithm.

Proofs of these numbers, and a great deal of other discussion can be found by
sifting through the archives (unfortunately they are spread all throught the
files).  I would urge people to sift through the archives before starting any
new discussions on the subject.