Date:  1 June 1982 2220-EDT (Tuesday)
From: Dan Hoey at CMU-10A
To: Cube-Lovers at MIT-MC
Subject:  Lower bounds for the 4x4x4
Message-Id: <01Jun82 222052 DH51@CMU-10A>

Well, since you insist, here are my lower bounds for the 4^3.  For
the "colored" 4^3, where only the color pattern matters, some
positions require at least 41 qtw to solve.  For the "marked" 4^3,
where center facets of the same color are distinguished so as to
force a unique home position for each, some positions require at
least 48 qtw.  The proof is in MC:ALAN;CUBE4 LB and is about 5K
characters long.

A qtw of the 4^3 is either a quarter-twist of a face relative to
the rest, or a quarter-twist of half of the puzzle relative to the
other half.  Note that this makes a slice twist into two moves.  I
like this metric because it is consistent with our conventions for
the 3^3.  One of these days I'll explain why I like those
conventions.