In response to Richard Pavelle's message, there are about 7.4*10^45
color combinations. See "mc:alan;cube 4x4x4" for more details on the
subject (especially the derivations) by Dan Hoey.
I bought a solution book from my favorite bookstore today. The title is
"The Winning Solution to Rubik's Revenge". It is a sequel to "A Winning
Solution to Rubik's Cube" by the same author (Minh Thai), who is billed
as the U.S. National Champion of the Rubik's Cube-A-Thon. I assume the
Revenge book is not REALLY a winning solution... at least not yet.
His method is to put all corners together, go for two opposite centers,
then the edges of said centers. Next he goes for the remaining edges
(numbering eight), and then the last four centers.
He also goes into many patterns, and has developed a notation which I
have not found immediately obvious. I haven't looked at the book more
than to scan it yet, however. He does list a few pretty-patterns, none
of which are checkerboards of any sort.
On a different subject, Paul Schauble accidentally found out how to take
apart the 4^3 today - twist an outer layer about 30 degrees so that an
edge of the twisted face is directly over the edge cube that now forms a
corner on the rest of the cube. Pop out the cube from the outer layer,
twist the face again so that the popped cube's partner is in a similar
position for popping, and then pop it. The corners now come out fairly
Unlike Plummer's design, the insides of this beastie is a sphere with
grooves running along it to make a kind of universal joint. The center
cubies ride in these grooves and hold all of the other pieces in. We
only took out about a half-dozen cubies because the cube was NOT in an
initialized state, but closer to a pretty pattern. I was definitely
not interested in getting it ALL apart in order to get it back together
correctly. (we did it correctly the first try!). Unlike the 3^3 cube,
the 4^3 does not seem to come apart easily after the first few are out.
I seem to have gone on for a bit longer than I intended. I will study
the pamphlet (published by Dell/Banbury by the way) and report in more
detail on notation later.
I just noticed that in the section on Cubology, the author lists the
number of "arrangements is something in excess of 3.7*10^45". Since
Hoey's number is larger, I guess the statement is correct.