I finally found a 4x4x4 cube a couple of days ago, and have a couple of
interesting observations. Forgive me if I repeat anything said so far,
but I have been ignoring everything on this list having to do with the
4^3 cube for fear of any sort of spoilers.
Using my 3^3 knowledge, I found it fairly easy to get it almost solved.
Half the time, however, I got it to the state where everything was
solved except that two adjacent edge cubies were flipped. I finally
convinced myself by means of a somewhat involved (and probably
fallacious) "proof" that I would have to exchange them before I could
solve the cube.
My first observation is simply a trivial proof of that fact that I
discovered immediately after I took the cube apart for the first time
to see what was inside -- it is mechanically impossible to put the cube
back together with the cubies flipped (but not exchanged). Some
similar parity-type arguments can be made about possible configurations
of the center cubies. What is interesting is that this presents a new
method of proving things about configurations -- if one can dream up a
mechanical model of a cube with different guts, it may be obvious that
some sorts of things are impossible. The cube simply has to behave the
same way externally. I wonder if there are nice ways to look at the
various parity-trinity features of the three-cube by looking at it
using a different model of the internal mechanics.
My second observation is that although a 5^3 and a 6^3 may someday
appear on the market, the 7^3 will be pretty tricky to build. When one
of the faces of a 7^3 is turned 45 degrees, the corner will lie
completely outside the original cube. Any mechanical linkage will be
complicated indeed. Maybe a cube could be built with little
microprocessors inside each cubie controlling little arms and hooks to
grab adjacent cubie faces ...
-- Tom Davis