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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