Date: Fri, 14 Jan 94 15:44:48 -0500
From: der Mouse <mouse@collatz.mcrcim.mcgill.edu >
~~~ Subject: Re: 4x4x4 and 5x5x5 cubes.

I've been getting a lot of requests for 4x4x4 cubes, and we're
looking into getting them.

I'd (probably) buy one - I don't know what's become of the one I had.
Depends on price, of course, but I found the 5-Cube price acceptable.

1) Why are 4x4x4 cubes so interesting? Do the additional symmetries
make for interesting questions, are they more fun, or easier to
solve?

I doubt it. If one ignores the center slice in each dimension on a
5x5x5, one has a 4x4x4. I think it's the completist instinct any
collector has. :-)

Now what I'd *really* like is something topologically equivalent to a
2x2x2x2 Cube. Obviously a 2x2x2x2 Cube can't really be built, but it
should be possible to build something topologically equivalent. (A
3x3x3x3 would be nice too, but perhaps too difficult.) The hard part
is designing an emulation that has some aesthetic appeal; it's easy
enough to write a program that lets you permute appropriate overlapping
4-cycles of objects without any intuitively-obvious structure.

2) It appears to me that if you know how to solve the 3x3x3 Rubik's
cube, then you can easily solve the 5x5x5 rubiks (i.e., the
solution is derivative).

No, not really. If you can do the 3-Cube *and the 4-Cube*, then the
5-Cube has no new challenges to offer (nor, I believe, does any size).

But the 4-Cube and 5-Cube do have challenges the 3-Cube doesn't, namely
edge cubies and face-center cubies. All the 3-Cube experience in the
cubies which are swapped. (Or rather, general Cube-type-puzzle
experience will help - for example, how to use conjugates - but
3-Cube-specific experience won't.)

For example, you can treat the inner 3x3 faces of the 5x5x5 as a
single 3x3x3 cube. Alternately, you can treat the edges/faces
along with the the middle three slices combined into a single
slice as its own 3x3x3 cube, and this would not really disturb the
"inner face" 3x3x3 cube. Is this really so, or am I missing
something?

You're missing something, but not much. :-) As you say, there are two
ways of emulating a 3-Cube on the 5-Cube, namely to paste slices 2-1-2
along each dimension and to paste them 1-3-1. (I hope that's not too
abbreviated to be comprehensible - I mean, along each dimension, paste
the 5-Cube slices together into three groups, taking two, then one,
then two slices, or one, three, and one.) However, it is entirely
possible to scramble the 5-Cube in ways that cannot be solved by
treating the 5-Cube as virtual 3-Cubes, except in the trivial sense
that any 5-Cube turn can be viewed as one or more turns of
appropriately-chosen virtual 3-Cubes.

For example, I can swap two edge cubies (and also permute center cubies
in invisible ways); alternatively, I can permute the face cubies so
that the 2-1-2 virtual 3-Cube has two identical corner v-cubies.

der Mouse

mouse@collatz.mcrcim.mcgill.edu