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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

world won't help you if you get your 5-Cube solved except for two edge

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