Date: Mon, 17 Jan 94 16:22:50 -0500
From: der Mouse <mouse@collatz.mcrcim.mcgill.edu >
~~~ Subject: Re: Higher Order Cubes

Perhaps it would be possible to build a 4-Cube that was internally a
5-Cube but for which the middle slice was not actually visible on
the surface? Or a 2-Cube that's internally a 3-Cube?

Yes, I think you could build such a 4-Cube. Likewise, you could build
a 2-Cube as a 3-Cube with invisible middle slices. But I don't
believe you'd want one: it could get completely jammed much too
easily.

The reason: If you take a 3-Cube and rotate its left and right slices
45 degrees each, you cannot rotate any of its other faces.

Duh, yeah; that never occurred to me.

There may be a way out, though. If you can anchor the place where the
three axes meet to one of the corner cubelets in some way, the
problem is solved: [...].

Yes. I think this may be possible, too...consider a normal 3-Cube, and
restrict yourself to R, U, and F turns. Then ignore the center and
edge cubies - the ones that get invisibilized. You're left with a
2-Cube. Three edge cubies never move with respect to the center cubies
or the corner cubie they surround; glue those together. Presto!

The same treatment is not possible for making a 4-Cube out of a 5-Cube,
but an alternative occurs to me, that I *think* will work for higher
cubes: key three of the (invisible) center cubies to the center
six-pronged piece, so that they can't turn. Then half the face turns
will cause the invisible center slice to turn with them; non-face
slices (which don't exist on the 2/3-Cube) work normally.

I notice with this construction for (say) a 4-Cube, the puzzle core
turns whenever certain face slices do. With the 4-Cube I owned (and
presumably still own, if I could find it), the puzzle core turns
whenever certain next-to-center slices do. I suspect the latter would
make for a smoother-turning puzzle. Perhaps someone will someday build
a 5-Cube-turned-4-Cube and this can be determined.

In the (IMO unlikely) event I originated any of the above ideas, I
hereby place it/them in the public domain. Go wild, Ishi Press. :-)

der Mouse

mouse@collatz.mcrcim.mcgill.edu