[geeks] Puzzle for the crowd

[der Mouse]

Those who are familiar with pentacubes can skip the next paragraph;
it's a very brief intorduction for those not familiar with them.

The pentacubes are the 29 different shapes that can be made by gluing
five cubes together by their faces.  There are numerous interesting
shapes that can be built with this set, or some subset of it, in a "3-D
jigsaw" sort of way.  (This is basically a 3-D analog to pentominoes,
if that means anything to you.)

The question: is there a shape that can be built from pentacubes in the
sense that there is a dissection of it into two or more
pentacube-shaped pieces, but which cannot be built from pentacubes in
the sense of physically assembling it from the pieces because every
possible assembly would involve moving one piece through another?

This can be taken to mean restricting assembly to moving one piece at a
time; it could also be taken to permit multiple pieces moving at once.
Obviously, the set of assemblies possible under the former paradigm is
a subset of those possible under the latter.  There's also the question
of whether motion means rectilinear motion of pre-oriented pieces
parallel to the grid axes or whether it can include arbitrary
rigid-body motions.

The question may be taken to refer to just a set of 29 pieces, one of
each shape; it may also be taken to refer to a set with an unlimited
number of each piece (though in that case it probably also needs a
restriction to finite shapes).

I haven't come up with a shape impossible of assembly even under the
restrictive-grid-motion and unlimited-pieces paradigms, which seems to
me to be the form most likely to produce an example.  But I also
haven't come up with a proof there is none.  Anyone?

Or is this a solved problem and I just don't know it?

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