I have lately been playing with several simple graphical-logic games,
including a couple of highly addictive versions of Uncross/Untangle:
  http://smartboards.typepad.com/smartboard/games/uncross-the-lines.swf
  http://www.nonoba.com/chris/untangle

and had the idea that maybe this could be done in 3D.

As far as manipulating points in 3D and having an OpenGL figure follow
those points, I did this several years ago with MockTurtle in the
Croquet SDK.
But exactly what would this game look like in 3D?  I'm thinking that
to keep it simple for an initial try, this would involve intersecting
hinged triangles.
The triangles would be hinged together along common edges, each edge
with a pair of common vertices, where those edges would NOT count as
intersections to be undone.  If anything other than the edges
intersects, then something needs to move in order to complete the
figure.

I started to build this in Panda3D and thought about trying it in a
current Cobalt image, but have now decided that the whole project is
beyond my
available time and effort threshold.  So if someone wants to try this,
go right ahead!  I may be able to help a little here and there.

Also, it might be fun and instructive to just build a clone of the 2D
games above in regular Squeak.

Cheers,
Dave