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Last updated: October 9, 2018.

Rubik's magic constructible configurations

 Unlike other classical puzzles like the Rubik's cube or the more basic fifteen puzzle of Sam Loyd, obtaining all possible constructible configurations of the Rubik's magic is not an easy task. Apparently simple configuration often require moves through quite involved 3D shapes.
Proving that some configuration can be constructed requires an actual construction.
Proving that some configuration cannot be constructed requires the definition of a suitable invariant that is violated.
In particular enforcing two such invariants (so-called metric and topological invariants) we obtain a set of feasible configurations; we cannot state that they are constructible unless we are actually able to construct them.
Moreover, the set of constructible 3D configurations is infinite; too much to be studied completely. We are then forced to restrict to special (finite) subsets.

3D octominoids

Generalizing the notion of polyominoes (the game tetris uses the complete set of tetrominoes) in 3D we can focus attention to those configurations of Rubik's magic corresponding to eight unit squares with mutual orthogonal angles (or mutually coplanar). The concept of octominoid configurations was first introduced by Jürgen Köller in http://www.mathematische-basteleien.de/magics.htm.
In this case all feasible configurations have been actually constructed, so that we have the complete set of constructible configurations corresponding to 3D octominoids (click on link above).

planar face-up configurations

Consist of all configurations with all tiles lying on a plane, possibly including stacked tiles, and all tiles oriented face-up (this excludes in particular the flat shapes below). In this set there are still a few configurations that are feasible but not (yet) actually constructed, so that we do not have the complete picture of constructible configurations.

Flat shapes (tetris-like)

Configurations consisting of four stacks of two superposed tiles, one face-up and one face-down, in the overall shape of a tetromino. These have been described in [Nourse], to which we add the U flat shapes. Click on link above for a list and instruction to construct them

Interesting configurations

the cube
armchair
hard-to-reach planar shape

References

Other resources

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