I have created a few different polyhedron puzzles whose parts are able to move along great circle planes as hemispheres, and are held together by small powerful magnets.  The objective is to move the parts to separate the 2 different colors into 2 hemispheres.  A great circle plane is a plane through a sphere or polyhedron that includes the center point of the shape and divides the shape into hemispheres.  All of the individual faces have edges that are on great circle planes, and each edge of each face has a magnet that holds the edge tight against the edge of the adjacent face.  The overall fit of the puzzles is very good because the beveled edges were cut very precisely to allow the parts to move around without binding. 

The shape of the smaller puzzle is known as a cuboctahedron and it has 14 faces - 6 squares, and 8 triangles.  There are 4 great circles (or great hexagons to be more precise) in this puzzle, you can see all four of them surrounding each of the squares.  This makes it move quite a bit different from Rubik's Cube which only has 3 planes or parallel planes of rotation that are perpendicular to the others.

The larger puzzle has a shape known as an Icosidodecahedron - or just "id".  It has 32 faces consisting of 12 pentagons and 20 triangles.  It has a total of 6 great circle planes of rotation making it rather complex to solve.  If all else fails, the puzzles can be easily taken apart and put back together.