# Alan Schoen’s Cubons (Solitaire XIV)

Alan Schoen is best known for the discovery of the gyroid, but he has also invented an enormous number of puzzles. The one I will discuss today he named cubons. Above you see his model of the 24 cubons which I currently have on loan for exploration.

Cubons of order n are obtained from a regular cube by dividing each edge into n+1 equal segments, and choosing on each edge one of the n subdivision points. These are then joined with the vertices on the same edge, the face centers on the adjacent faces, and the center of the cube. Adding faces results in eight polyhedra that can be assembled into a cube, obviously. There are n^2+n(n-1)(n-2)/2 different cubons of order n, which gives the sequence 1, 4, 11, 24, 45, 76, …

The 24 cubons for n=4 are particularly interesting because they might be used to assemble three cubes, using every cubon just once.  Finding a single solution is not so easy, because there are 735471 ways to select 8 from the 24 cubons, but only 18844 will allow themselves to be assembled into a cube, many of them in several different ways.

Still, there are a mind-blowing 1050759 different solutions to partition the 24 cubons into three groups of 8, each of which can be put together into a cube. One might want to put additional restrictions on the solution. For instance, one could ask that each cube has an equator, i.e. four consecutive unbroken segments. The picture shows front and back side of each cube on top of each other, the back is obtained by rotating the fron by 180º about a horizonta axis parallel the screen. There are still 1887 solutions of this case. One can also aim for the opposite, namely insisting that there is no straight dividing edge on any face. This allows 6361 many solutions. An esthetically pleasing limitation asks to have parallel edges being subdivided the same way. There are only two different solutions, using 16 cubons. Unfortunately, the remaining 8 cubons do not fit together. One can also us the color of the faces to impose restrictions. I am using here Alan’s color scheme (which employs 5 colors, one for two of the twelve possible cubon faces that are visible in a cube), but I am sure there are many possibilities.

For instance, below is a solution so that each face uses two different colors. There are 2544 of those… 