Take a perfect cube. While keeping the cube intact and all the pieces together, make as many planar slices as possible through exactly three vertices.

After doing so, separate the resulting pieces. What shapes result and how many are there of each and in total?

To what do these numbers correspond?

Note: Please do NOT punch the problem into a 3D graphics program and then rush to post the solution here so you can be first. This remains my most enjoyable solution to date because I attempted it without pen and paper, computer or polyhedron.