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.
Here's my guess. I was inspired by charlie's note to try it in my head, so i'm not going to rigorously prove it ;).
- There will be 1 regular octahedron (equilateral triangles for faces). It came from the center of the original cube, and each face points to a different vertex. The edges of the octahedron are the same length as the edges of the original cube, and the vertices of the octahedron were the centers of teh faces of the cube.
- There will be 8 triangular pyramids. The bases of the pyramids correspond to the faces of the octahedron, and their peaks were at each vertex of the original cube. The pyramids aren't regular: the edges of the base are longer than the remaining edges by a factor of sqrt(2).
- There will be 12 triangular tetrahedrons (Technically, a triangular tetrahedronis equivalent to a triangular pyramid). One edge of each tetrahedron corresponds to an edge of the original cube. for each tetrahedron, the edge corresponding to the cube's edge and the edge opposite it are the same length (note: two edges of a tetrahedron are opposite iff they don't meet). The remaining edges are smaller by a factor of sqrt(2).
nice problem =)
guess
