How many faces does the resulting polyhedron have? Take a regular tetrahedron of edge one.

Also take a square-based pyramid, whose edges are all one (therefore the side faces are equilateral triangles of same size as the faces of the tetrahedron).

Glue a face of the tetrahedron to a triangular face of the pyramid so that their edges match up.

Considering the volume taken up by the two pieces as a single polyhedron, how many faces does it have?
I visualized the solution Nick provided:




(I found it because it had used this arrangement to build boats models with a magnet game)
Here's a variation on your solution to the problem.
If you truncate the regular tetrahedron at the midpoints (instead of the 1/3 points) this gives a regular octahedron. The above problem looks at 1/2 of the octahedron (a square pyramid) and puts back one of the tetrahedra that were truncated and removed in the first step. The two faces were coplanar when they were together originally.
To see that the above truncation gives an octahedron notice that each face on the original tetrahedron loses corner triangles but keeps an &quot;old&quot; equilateral triangle. The removal of the corner tetrahedron gives a new equilateral triangular face. 4 old equilateral faces + 4 new equilateral faces yield an 8 faced solid which has 4 faces at each vertex--a regular octahedron.