Imagine taking two solids, a tetrahedron and a square pyramid. The square pyramid has five (5) faces, one of them is a square while the other four are triangles. The tetrahedron has only four (4) faces, all of which are triangles.
If you attached two tetrahedron's face to face you would have an object with 6 faces. Each tetrahedron had 4 faces, added together is 8 faces minus the 2 faces that are now inaccessible since they are attached to each other.
If you attached the square pyramid’s square face to one of the square faces of a cube you would have an object with 9 faces. The cube had 6 faces, the pyramid had 5 faces, added together is 11 faces minus the 2 faces that are now inaccessible since they are attached to each other.
Now, if the triangles of the tetrahedron exactly match the triangles of the square pyramid, in other words they are congruent, then you could create a new solid by attaching one triangular face of the pyramid to one of the triangular faces of the tetrahedron.
The puzzle is: How many faces does your new solid have?
Try this is in 3D. Click here to get a pattern you can cut out and fold to make a tetrahedron, and click here to get a pattern you can cut out and fold to make a square pyramid with triangular faces congruent to those on the tetrahedron. You may be surprised!
You can have fun making new solids with two square pyramids attached by different faces or two tetrahedrons attached together. These are not as interesting as the puzzle, but still fun.