16 June 1999 Version
Using analytic geometry and determinants yields the following expansion of Adjoining Two Kinds of Pyramids, i.e., the material that begins:

The following problem appeared in the 1980 PSAT. The answer believed to be correct by the Educational Testing Service experts was in fact incorrect.

Construct two pyramids with every edge of equal length, L. The first pyramid has a square base and four equilateral triangle faces. The second pyramid has an equilateral triangle base. It has three more equilateral triangles as faces. This object is a Tetrahedron. The task is to glue any triangular face of the first pyramid to one of the second, the tetrahedron. In doing this, completely match the two glued triangles together. This creates a single solid object. How many faces does the resulti ng solid have? Helpful Diagrams. Reasoning from visual images can be tricky. Hence the following discussion. It was sent by electronic mail by a UCLA mathematician.

To avoid fractions assume that the triangles all have side 2. Place the square base of the pyramid on the x-y plane with its center at the origin and the vertex pointing in the positive z direction. [This means that the base edges of the pyramid are parallel to the coordinate axes.] Then the coordinates of the four base points are (-1, -1, 0), (-1, 1, 0), (1, -1, 0), (1, 1, 0) and the coordinate of the vertex is (0, 0, sqrt 2). You can check this by computing the distance between the various points using the Pythagorean theorem. If you place the tetrahedron on the face which contains the vertices (1, -1, 0), (1, 1, 0) and (0, 0, sqrt 2), then you can check that the other vertex is (2, 0, sqrt 2). Again use the Pythagorean theorem to verify all distances.

Next you must verify that the four points (1, -1, 0), (0, 0, sqrt 2), (1, 1, 0), (2, 0, sqrt 2) all lie in a plane. Call these points, in order, P₁, P₂, P₃, P₄. There are many ways to verify that these lie in a plane. For example, check that the three vectors P₁-P₄, P₂-P₄, P₃-P₄ are linearly dependent. These are the vectors from the origin to the points (1, 1, sqrt 2), (0, 0, sqrt 2), (-1, -1, sqrt 2). To check, look at the 3 by 3 determinant whose rows are these three vectors. It is immediate that it is 0, completing everything.