The following problem appeared in the 1980 PSAT and 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 side. The second pyramid has an
equilateral triangle base and three more equilateral triangle sides. Glue any
triangular side of the first pyramid to any triangular side of the second
pyramid, completely matching the two glued triangles together, and creating
a single solid object. How many faces does the resulting solid have?
For hints and solutions:
Steven Young's "pup tent" explanation