enigma1376.txt (c) 2006 by Charles Petzold
"Square Facts and Faces" by Susan Denham, New Scientist, 28 January 2006, page 50
Regular polyhedron:
1. Number of vertices (V) is square.
2. Number of edges (E) is square.
3. Number of vertices plus edges is square.
4. Number of faces (F) is square.
5. More than half of the faces are squares.
How many faces? How many triangles?
Clues 1, 2, and 3 tell us that V = 9, E = 16, or V = 36, E = 64, or V = 81, E = 144, etc.
Let T be the number of triangle faces, S the number of squares, P the number of pentagons, etc.
Then, F = T + S + P + ...
Also, E = (3*T + 4*S + 5*P + ...) / 2 because each edge is shared by two faces.
Clue 4 tells us F = 4, or F = 9, or F = 16, or F = 25, etc.
F = 4 doesn't work so try F = 9. Based on Clue 5, suppose S = 5, T = 4.
Then E = (3 * 4 + 4 * 5) / 2 = 16, a square.
Could V = 9? Consider the polyhedron that looks like a square pyramid on top of a cube.
It satisfies all conditions.
Therefore, there are 9 faces and 4 of them are triangles.