One of the five faces of the triangular prism shown here will be used as the base of a new pyramid. The numbers of exterior faces, vertices, and edges of the resulting shape (the fusion of the prism and pyramid) are added. What is the minimum value of this sum?
Using one of the rectangular faces as a base for the next pyramid, there would be the addition of 1 vertex, 4 edges, and 4 faces.
Using one of the triangular faces as a base, there would be the addition of 1 vertex, 3 edges, and 3 faces.
Therefore for a minimal sum, one of the triangular faces must be the base of the pyramid.
The existing figure has 6 vertices, 9 edges, and 5 faces.
6+9+5 = 20
With the addition of the pyramid, there would be an extra 1 vertex, 3 edges, and 3 faces.
1+3+3 = 7
But note that the base of the pyramid would no longer be a face - so we subtract one from our total:
Thus, the minimal sum is 26.