Is it possible to assign eight distinct prime numbers to the eight vertices of a cube so that the sum of the four vertices on each of the six faces is the same? If this is not possible, show why not; if this is possible, find an example with the smallest sum of all eight primes.


Extra credit problems:

Same question for the other Platonic solids.

Same question for semi-regular solids (Archimedean solids, prisms, and anti-prisms).

Note: For the semi-regular solids the sum of the vertices of each face must be in porportion to their number of vertices. For example, with a triangular prism, the sum of the vertices for the triangular faces will be 3/4 of the sum of the square faces.