The graphic represents the partially exploded net of a stellated cube as viewed in its cubic layout; each face is a right pyramid viewed from above.

As with a problem of similar concept the letters are placeholders for numerals; in this case they are 1 through 24, [A=1].

The apices of the pyramids form 'external' vertices
while those which describe the foundation cube are the 'interior' vertices.

1. What is the global sum of the 6 'external' vertices and then their average?
2. What is the global sum of the 8 'interior' vertices and then their average?
3. How can the faces be labeled to optimise both #1 and #2 simultaneously?

The optimum outcome would be for all vertices to have the value of their type, 'external' and 'internal'.

If we make pairs across edges of the original cube add up to 25, such as L and V, G and I, and H and U, then the interior vertices will all add up to 75.

But that of course prevents, say, L and J from adding to 25 also, but maybe there's a way to get the apices to add up right another way.

 

Posted by Charlie on 2007-06-22 12:30:30