The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is m+\sqrt{n}+\sqrt{p}, where m, n, and p are integers. Find m+n+p.
The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is m+\sqrt{n}+\sqrt{p}, where m, n, and p are integers. Find m+n+p.