Which rational numbers cannot be written in the form x + 1/(y +
1/z) where x, y and z are integers?

Two Cubes

Age 14 to 16 Challenge Level

Two cubes, each with integral side lengths, have a combined
volume equal to the total of the lengths of their edges. How big
are the cubes? [If you find a result by 'trial and error' you'll
need to prove you have found all possible solutions.]