by Bill Graham

A solid wooden cuboid is made which measures a whole number of centimeters along each of its edges. (I think a cuboid must be a British term for what I would call a rectangular prism or more simply, something shaped like a shoe box or a cereal box.)

It is then painted black all over its exposed faces and, when the paint is dry, the cuboid (prism) is cut up into 1-cm cubes. These small cubes are counted and it is then found that the number having no paint on them is exactly the same as the number which do have some paint on them.

What is the least number of small cubes there could be in total? What are any other possible answers other than the least number?

Suppose the number of unpainted cubes is 1/10 of the total number of cubes. What are the possible cuboids then?