Show that it is possible to fill a rectangular box with rectangular blocks so that:
(1) the dimensions of all of the blocks are integers,
(2) no edge of a block coincides with an entire edge of the box, and
(3) no two blocks have identical faces (that is, you could not have blocks of size A×B×C and A×B×D).
Find the box of least volume for which this is possible.
There is a classic packing puzzle in which you pack six 1x2x2 bricks and three 1x1x1 cubes into a 3x3x3 box. The solution there fits parts 1 and 2 of this puzzle.
So then I thought all I needed to do was to adjust the sizes to satisfy part 3. So just sketching stuff out on paper I created a set of nine blocks:
1x5x8, 2x3x8, 1x3x6, 2x6x7, 1x4x7, 2x4x5, 1x1x1, 2x2x2, 2x4x6.
These pack into a 5x7x9 box.
Answer
