1997 PMWC Problems/Problem T7

Problem

Color the surfaces of a cube of dimension 5*5*5 red, and then cut the cube into smaller cubes of dimension 1*1*1. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?

Solution

The number of cubes with at least one red face is . The smallest cube below that is 64, thus we need to prove that we can make a 4-4-4 cube with a totally red surface.

The corners of the 5-5-5 must be the corners of the 4-4-4, because they have the most colored faces and there are only 8 of them. The edges of the 4-4-4 are the edges of the 5-5-5 minus one, and the faces of the 4-4-4 are the faces of the 5-5-5 minus 9. Then we can fill in the center of the 4-4-4 with anything and it will work. Thus, works.