Problem:
An rectangular box is built from unit cubes. Each unit cube is colored red, green, or yellow. Each of the layers of size parallel to the -faces of the box contains exactly red cubes, exactly green cubes, and some yellow cubes. Each of the layers of size parallel to the -faces of the box contains exactly green cubes, exactly yellow cubes, and some red cubes. Find the smallest possible volume of the box.
Solution:
Let the box contain red cubes, green cubes, and yellow cubes. The given information implies that . Thus every layer contains yellow cubes, and each layer contains red cubes. It follows that and , so . The value of is minimized when is chosen to be as large as possible, that is, when . The corresponding values of and are and , respectively, and the minimum volume of the box is . Note that this can be done if each of the five layers is colored in the pattern
The problems on this page are the property of the MAA's American Mathematics Competitions