Earlier this month (December 1), the 62nd William Lowell Putnam Mathematical Competition was held at universities throughout the United States and Canada. In commemoration, this month's problem is from a previous Putnam.
A man has a rectangular block of wood m by n by r inches (m, n, and r are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half of the cubes are completely unpainted. Find all possible values of m, n, and r.
Source: 1952 Putnam Competition