Consider the subcubes to form a three dimensional checkerboard composed of subcubes of swiss and cheddar cheese. Assume the corners and centers of each face are swiss cheese and the rest of the subcubes, including the center, are cheddar cheese. Assume the mouse eats two subcubes an hour. At the beginning of every hour the mouse will be starting on a swiss subcube and will end the hour with a subcube of cheddar cheese. After 13 hours the mouse will have just finished a cheddar subcube. However the center subcube is also cheddar. Since two adjacent cubes must be of different kinds of cheese the mouse can not eat the center cube last.

Thanks to Nick Hobson for suggesting this one.

Michael Shackleford, A.S.A.