A cube of cheese is divided into 27 subcubes.

geometry/coloring/cheese.cube

Problem

A cube of cheese is divided into 27 subcubes.  A mouse starts at one
corner and eats each subcube, one at a time.  Can it finish in the middle?

Show solution

Give the subcubes a checkerboard-like coloring so that no two adjacent
subcubes have the same color.  If the corner subcubes are black, the
cube will have 14 black subcubes and 13 white ones.  The mouse always
alternates colors and so must end in a black subcube.  But the center
subcube is white, so the mouse can't end there.