This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
A cube of cheese is divided into 27 subcubes. A mouse starts at one
corner and eats through every subcube. Can it finish in the middle?
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.
Cut the 3*3*3 cube into single cubes. At each slice you can
rearrange the blocks. Can you do it with fewer than 6 cuts?