A mouse eating a 3×3×3 block of cheese

Question

A cube of cheese is cut into a 3 	imes 3 	imes 3 arrangement of 27 little cubes. A mouse starts by eating the cube at one corner. Then, over and over, it moves to a little cube that shares a face with the one it just ate (so it steps to a cube directly beside, above/below, or in front/behind, never diagonally), and eats that one too. It never revisits a cube it has already eaten. The mouse hopes to eat all 27 cubes and finish with the center cube as its very last bite. Is that possible? If not, prove it can't be done.

Accepted Answer

It is impossible, and a three-dimensional checkerboard coloring shows why. Color the little cubes. Give each little cube coordinates (x, y, z) with each of x, y, z in \{0, 1, 2\}. Color a cube black if x + y + z is even and white if x + y + z is odd. Every step the mouse takes changes exactly one coordinate by 1, which flips x + y + z between even and odd. So each bite is the opposite color of the bite before it, the mouse strictly alternates black, white, black, white, ... Count the colors. Working through all 27 cubes, the ones with an even coordinate-sum number 14 (black) and the ones with…

A mouse eating a 3×3×3 block of cheese

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