A piece of cheese has the shape of a 3×3×3 cube, from which the central cube has been cut out. A mouse starts gnawing at this piece of cheese. First, it eats some 1×1×1 cube. After the mouse eats another 1×1×1 cube, it proceeds to eat one of the adjacent (by face) cubes to the one just eaten. Will the mouse be able to eat the entire piece of cheese

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Paint the $1 * 1$ cubes in a checkerboard pattern. After eating a black cube, the mouse can only eat a white cube, and vice versa.

## Solution

We will paint the $1 * 1$ cubes in a checkerboard pattern. Notice that there will be two more cubes of one color than the other (12 of one color, 14 of the other). But after a black cube, the mouse can only eat a white cube, and vice versa, meaning the colors of the eaten cubes strictly alternate. Thus, at any given time, the number of white cubes eaten differs from the number of black cubes eaten by no more than 1. Therefore, it is impossible for the mouse to eat the entire piece of cheese.

## Answer

no.