KöMaL 2017 K543

In a food store, 27 cubical boxes are arranged to form a large cube. In each box, there is a large piece of cheese. A mouse crawls into one of the boxes lying on the floor and eats the cheese he finds there. Then he moves to another nonempty box that has a face in common with the first box, and eats the cheese again. He continues in this way. Is it possible for the mouse to organize his moves so that he should finish $a)$ in the box at the centre of a face of the large cube? $b)$ in the box in the interior of the large cube?