A $3 \times 3 \times 3$ cake cube is made of $1 \times 1 \times 1$ little cake cubes.
A rat eats one of the top corner little cubes. After eating any little cube , the rat can go on to eat any adjacent little cube.
Can the rat eat through all cake cubes finishing last with the center cube ?
Alternately colour the cubes black and white so that the rat starts at a white cube. There are $14$ white cubes and $13$ black cubes, so any path through all cubes must end at a white cube. But the centre cube is black by the colouring, so the rat cannnot eat all cubes and end up at the centre.