A cube of dimension $3×3×3$ is made of sugar and consists of $27$ small cubical sugar pieces arranged in the $3×3×3$ pattern. An ant is eating the sugar in such a way that it starts at one of the corners and eats smaller pieces one by one. After the ant finishes one piece, it moves to the adjacent piece (pieces are adjacent if they share a face). Is it possible that the last piece the ant has eaten is the central one?
Remark: Pieces don‘t fall down if a piece underneath is eaten first.
After many attempts to find a route for the ant to follow so that the last piece it eats is the central one, it seems like it is impossible.
Now to try and explain why is where I need help, I tried to work backwards and see where the ant needs to end up in order to be in a position where it is adjacent to the central piece but I just do not know how to prove that this is impossible.