The teacher wants to alter the seating by having every student move to an adjacent seat (just ahead, just behind, on the left, or on the right). Show that such a move is impossible.
I just want to make sure that I understand what this question is asking.
This question is (in subtle terms) asking you to construct a hamilton circuit and then asking you to show that it's impossible correct?
Paint the desks black and white as if it was a chessboard. There will be $13$ black and $12$ white desks. After a move, if it existed, every student would move from a black desk to a white desk and vice versa. This cannot happen as it would suddenly require $13$ white desks.