Suppose that you have a 7x7 chessboard that has a single knight upon each square. Is it possible for every knight to move, legally, to another square so that each knight has it's own square?
Correct. There is another way to show that. Here it is. Suppose that it was possible, then to each knight there corrosponds exactly one knight that will take its square. There are 24 such pairs, which leaves one knight that isn't paired up. This knight moves, and its square is left empty. Hence there is at least 1 square with 2 knights on it. Thus, it's impossible.