Therefore let consider sliding a counter vertically. Let's assume, for example, that the counter a is located directly over the empty square. Sliding it down changes the parity of the row number of the empty square. Now consider the total number of inversions. Two additional cases obviously lead to the same result.
N is always even.
This is precisely what we have set out to show. In the general framework of Graph Theory , R.Wilson gave a complete treatment of puzzles on graphs of which Fifteen is a particular case. However, a variant of the puzzle seems to contradict the general theory. If we allow the counters wrap around the puzzle in, say the vertical direction, we may look at the puzzle as played on the surface of a cylinder . And, similarly, we may imagine playing on a Moebius strip .