AUSTRIAN-POLISH 1978 Problem 7

Let $M$ be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point $P=(x,y)\in M$ we call the points $(x-1,y)$ , $(x+1,y)$ , $(x,y-1)$ , $(x,y+1)$ neighbors of $P$ . Let $S$ be a finite subset of $M$ . A one-to-one mapping $f$ of $S$ onto $S$ is called perfect if $f(P)$ is a neighbor of $P$ , for any $P\in S$ . Prove that if such a mapping exists, then there exists also a perfect mapping $g:S\to S$ with the additional property $g(g(P))=P$ for $P\in S$ .