Problem 1470

1. Define the problem and the given conditions:
- Let $M$ be the set of all lattice points in the plane, i.e., points with integer coordinates $(x, y)$.
- For any point $P = (x, y) \in M$, the neighbors of $P$ are the points $(x-1, y)$, $(x+1, y)$, $(x, y-1)$, and $(x, y+1)$.
- 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 $P \in S$.

2. Objective:
- Prove that if such a perfect mapping $f$ exists, then there also exists a perfect mapping $g: S \to S$ with the additional property $g(g(P)) = P$ for all $P \in S$.

3. **Consider the orbit of a point $P$ under $f$:**
- Define the orbit $O(P)$ of a point $P \in S$ as $O(P) := \{ y \in S : y = f^n(P), n \in \mathbb{Z}_{\geq 0} \}$.
- Since $S$ is finite and $f$ is one-to-one, the sequence $P, f(P), f^2(P), \ldots$ must eventually repeat, implying that $f^n(P) = P$ for some $n$.

4. **Partition $S$ into disjoint orbits:**
- The set $S$ can be partitioned into disjoint orbits $O(P)$ for $P \in S$.
- Each orbit $O(P)$ is a cycle of points under the mapping $f$.

5. Properties of the orbits:
- Since $f$ is a perfect mapping, each point in an orbit $O(P)$ is mapped to one of its neighbors.
- The length of each orbit $|O(P)|$ must be even. This is because each point has exactly one neighbor it maps to, and the cycle must return to the starting point, requiring an even number of steps.

6. **Construct the mapping $g$:**
- For each orbit $O(P)$, define $g$ such that $g(P) = f(P)$ and $g(f(P)) = P$.
- This ensures that $g(g(P)) = g(f(P)) = P$ for all $P \in S$.

7. **Verify the properties of $g$:**
- $g$ is a one-to-one mapping of $S$ onto $S$ because it is defined in terms of the one-to-one mapping $f$.
- $g(P)$ is a neighbor of $P$ because $f(P)$ is a neighbor of $P$.
- $g(g(P)) = P$ by construction.

Thus, we have constructed a perfect mapping $g$ with the desired properties.

$\blacksquare$