Question 127
Let $n \ge 2$ be a positive integer. There is an $n \times n$ chessboard-shaped region, each unit cell of which is a park. Each park contains some cats (the number of cats is a non-negative integer). To manage the cats, the administration performs operations on the parks. In each operation the administration chooses one park:
(1) the chosen park must contain at least as many cats as the number of parks adjacent to it;
(2) after park $A$ is chosen, for each park $B$ adjacent to it, the administration drives one cat from $A$ to $B$ (two parks are called adjacent if and only if they share a common edge).
Let $m$ be the total number of cats in all parks. Find the smallest $m$ for which there is an initial distribution of cats such that, by choosing the park for each operation appropriately, the administration can perform infinitely many operations.
(1) the chosen park must contain at least as many cats as the number of parks adjacent to it;
(2) after park $A$ is chosen, for each park $B$ adjacent to it, the administration drives one cat from $A$ to $B$ (two parks are called adjacent if and only if they share a common edge).
Let $m$ be the total number of cats in all parks. Find the smallest $m$ for which there is an initial distribution of cats such that, by choosing the park for each operation appropriately, the administration can perform infinitely many operations.