OMM 2012 Problem 2

Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ( $1 \times 1$ squares) are neighbors if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors. An integer from 1 to 4 is written inside each square according to the following rules:

• If a cell has a 2 written on it, then at least two of its neighbors contain a 1.
• If a cell has a 3 written on it, then at least three of its neighbors contain a 1.
• If a cell has a 4 written on it, then all of its neighbors contain a 1.

Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?