MathDB

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?

Problem Statement