Maximise sum of "non-overlapping" numbers in square array - help with proof
Actually, I don't believe your proposed solution is correct. An example where you might need this case

|   #    |
| #   # #|
| #   # #| 
| #   # #|


might be

| 1 2 1 1|
| 2 1 1 1|
| 1 1 1 2| 
| 1 1 2 1|


The optimal solution consists of all 2s, while there are only 1s in the corners; therefore, you cannot find an optimal solution just by starting at the corners.