Maximise sum of “non-overlapping” numbers from matrix

Question

Just looking for a bit of direction, I realise that the example given is possible to solve using brute force iteration, but I am looking for a more elegant (ie. mathematical

Answer 1

This is related to the n Queens problem, except that you do not care about the diagonal and you have weighted solutions. As the Queens problem, you can solve it by (multiple) backtracking.

I.e., once you find a solution you remember its weight, mark the soulution as invalid, and start over. The (a) solution with the highest weight wins.