There are a couple of ways to try this problem . . .

1. The way you probably have been trying is guess and check: P is probably a 7 since it is shared by the 4 largest numbers etc.

2. Setting up simultaneous equations with all 40 variables. You would need 40 equations to solve it easily but I can only see the following:

4 vertical rows that add up to the sum of (1 through 7)

4 horizontal rows that add up to the sum of (1 through 7)

9 sum of 8 variables that are surronding grey squares.

so you have 17 equations and 40 unknowns.

I think a combination of the two strategies will help you out. for example set up the equations and make some intelligent guesses and see where it takes you.