I don't have the patience usually to figure out how to solve the hardest puzzles without using hints. When I used to do these on paper, sometimes I would go so far as to make a copy of it and then see "well if I assume that THIS square is black, then.....". If there's a reasonably large number on an edge with only tiny numbers in the next row/col over, that often helps me figure out spots that can't hold a tile. For example, if there top row's rightmost number is a 4 and the second row has nothing but 1s in it, then I can eliminate spots starting at the right that would require adjacent tiles in the second row.