"middle", & "Last- most" are good places to start in your sudoku-like quest to fill in the rest of the blocks
Stage 2: "focus on inside Squares"–ignore the blocks on the outside edge and focus on the 15 inside; also, Dumbledore wouldn’t capitalize a word like that without reason
Stage 1: "Last-most…." "Last" means the last column, which you might have guessed since there are seven values described, and seven spaces in a column. "Most" means the number that is the most, as in, the highest value. Since there are 35 spaces and you are entering unique numbers into each block, starting wth "1,2,3," that means the top right block is 35.
Stage 2: "…Three digits each…." what could Dumbledore’s hasty note mean? Use each of the inside Squares for each snake? Make each Square three digits? Use each of the three digits? (D) All of the above–or should I say, each of the above?
Assume N39 27. W77 29. When it is time to "drive out the snakes", find six to complete the coords.
Desperate for another clue towards unraveling this mystery, you decide to use the Pensieve and browse a few of Dumbledore’s memories from around the time he would have been working on this puzzle. In one vision, you notice a scroll on his desk with a grid strikingly similar to the one you have been studying. Once released from the mystical flashback, you jot down the grid from that scroll as best you could: