[will_pseudonym]

Another way to think about it using the square board concept would be to figure out how many ways you can get not the result you're looking for, and take 36 minus that number for the number of possible squares out of 36 squares. So getting "no 6's" on either die would be the square on the board of 1 through 5, by 1 through 5, or 25 squares. So inverting that we'd arrive at the 11 squares.

In studying probability, I found that accounting for the "overlap" as you described it was more tedious in more complicated problems than just always calculating the joint probabilities and inverting them.