I have an $8 \times 8$ board. On the board, I want to choose 2 unit squares in each column and row such that none of the chosen squares are touching. This means they cannot share a side or a corner. How many ways can I choose the unit squares? What if the board is $9 \times 9$?
Now, I want to choose 3 unit squares in each column and row such that none of the chosen squares are touching (touching defined similarly). What is the minimum dimensions of the square board so that this is achievable, and on that board, how many ways is it doable?