Considering a $n\times m$ board and all possible 3x3 squares on the board, 1 square is chosen and random and excluded from the $n\times m$ board. How many $3x3$ are possible on the remaining board?
One of the key notes is in a 5x5 board, there are 9 possible square locations, but any one will block out all the others.
Also notice that a 6x6 has the same area as a 4x9 and 3x12.
A 6x6 has 16 possible squares, and max squares (S) $1\leq S\leq 4$
A 4x9 has 14 possible squares, and max squares (S) $2\leq S\leq 3$
A 3x12 has 10 possible squares, and max squares (S) $2\leq S\leq 4$
The main question was, what is the expected number of squares on average?
Would it help if n and m were bounded by 100?
Is the optimal solution a multiple of 3 for m and n?
What would be the best way to learn more about this kind of question/topic?