I am trying to find a proof for the general case of the solution to the 'Squares' Problem.
This is what I have managed to figure out: If n is the number of squares in the top row, then the number of 3-by-3 squares is given by the formula $(n-2)^2$ I know this works for $n=4$ (the case in the diagram), because there are four 3-by-3 squares. It also works for $n=5$ and $n=6$, because I have drawn the cases and counted the 3-by-3 squares.
I know empirical evidence such as this does not constitute proof, so my answer to the $n=60$ case, that there are $58^2$ 3-by-3 squares remains unproven.
I have used the induction tag only because I tried (and failed) to use this method this afternoon, I will be happy with any kind of proof.