jmc

Identify all the possible sizes of squares and count the number of squares of each size separately. $$\begin{array}{cc}\text{Size}& \text{number of squares}\\ 1\times 1& 16\\ 2\times 2& 9\\ 3\times 3& 4\\ 4\times 4& 1\\ \sqrt{2}\times \sqrt{2}& 9\\ \sqrt{5}\times \sqrt{5}& 8\\ \sqrt{8}\times \sqrt{8}& 1\\ \sqrt{10}\times \sqrt{10}& 2\end{array}$$ The sum of the numbers in the second column is $\boxed{50}$.

Note: the possible side lengths of a square drawn on a square grid with $n^2$ dots are the real numbers of the form $\sqrt{x^2+y^2}$ where $x$ and $y$ are nonnegative integers satisfying $x+y\le n-1$.