Problem 643

Consider a square of side length $d$, and the position of its bottom-right unit square. It has $d-1$ squares above, and $d-1$ squares to the left of it. Hence there are $(21-d)(16-d)$ possibilities for the bottom-right corner, resulting in $(21-d)(16-d)$ squares of size $d$ appearing in the grid. The total number of squares is
$$\begin{aligned} S &= \sum_{d=1}^{15} (21 - d)(16 - d) \\ &= \sum_{d=1}^{15} (336 - 37d + d^2) \\ &= 15 \cdot 336 - 37 \sum_{d=1}^{15} d + \sum_{d=1}^{15} d^2 \\ &= 15 \cdot 336 - 37 \frac{15 \cdot 16}{2} + \frac{15 \cdot 16 \cdot 31}{6} \\ &= 1840. \end{aligned}$$