How Many?

Consider the large square shown below (if it does not come up as a square, assume it is one!):

Each side is divided into 4 equal segments and opposite points are connected (as shown) to form 16 small (1x1) squares. But there are now squares of other sizes (such as 2x2, and 3x3) “hiding” in this figure.

1. What is the total number of squares of ALL sizes (1x1, 2x2, etc.) that exist in this figure? Don’t guess. Find a methodical and convincing answer to the question. Explain your method.
2. If instead of dividing the sides into 4 equal segments, we divided them into N equal segments, what would be the total number (in terms of N) of squares of ALL sizes that exist in the figure?