You are making sugar cookies from a batch of dough in the following way: You take your dough and roll it into a perfect square of just the right thickness (you are a very precise baker). Then you cut out 1-inch radius, perfectly circular, cookies arranged in a square grid so that the cookies are tangent to each other and to at least two of the edges of the grid, as shown schematically in Figure 2. You bake those cookies, and then gather up all of the unused dough and repeat the process with what remains. You continue as long as the square formed from the scraps of the last iteration has enough room for at least one more cookie.

Suppose you start with the exact amount of dough needed to fit a 4-by-4 grid of cookies the first time you roll out the dough. How many cookies do you end up with in all? What about if you start with dough that just fits an $n\times n$ square of cookies to begin with?