Sam Fairchild

PDL C36
Suppose you are going to remodel your circular kitchen of radius with tiles which have area 1 square inch. Siegel showed that the expected number of tiles you need to order is .
The strategy in computing the expected number of tiles is to count the number of vertices of the tiles which lie in the kitchen. This corresponds to counting the average number of points in the ball of radius over a family of discrete subgroups of the plane. Since we are all mathematicians, we will also explore some other families of discrete subsets which may not correspond to tiling your kitchen, but will still give (relatively) nice expected value and variance results.
This will serve as a practice general exam.