Suppose that one has an infinite two-dimensional regular grid of spacing one. When laid on the plane it cuts it into unit squares. Now take a second (identical) grid and place it with random shift and orientation on the plane, cutting the squares into smaller pieces. Continue in this way with $n$ grids.

Can anything be said, statistically, of the areas of the pieces chopped up in this way? Does this problem have a name?

Square Line Pickingmay be a useful search term: mathworld.wolfram.com/SquareLinePicking.html – Joseph O'Rourke Dec 3 '10 at 12:57