I couldn't get all the areas Larry did without some research. But I think you can see it without calculating the areas.
The four small circles have exactly the same area (4n in Larry's explanation) as the large circle. When you overlap those small circles to fit into the large circle, you are reducing the area covered by those circles by the area of the overlaps. So the overlapped small circles in the drawing cover an area of 4n-red. The blue area is 4n-(4n-red) = red.