# Number of regions created by overlapping circles

Given one circle, the number of regions is 2, the region outside the circle, and the region inside. Given two circles with the same size, the maximum number of regions is 4. The region outside both circles, the region inside the first circle but not the second, the region inside the second but not the first, and the region where the two circles overlap. How many regions can we make with 3 circles of the same size? 4? n? What if we allow the circles to differ in size? For some n, can we make all number of regions between the minimum and maximum, or are there certain numbers of regions which cannot be made? What about overlapping spheres? What about other shapes, and if applicable, their 3D counterparts. Here's the solution to the first two questions.