Most regions from four circles

Question

One circle divides the plane into 2 regions (inside and outside). Two overlapping circles make 4.

What is the largest number of regions that four circles can divide the plane into? Find the general formula for n circles and explain where it comes from.

Accepted Answer

Fourteen regions. For n circles the maximum is n squared minus n plus 2. Adding one circle at a time. Suppose k circles are already drawn and you add the (k + 1)-th. The new circle can cross each existing circle at most twice, so it has at most 2k crossing points. Those crossing points cut the new circle into at most 2k arcs, and each arc slices an existing region into two. So the new circle adds at most 2k regions. Running total. Start with 1 region (no circles). - Circle 1 has no crossings, so it is a single closed arc that splits the plane once: it adds 1 region. Total 2. - Circle 2 adds 2…

Most regions from four circles

Your answer

Discussion

Learn the concepts

Related questions