This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.
A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?
This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)
OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane. There’s a standard way of doing that, used both by map-makers and mathematicians. Place the sphere with the south pole on the plane at the origin. Then for each point on the sphere, run a line from the north pole through that point to the plane. This gives a 1-1 mapping of sphere to plane. The diagram below shows this mapping, with the points A and B on the sphere mapping to the points A’ and B’ on the plane respectively.
Notice that in the mapping above, the south pole is mapped to the origin (“straight down”), while the north pole itself cannot be mapped. We call the north point the “point at infinity.” Also notice that the equator gets mapped to a circle. And, any circle around the sphere that goes through the north pole will also go through the south pole, and so becomes a line in the plane.