Solving a Math Competition Problem: Part 2

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?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

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.)

At this point, various methods suggested themselves. Perhaps we could use recursion: let N_n be the regions created by n planes, and then we could examine the number of additional regions formed by n+1 planes?

Or, related to this, perhaps we needed to find the number of intersection points of each of the planes, and then relate the number of intersection points to the number of regions. But how to describe the intersection points?

It did occur to me that if we have n planes situated for maximal regions, they will divide the equator up into 2n subintervals, and adding another plane will divide up two of those subintervals into 4. Did that help? Well, it could help count the number of regions touching the equator: two for each subinterval (one north of equator, one south). But what about the regions not touching the equator? Hmph.

One possible way to visualize this problem is to project the plane onto a sphere. I know how to
do that, but counting the regions still seems hard.

For a while, I toyed with the situation where we have

  • Plane 1 – equator (this always happens: Just make plane 1 the equator) 𝑃1(0𝑁, 0𝐸).
  • Plane 2 – Prime Meridian 𝑃2(90𝑁, 0𝐸)
  • Plane 3 – Intl Date Line 𝑃3(90𝑁, 90𝐸)
  • Plane 4 – at an angle to all of those 𝑃4(45𝑁, 45𝐸)

I looked at my daughter’s wall map of the world: P4 goes through Tblisi Georgia and south of French Polynesia.

Where does P4 intersect the others? Could I make a formula to find the intersection points?

 

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