Closed Hemisphere with Points

If you pick any five points on Earth, can you find some closed hemisphere that includes at least four of those five points?

This problem is an interesting demonstration of the pigeonhole principle, a fundamental concept in combinatorics. The pigeonhole principle posits that if you have more items than containers, at least one container must hold more than one item. Applied to this problem's context, the Earth can be considered a sphere divided into two hemispheres, each acting like a 'container.' When you select any five points on the globe, you can think of them as your 'items.' Since there are more items (points) than containers (hemispheres), by the pigeonhole principle, at least one hemisphere must contain more than half of the points, which translates to at least three out of the five points.

In fact, due to the geometric properties of a sphere, we can do better than what the pigeonhole principle directly suggests. An extension shows that you can always find a hemisphere that contains at least four of the points, no matter how they are positioned on Earth. This is because you can always rotate the hemisphere such that it captures four points. Understanding this outcome not only enhances one's grasp of combinatorics but also explores the interplay between geometry and discrete mathematics. It is a clear reminder of how theoretical math principles, like pigeonhole and geometry, connect and function together in practical, although often abstract, scenarios.