Choose any 5 cities on Planet Earth. It is possible to cut the Earth exactly in half, such that the half including the slice (or boundary) contains 4 of the 5 cities. Can you prove it?
Often mathematical problems are easier to solve when we abstract. Abstracting is when you remove all information that will not help you solve a problem. Then you are left with only the important bits, or the abstraction. Above is an image of a mathematical abstraction of a sphere. During the abstraction process the we removed lots of information, such as the background of space, topography of Earth, and the distracting context. Imagine randomly placing 5 points on the sphere. How can you explain or show that it is always possible to cut the sphere in half, such that the half containing the boundary includes 4 out of 5 points?
Hint: Try to find a counterexample, such that it is impossible to split the sphere in half and still contain 4 points in one half. Start by adding one point at a time, as far apart as possible. Then explain why it is impossible to create a counter example. Good luck!
There are many different solutions!
MathSantaCruz 