Imagine you're a mapmaker, ready to finish your masterpiece map of the world. The only thing left to do is add the colors. But how many colors do you need for the countries if you want to make sure no two countries of the same color are touching? Six? Five? It turns out that your world map, along with every other possible map, can be completely colored using only four different colors.

This is not immediately obvious. In fact, this particular mathematical problem, called the four color theorem, remained unproven for more than a hundred years. In the end, it took two clever mathematicians and a powerful computer to solve it, as Numberphile explains.

As with most difficult math problems, the first step in solving one is to solve a simpler version first. In this case, that means solving the five and six color problems. To solve these, you only need to know one crucial fact: It's impossible to create a map where every region has six or more neighbors. In other words, in any map there must be at least one region that has five or fewer borders.

This fact lets us determine that there are no maps that require more than five colors. But proving such a thing for *four* colors eluded mathematicians for over a century. The proof was finally found in 1976 by two mathematicians from the University of Illinois. It involved a set of more than a thousand small maps, along with the proof that each map could be colored with only four colors and that every possible map had to contain at least one of these smaller maps.

This was enough to prove the four color theorem, but getting there was not easy. The two mathematicians had to use a computer to check each one of the small maps to ensure it could indeed be colored with only four colors. At the time, this was controversial, as a computer had never been used to solve a math problem before.

Today, computers are commonplace in math proofs, but the four-color problem broke the first ground. With this one proof, mathematicians created an entirely new way to solve problems that dramatically changed how we view math, science, and our entire world.

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