Coloring the World
Have you ever looked at the map of the United States and noticed all of the shared boundaries? Or maybe you live somewhere with counties and notice all of the little blocks that cover a large body of land. You might have tried to color one of those maps and noticed that you may have only needed 4 or 5 colors so no same-color touched. It turns out that there is a rule to figure out how many colors you need to color it.
Take the map of the united states:
How many colors maximum do you need to fill it?
It turns out that the Four Color Theorem applies to any map on a plane stating only 4 colors are needed to color it.
Note: Depending on the map it may only require 1, 2, or 3, but it will never need more than 4. Here is the map filled only using four colors:
This solution was first discovered in 1852 when someone was trying to color a map. Francis Guthrie, the man who discovered this, then looked at other maps and saw that for them, only four colors were required too. Though the Four Color Theorem was proposed, no one suggested a proof for over a century. There have been many proofs for this problem over the centuries, but they were all found to be flawed. A mathematician by the name Heawood provided proof and an equation to show that four colors suffice for map coloring, however, it wasn't until 1977 that a more solid proof was discovered by computers. By the '70s, computers began to be used for proving theorems. They were somewhat reliable and could take large cases and test them multiple times. Appel and Haken took it upon themselves to find another proof for this problem using computers. They used many different maps which would then be solved by computers over the course of hundreds of hours. Even though this proof does stand firm today, because of the use of many discrete cases by a computer, it is not always accepted. This theory also scales up for maps on surfaces with different genuses. For example, a map on a torus requires 7 colors to color it. Even though this might not be the most useful theorem, it is still one step closer to explaining the world around us. Until a better proof is found, a computer solution will have to do. So next time you find a map, think to color it just to show that the Four Color Theorem will be held valid for that map too.