Schnyder's theorem

Okay, so Schnyder's theorem is a really cool math idea that helps us understand how we can color different shapes. You know how you like to color things in coloring books, right? Well, imagine we have a shape with lots of sides, like a triangle or a square. If we want to color in all of the sides with just three colors (let's say we have a red, a green, and a blue crayon), then Schnyder's theorem tells us that we can always do it in a certain way.

Basically, the theorem says that we can choose one of the sides to be special, and then we can color in all of the rest of the sides so that they alternate between two of the colors (let's say red and green). Then, we can color in the special side so that it's always a different color from the color on the adjacent sides (so if the special side is green, then the sides next to it are red and blue).

Now, you might be wondering why this is such a big deal. Well, it turns out that this theorem can be used to solve lots of other math problems too! For example, it can help us figure out things like how to divide up a region into different parts that are all connected. So, even though it seems like just a simple coloring idea, Schnyder's theorem is actually a really powerful tool that can help us solve all kinds of math puzzles!