Skeptic’s Play has a couple of interesting geometry puzzles I’ve repeated below:
Let’s say I’ve painted a plane — the flat kind, not the flying kind. Every single point on the plane is assigned the color red or blue. Prove that there must exist two points of the same color that are exactly one unit apart.
Every single point on the plane is assigned the color red, blue or green. Prove that there must exist two points of the same color that are exactly one unit apart.
What makes this interesting is that you can’t assume anything about how a plane could be painted. Mathematically, there can be all sorts of ways one could color a plane that are literally unimaginable. For example, if you color all points with rational number Cartesian coordinates with red, then the entire plane is colored red and yet it is at the same time almost completely colorless. Fractals are another example, where you can imagine large scale structures, but it’s not really possible to imagine it with all the small scale structures repeating no matter how much you zoom in. To prove the above statements, one needs to use a method that would still ...