# Coloring a plane

Skeptic’s Play has a couple of interesting geometry puzzles I’ve repeated below:

1. 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.

2. 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 apply no matter how bizarrely the plane is colored.

Anyways, the first statement is pretty easy to prove. Let’s assume that there are no two points that are one unit apart that have the same color. Then there are both red and blue points in the plane. If there was only one color , then any two points at unit distance would obviously be the same color. Let’s pick any blue point. Then the circle centered around the blue point with a unit radius must be completely red. Any two points on the circle that are a unit distance apart would both be red, which would be a contradiction.

That was easy, but would the second statement be as easy to solve? Let’s try to repeat our approach and assume that no two points a unit distance apart has the same color. Without loss of generality, let’s pick any point A and say that it’s blue. Then the unit circle centered around A must be made of points that are either red or green. Again without loss of generality, let’s pick any point B on the circle and say that it’s green. Then the unit circle around B must be blue or red. Then point C at where the two circles intersect must be red. And the unit circle around C must be blue or green. I show this in the figure below, where a magenta circle is made up of points that can be either blue or red, a yellow circle is made up of points that can be either red or green, and a cyan circle is made up of points that can be either blue or green.

The obvious path to take is to repeatedly draw a circle around each intersection and hope there is a point which must result in a contradictory coloring. This would happen if magenta, yellow, and cyan circles all intersected at a single point. Unfortunately, this never happens no matter how far you go. In fact, we get a repeating pattern for the intersections that is consistent with our initial assumption. We need another approach.