Suppose you have an infinite plane, and each point on the plane has been arbitrarily painted one of two colors.

Prove that there exists an equilateral triangle whose vertices are all the same color.

What is the fewest number of points needed to prove this?

Start with any two points, one red (R) and one blue (B). The point midway between them will obviously match the color of one point or the other. For the sake of argument, let's suppose it is red.

Now, consider a point equidistant from the two red points. Either it is also red, meeting our condition, or it is blue. If it is blue, we can consider the other point X that is equidistant from the two red points:

 B


R R B


 X

Note that point X is equidistant from the two blue points as well as from the two red points. Whichever color it is, we have an equilateral triangle.

Posted by Bryan on 2003-08-26 10:59:20