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?

(In reply to re: 5-point proof by Kelsey)

No; if the problem were asking for the fewest number of points so that you can rearrange the colors any way you want, that's what it would have said.

We are looking for the fewest number of points needed to prove that such a triangle must exist, which is exactly what Bryan has done.

By the way, 5 points (two of one color and two of another, with a fifth point that makes a triangle with both pairs) must be the minimum number of points.

Posted by DJ on 2003-08-26 16:51:48