Imagine that a painter went down to a mathematical plane and colored all of the points on that plane one of three colors.

Prove that there exist two points on this plane, exactly one meter apart, that have the same color.

Consider two points in cartesian space: A (0,0) and B (√3,0), where one unit equals 1m of distance. Points C (√3/2,1/2) and D (√3/2,-1/2) are each 1m from both A and B, and C and D are 1m from each other.

If A and B are different colors, both C and D will be the third color, and will satisfy the condition that two points 1m apart are the same color. To avoid the situation where two points √3m apart are different colors, for a given point, every point √3m from it would have to be the same color as the center point, in which case two points 1m apart on the resulting circle will satisfy the condition.

Posted by Bryan on 2003-05-01 13:01:09