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.

(In reply to Two-Dimensional Plane by Ravi Raja)

Apparently you and I were posting at the same time. Again, consider 60-degree arcs along the circumference of alternating color. The endpoints of the arcs are 1 meter apart, and one of the endpoints of any arc is the opposite color as the other, since we always choose to include the endpoint as a specific (ie cw or ccw) end of each arc.

Posted by Charlie on 2003-05-01 05:29:07