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
ThreeDimensional Plane by Ravi Raja)
In the threedimensional case (which I don't think is actually the problem) your logic about the sphere is just as faulty as in the 2d case, though in fact factually correct, you haven't proved it.
In the 3D case, the best way is to expand upon the solution of the 2d 2color problem. Instead of a triangle, construct a regular tetrahedron with edges 1 meter long. Then as there are only 3 colors to go around, and there are 4 vertices to a tetrahedron, two vertices must be the same color, and since it's a regular tetrahedron the points are 1 meter apart.

Posted by Charlie
on 20030501 05:32:54 