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 Three-Dimensional Plane
by Ravi Raja)
In the three-dimensional case (which I don't think is actually the problem) your logic about the sphere is just as faulty as in the 2-d case, though in fact factually correct, you haven't proved it.
In the 3-D case, the best way is to expand upon the solution of the 2-d 2-color 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 2003-05-01 05:32:54