Every point in 3D-space is colored either red, green or blue. Let R (resp. G and B) be the set of distances between red (resp. green and blue) points. Prove that at least one of R, G, or B, consists of all the non-negative real numbers.

Choose a real number x.

Take any point in space and find three additional points that are each distance x away from the original point and distance x away from each other.  This should take the shape of a tetrahedron.

Two of these points will be of the same color. 

Rinse and repeat for all real values of x.

Reminds me of a previous 2D problem

Posted by Leming on 2006-08-25 14:54:16