Every point in 3Dspace 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 nonnegative 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 20060825 14:54:16 