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 re: Proof by contradiction by Ryan)

In Brian Smiths solution E and F were in the same 1-m equilateral triangle with D and so could not be the same color as D or each other and so must be the same as B and C, which were also in a (different) 1-m equilateral triangle with D. Therefore G is the same color as D and that's the same color as A. Consider A and G opposite ends of two diamonds each composed of back-to-back eq. triangles, swivelling about pt D.

Posted by Charlie on 2003-05-07 06:15:40