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.
Suppose that the Mathematical Plane is a Three-Dimensional Plane, then we consider a SPHERE of Unit radius. Then there are Infinitely many points lying on the circumference of the sphere, all of which are at Unit distance from the centre of the sphere. Suppose the centre of one color and there is AT LEAST one point on the circumference which is of the same color as that of the centre, then there is nothing left to prove. If suppose the circumference consists entirely of points of ther other two colors ONLY, then obviously there exists AT LEAST one pair of points on the circumference which will satisfy the given conditions.
Hence Proved.
(I do not know whether this reasoning of mine is correct or not. If not then someone please do correct me).
Three-Dimensional Plane
