A sphere is painted black and white. Show that there are 3 equidistant points of the same color.

An arbitrarily small portion of a sphere looks like a piece of a euclidiean plane.  So the problem can be simplified to show there are 3 equidistant points of the same color on a finite region of a plane.  (Is this true or am I cheating?)

Assume it is possible to paint a portion of the plane so that no 3 equidistant points are the same color and try to do so.

Choose any two points, call all them A and B and paint them Black.  Find the two points that would form an equilateral triangle with A and B and call them C and D.  These must be painted White. 

Find the point that would form an equilateral triangle with C and D on the same side as A an call it E.  This point must be Black.

Find the point that would form an equilateral triangle with E and B on the same side as C and call it F.  This point must be White.

There is a point the would form an equilateral triangle with E and A (Black points) but also with C and F (White points).  This point cannot be painted without a contradiction.

Posted by Jer on 2007-02-13 15:41:03