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