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.

This one came to me and I tried to write it more clearly than some of the others had been written, since people are still talking about this one, even though it's solved. Let me know what you think.

Begin proof:

Assume no point is one meter from another point of that same color.

Remember that every triangle has three vertices, and therefore interpret that every equilateral triangle on this plane must have one vertex of each color.

Using a Cartesian Plane, start at (0,0) and go right along the x-axis, draw three equalilateral triangles with sides of one meter.

The bases of the three triangles should be the following line segments: (0,0) to (0,1), (0,1) to (0,2), and (0,2) to (0,3).

Chose colors for these four points (0,0), (0,1), (0,2) and (0,3), and the three tops of the triangles, making sure each triangle has three different colored vertices.

Conclude: Points (0,0) and (0,3) have to be the same color.

Interpret: Following this procedure, all points 3 meters from (0,0) must be the same color.

Graphing all points three meters from (0,0) forms a large circle with radius 3 meters.

Choose any point on the circumference of the large circle. Graph a circle with radius 1 meter around that chosen point. It will intersect the larger circle twice.

Interpret: This can be done for each point on the large circle, so each point on the large circle is 1 meter away from two other points on the circle, all of the same color.

Conclude: There must exist more than zero points that are 1 meter away from a point of the same color on a plane with only three colors.

Note: This is obviously true for any measure of distance.
Note: One or two colors could have no points as described in the original problem (imagine a plane with one red point, one blue point, and the rest green), but a point of at least one of the three colors will have a point of the same color 1 meter away.

End Proof.

Posted by Ryan on 2003-05-07 15:46:43