🚀 go-pugleaf

In article <1993Jan8.193314.3289@zip.eecs.umich.edu> kanad@quip.eecs.umich.edu (Kanad Chakraborty) writes:
>Suppose you are given 3 colors and asked to assign a color
>to each point in a plane such that (a) you *must* use more than one
>color for the entire plane; (b) *every* pair of points 1 inch apart
>are colored alike.  Can you make such an assignment ? The plane is
>a continuum of points, it is not a grid.

No.  Consider a single point of the first color.  All concentric rings of
integral radius around it must also be of the same color.  But since the
same holds true for each point on the ring of radius 1, we get that the
entire disk of radius 2 around any point must be exactly the same color!
This imples that the entire plane must be that color.

Conversely, if there WERE a point of the other color, the ring of radius 1
around it would be the other color too.  But where could you place this ring
without intersecting one of the concentric rings mentioned in the previous
paragraph?

	Howard A. Landman