🚀 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 ?

<SPOILER> ...





For the following, assume unit = 1 inch for simplicity.

Pick a color for the origin, say, red.

Then since every point of the unit circle about the origin is exactly
1 inch from a red point, all points on the unit circle about the
origin are red.

Then since every point of the unit disk about the origin is exactly
1 inch from some point of the unit circle about the origin, all
points of the unit disk about the origin are red.

Now assume every point of the disk of radius N about the origin is red.
Every point of the disk of radius N+1 about the origin is exactly
1 inch from some point of the first disk, so all points of the
disk of radius N+1 about the origin is red.

By induction, all disks of integer radius about the origin are
completely red, and since every point of the plane is within such a
disk (I assume this is NOT a projective geometry) the entire plane is
red.

Ergo it is impossible to use two colors.

-- David Karr (karr@cs.cornell.edu)