http://www.ics.uci.edu/~eppstein/junkyard/geom-color.html http://www.ics.uci.edu/~eppstein/junkyard/geom-color.html Yes, there are an infinite number of points

http://www.ics.uci.edu/~eppstein/junkyard/geom-color.html http://www.ics.uci.edu/~eppstein/junkyard/geom-color.html Yes, there are an infinite number of points at a distance 1 from the original point. However, that doesn't matter for the problem. All you want to do is color regions of the plane such that, for any given point, every point a unit away is in a different colored region.