An infinite plane is colored in red and blue, each spot of color being finite.

geometry/coloring/two.colored.plane New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

An infinite plane is colored in red and blue, each spot of color being
finite.  Show that for a given length r there is a place in the plane
where you can put a stick of length r so that its two ends touch the
same color.

Part b (a bit more difficult): the same, but with the two ends touching
different colors.

-- Lovett Shahar (Shalov), May 1996

Show solution

Suppose there is no such point.  This means that for each point in the
plane, the circle of radius r is of the opposite color.  Pick 2 points
on this circle of distance r and they will be of the same color - hence
a contradiction.

-- Elisha Yalin-Mor <eym@netvision.net.il>

The proof for different colors (part b) seems trivial.  Select 2 spots
of different colors anywhere in the plane.  Using "toothpicks" of
length r, form a chain from one point to the other.  One of the
toothpicks must have ends on different colors.

-- Gene Wagenbreth <genew@netcom.com>