MathDB

Let

P

is a regular

(2n+1)

-gon in the plane, where

n

is a positive integer. We say that a point

S

on one of the sides of

P

can be seen from a point

E

that is external to

P

, if the line segment

\overline{ES}

contains no other points that lie on the sides of

P

except

S

. We want to color the sides of

P

3

colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to

P

, at most

2

different colors on

P

can be seen (ignore the vertices of

P

, we consider them colorless). Find the largest positive integer

n

for which such a coloring is possible.

Problem Statement