Coloring regular 2n-polygon
Source: Vietnam MO 1995
February 20, 2004
geometrygeometric transformationrotationinvariantcombinatorics unsolvedcombinatorics
Problem Statement
Given an integer and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that:
(i) Each vertice is colored by exactly one color.
(ii) Two vertices don't have the same color.
Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring.
Alternative statement:
In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.