Polygonal chain on a chess board
Source: Problem 3 of Russian Regional Olympiad 2011, grade 9
September 1, 2011
geometrycombinatorics proposedcombinatorics
Problem Statement
A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on the chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. (Author: D. Khramtsov)