Problem

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

Tags: combinatorics, Coloring, cells, table



A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.