MathDB

Initially, each unit square of an

n \times n

grid is colored red, yellow or blue. In each round, perform the following operation for every unit square simultaneously:
[*] For a red square, if there is a yellow square that has a common edge with it, then color it yellow. [*] For a yellow square, if there is a blue square that has a common edge with it, then color it blue. [*] For a blue square, if there is a red square that has a common edge with it, then color it red.
It is known that after several rounds, all unit squares of this

n \times n

grid have the same color. Prove that the grid has became monochromatic no later than the end of the

(2n-2)

-th round.

Problem Statement