Question 171
On a screen formed by $n \times n$ ($n \ge 2$) squares, every square initially displays one of the three colors: red, yellow, and blue. Every second, the screen changes the color of each square following the rules below:
* for each square $A$ whose current color is red, if there is a yellow square sharing a side with this square, then $A$ turns yellow in the next second; otherwise, $A$ remains red;
* for each square $B$ whose current color is yellow, if there is a blue square sharing a side with this square, then $B$ turns blue in the next second; otherwise, $B$ remains yellow;
* for each square $C$ whose current color is blue, if there is a red square sharing a side with this square, then $C$ turns red in the next second; otherwise, $C$ remains blue.
Prove that if the screen does not change to a single color after $2n - 2$ seconds, then it will never change to a single color.
* for each square $A$ whose current color is red, if there is a yellow square sharing a side with this square, then $A$ turns yellow in the next second; otherwise, $A$ remains red;
* for each square $B$ whose current color is yellow, if there is a blue square sharing a side with this square, then $B$ turns blue in the next second; otherwise, $B$ remains yellow;
* for each square $C$ whose current color is blue, if there is a red square sharing a side with this square, then $C$ turns red in the next second; otherwise, $C$ remains blue.
Prove that if the screen does not change to a single color after $2n - 2$ seconds, then it will never change to a single color.