Question 3
Let $n \geq 3$ be an integer. There are $n$ cells on a circle, and each cell is assigned either $0$ or $1$. There is a rooster on one of these cells, and it repeats the following operations:
- If the rooster is on a cell assigned $0$, it changes the assigned number to $1$ and moves to the next cell counterclockwise.
- If the rooster is on a cell assigned $1$, it changes the assigned number to $0$ and moves to the cell after the next cell counterclockwise.
Prove that the following statement holds true after sufficiently many operations:
If the rooster is on a cell $C$, then the rooster would go around the circle exactly three times, stopping again at $C$. Moreover, every cell would be assigned the same number as it was assigned right before the rooster went around the circle $3$ times.
- If the rooster is on a cell assigned $0$, it changes the assigned number to $1$ and moves to the next cell counterclockwise.
- If the rooster is on a cell assigned $1$, it changes the assigned number to $0$ and moves to the cell after the next cell counterclockwise.
Prove that the following statement holds true after sufficiently many operations:
If the rooster is on a cell $C$, then the rooster would go around the circle exactly three times, stopping again at $C$. Moreover, every cell would be assigned the same number as it was assigned right before the rooster went around the circle $3$ times.