Question 4
Let $n\ge3$ be an integer. There are $n$ cells on a circle, each assigned $0$ or $1$. A rooster stands on one cell and repeatedly does the following:
- On a cell assigned $0$, it changes the number to $1$ and moves to the next cell counterclockwise.
- On a cell assigned $1$, it changes the number to $0$ and moves to the cell after the next cell counterclockwise.
Prove that after sufficiently many operations the following statement holds: if the rooster is on a cell $C$, it will go around the circle exactly three times and stop again at $C$, with every cell assigned the same number as immediately before these three laps.
- On a cell assigned $0$, it changes the number to $1$ and moves to the next cell counterclockwise.
- On a cell assigned $1$, it changes the number to $0$ and moves to the cell after the next cell counterclockwise.
Prove that after sufficiently many operations the following statement holds: if the rooster is on a cell $C$, it will go around the circle exactly three times and stop again at $C$, with every cell assigned the same number as immediately before these three laps.