AUSTRIAN-POLISH 1981 Problem 4

Let $n \ge 3$ cells be arranged into a circle. Each cell can be occupied by $0$ or $1$ . The following operation is admissible: Choose any cell $C$ occupied by a $1$ , change it into a $0$ and simultaneously reverse the entries in the two cells adjacent to $C$ (so that $x,y$ become $1 - x$ , $1 - y$ ). Initially, there is a $1$ in one cell and zeros elsewhere. For which values of $n$ is it possible to obtain zeros in all cells in a finite number of admissible steps?