Question 68
Let $n$ be a natural number, $n \ge 2$. There are $n$ lamps on a circle. The lamps are labeled clockwise by natural numbers from $1$ to $n$. Each lamp can be either on or off. A switch between every two adjacent lamps enables one to change the state of both lamps simultaneously. In the beginning, all lamps are off. How many distinct configurations of states of lamps is it possible to achieve using these switches?