Question 79
Some $n\gt 2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ..., lamp $k$ with $k+1$, ..., lamp $n$ with lamp $1$. At the beginning, all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice-versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.