OPM 2011 Problem 3

A set of $n$ lights, numbered $1$ to $n$ , are initially off. At every moment, it is possible to perform one of the following operations: $\bullet$ change the state of lamp $1$ , $\bullet$ change the state of lamp $2$ , as long as lamp $1$ is on, $\bullet$ change the state of lamp $k > 2$ , as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.