## Spring 2018, problem 64

We have $n \geq 2$ lamps $L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $L_{i}$ and its neighbors (only one neighbor for $i = 1$ or $i = n$, two neighbors for other $i$) are in the same state, then $L_{i}$ is switched off; Ð otherwise, $L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on.

$(a)$ Prove that there are infinitely many integers $n$ for which all the lamps will eventually be off. $(b)$ Prove that there are infinitely many integers $n$ for which the lamps will never be all off.