We have lamps 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 and its neighbours (only one neighbour for or , two neighbours for other ) are in the same state, then is switched off; – otherwise, is switched on.
Initially all the lamps are off except the leftmost one which is on.
Prove that there are infinitely many integers for which all the lamps will eventually be off.
Prove that there are infinitely many integers for which the lamps will never be all off.