Question 181
In a row there are $n$ lamps, $n \geq 3$, numbered from $1$ to $n$. At the beginning every odd-numbered lamp in the row is on and every even-numbered lamp is off. In each move we may simultaneously change the state of three consecutive lamps in the row (those that are off are switched on, those that are on are switched off).
a. Prove that the order in which the moves are made does not matter for the final state of the lamps.
b. For which numbers $n$ can we, in finitely many moves, reach a state in which every odd-numbered lamp in the row is off and every even-numbered lamp is on?
a. Prove that the order in which the moves are made does not matter for the final state of the lamps.
b. For which numbers $n$ can we, in finitely many moves, reach a state in which every odd-numbered lamp in the row is off and every even-numbered lamp is on?