MathDB

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 neighbours (only one neighbour for i \equal{} 1 or i \equal{} n, two neighbours for other

i

) are in the same state, then

L_{i}

is switched off; – 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.

Problem Statement