# Can All Bulbs Be On?

Let $$n$$ be a positive integer $$\leq 1000.$$ Suppose $$n$$ bulbs are arranged in a row. Initially the first bulb is switched on and all other bulbs are switched off. Each second, we change the states of the bulbs according to the following rules:

• If the state of a bulb is identical to the state of all its neighbors (one neighbor for the bulbs at the edges, two neighbors for the other bulbs), it is switched off.

• Otherwise, it is switched on.

It turns out that eventually (after a finite amount of time) all bulbs are switched on. Find the largest possible value of $$n.$$

Details and assumptions

• The state of a bulb refers to whether it is switched on or off.

• This problem is inspired by ISL 2006 C1.

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