There are n ≥ 2 lamps L₁, L₂, ..., L_n in a row. Each of them is
either on or off. Initially L₁ is on and all of the others are off.
Each second the state of each lamp changes as follows:

if the lamp and its neighbors (L₁ and L_n have one neighbor,
any other lamp has two neighbors) are in the same state,
then it is switched off; otherwise, it is switched on.

Prove or disprove that all of the lamps will eventually be switched off
if and only if n is a power of two.

Note: This is a problem that I modified from one proposed but not used at
the 47^th IMO in Slovenia 2006.

(In reply to solution by Dej Mar)

That part is interesting but it's phrased more as an observation rather than a proof. Regardless, you chose the easy half to prove -- the statement says "if and only if".

To prove the other half, one needs to show at least one light will always remain on for every value of n which is not a power of two. Thus the only values of n where they will all turn off at some point on the future are powers of two.

Posted by Gamer on 2010-09-03 04:17:50