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 re: Almost proved: two gaps by Bractals)

Even small numbers (such as 13) can create a pattern which is relatively long (the first 64 cycles are all different).

Of course, any finite number of lamps is a finite state machine so it must *eventually* repeat (even if that repetition is just all lamps being off).

Posted by Gamer on 2010-09-03 23:52:40