_{1}, L

_{2}, ..., L

_{n}in a row. Each of them is

either on or off. Initially L

_{1}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 (L

_{1}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.