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Lamps in a row (Posted on 2010-09-03) Difficulty: 4 of 5
There are n ≥ 2 lamps L1, L2, ..., Ln 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 Ln 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 47th IMO in Slovenia 2006.

No Solution Yet Submitted by Bractals    
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Comments: ( Back to comment list | You must be logged in to post comments.)
How problem was modified | Comment 8 of 11 |

Instead of

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

the original probrem wanted

"Prove that there are

(a) infinitely many n for which all the lamps will eventually be off,

(b) infinitely many n for which the lamps will never be all off."

Clearly, if you can prove what I ask for, then you can prove what the original asked for.


  Posted by Bractals on 2010-09-04 01:49:20
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