All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Almost proved: two gaps | Comment 7 of 11 |
(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

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information