Highly Dependent Lightbulbs
For integer
, I have
lightbulbs arranged on the circumference of a circle, each lightbulb can be either on or off. When I press a button, the lightbulbs will either:
- once every minute, be turned on if it is in the same state as the one of its left (both off or both on),
- once every minute, be turned off if it is in the different state as the one of its left (one on and one off).
How many integers are there such that after at most minutes, all the lightbulbs will be on simultaneously regardless of their initial states?
Image Credit: Flickr patrick etter .
The answer is 19.
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Consider 4 bulbs arranged in a circle, all off or 1,2,3 or all 4 on. wherever you begin you have to go clockwise and no matter what was their initial state, within 5 min all of them will be on. try the same with 3,5,6,7 bulbs, it won't work, but it works with 8 [take into account the number of bulbs that can be on and therefore the rest off]. It also works with 2 bulbs. The pattern can be easily seen that it is a geometric progression as follows: 2,4,8,.... use the formula for geometric progression to find out the largest term that is less than 10^6 [see below].
Tn = a r^(n-1) where a is the 1st term and r is the common ratio Tn : 2 (2)^n-1 < 10^6 ; 2^n-1 < 500 000 ; n-1 ln 2 < ln 500 000 ; n-1 < 18.9 ; n < 19.9 ;
therefore n is 19