This puzzle has been requested as an interview query at tech corporations like Google.
There are 100 lights numbered 1 to 100, all beginning within the off place. There are additionally 100 folks numbered 1 to 100. First, particular person 1 toggles each gentle change (toggle means to alter from off to on, or change from on to off). Then particular person 2 toggles each 2nd gentle change, and so forth, the place particular person i toggles each ith gentle change. The final particular person is particular person 100 who toggles each one hundredth change.
In any case 100 folks have handed, which gentle bulbs will likely be turned on?
As standard, watch the video for an answer.
Or hold studying.
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“All will likely be effectively for those who use your thoughts on your choices, and thoughts solely your choices.” Since 2007, I’ve devoted my life to sharing the enjoyment of sport idea and arithmetic. MindYourDecisions now has over 1,000 free articles with no adverts because of neighborhood assist! Assist out and get early entry to posts with a pledge on Patreon.
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Reply To 100 Gentle Bulbs Puzzle
(Just about all posts are transcribed shortly after I make the movies for them–please let me know if there are any typos/errors and I’ll right them, thanks).
The lights turned on on the finish are the sq. numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Particular person n will toggle all multiples of n. This equivalently means a light-weight change is toggled for every of its elements d.
Since every gentle begins within the off place, a light-weight will likely be on if and solely whether it is toggled an odd variety of occasions. Which numbers have an odd variety of elements? We are able to work out some examples:
1 – issue 1
2 – elements 1, 2
3 – elements 1, 3
4 – elements 1, 2, 4
5 – elements 1, 5
6 – elements 1, 2, 3, 6
7 – elements 1, 7
8 – elements 1, 2, 4, 8
9 – elements 1, 3, 9
The numbers with an odd variety of elements are 1, 4, 9. These are sq. numbers. And in reality this can be a property of sq. numbers.
In a non-square quantity, each issue d pairs with a definite different issue n/d, as d(n/d) = n.
However in a sq. quantity, the issue √n pairs with itself, whereas the opposite elements do have distinct pairs, so the full variety of elements is odd.
The sq. numbers from 1 to 100 are exactly the lights that will likely be on on the finish. The lights turned on on the finish space the sq. numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
References
Geeks for geeks (100 doorways puzzle)
https://www.geeksforgeeks.org/puzzle-16-100-doors/
Joe Howard video
https://www.youtube.com/watch?v=SFNMflql5q8
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