In a jail with 100 rooms, all locked initially, 100 rioters break in and disturb the rooms in the following way. First one stops at all rooms and opens them all. Second rioters stops at rooms numbered 2, 4, 6, . . . and locks open rooms, and leaving the other rooms as they were. The third rioter stops at rooms numbered 3, 6, 9, . . . and again opens a locked room and locks an open room, leaving others undisturbed. And this process continues. After all the 100 rioters have left which rooms would be open?

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In a jail with 100 rooms, all locked initially, 100 rioters break in and disturb the rooms in the following way. First one stops at all rooms and opens them all. Second rioters stops at rooms numbered 2, 4, 6, . . . and locks open rooms, and leaving the other rooms as they were. The third rioter stops at rooms numbered 3, 6, 9, . . . and again opens a locked room and locks an open room, leaving others undisturbed. And this process continues. After all the 100 rioters have left which rooms would be open?

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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i dont know! :P any good reasons ?
the ones that will be open are the squares (i.e. 4, 9, 16, 25, etc)
the ones that will be closed are the not squares

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Other answers:

@lgbasallote I think room 1 will be surely opened..
yes it will
1 is a perfect square isnt it
1 opens 4 2 closes 4 4 opens 4
100 th room
for 9 1 opens 9 3 closes 9 9 opens 9
for 16 1 opens 16 2 closes 16 4 opens 16 8 closes 16 16 opens 16
for 25 1 opens 25 5 closes 25 25 opens 25
for 36 1 opens 36 3 closes 36 6 opens 36 12 closes 36 36 opens 36
for 49 1 opens 49 7 closes 49 49 opens 49
I guess the number which has odd number of factors will be open
in 36 you missed some in between but result remains open,,hmm i see maybe you're correct..
And the room number which has even number of factors will be closed
for 64 1 opens 64 2 closes 64 4 opens 64 8 closes 64 16 opens 64 32 closes 64 64 opens 64
what comesafter 64 again?
81 1o 3c 9o 27c 81o
yes that
finally 100 1o 2c 4o 5c 10o 20c 25o 50c 100o nice.. thanks..
welcome
can there be some reason why its always sq. no ? or its just a pattern ? ! ?
it's a tedious proof... lol
i just know the trial and error proof of it
I guess only sq. no have odd number of factors
aha..yes it seems so..lol..might make up another good ques that only sq no. have odd no. of factors! :D
damn i closed the quess,,wait i shall tag,,
Oh..... got it....... I think I can prove it
IF its not an sq. number.... for example 12 then the factors are 1,2,3,4,6,12
right?
yes..
|dw:1345558048314:dw|Now,
Now, lets look a sq. number ... for example 25 its factors are 1,5,5,25|dw:1345558167578:dw|
hmm..so every no. will have even no. of factors if we count same no. twice..really ? that quite amazing!
If it is not a sq. number than there will always be a unique number for its factor.... so that their product will be that number
TO make it more clear......|dw:1345558421778:dw|
and sum of even numbers is always a even number
cool..i get it! nice.. :)
now lets return to my ques,,why only sq. nos should be there?
To...... write and make other understand is a very difficult......
without speaking
Am i right?
lol..you did a pretty good job sir.,thanks..
I mean it is easy for u guys........... but for others its really hard
am not very good at the subject sir,,so explaining to me always takes some effort! see,,am a bad student! hmm.. nevermind,,back to the ques please! why only sq. nos should be there?
*bookmark

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