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?

- shubhamsrg

- chestercat

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- shubhamsrg

i dont know! :P
any good reasons ?

- lgbasallote

the ones that will be open are the squares (i.e. 4, 9, 16, 25, etc)

- lgbasallote

the ones that will be closed are the not squares

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## More answers

- anonymous

@lgbasallote I think room 1 will be surely opened..

- lgbasallote

yes it will

- lgbasallote

1 is a perfect square isnt it

- lgbasallote

1 opens 4
2 closes 4
4 opens 4

- anonymous

100 th room

- lgbasallote

for 9
1 opens 9
3 closes 9
9 opens 9

- lgbasallote

for 16
1 opens 16
2 closes 16
4 opens 16
8 closes 16
16 opens 16

- lgbasallote

for 25
1 opens 25
5 closes 25
25 opens 25

- lgbasallote

for 36
1 opens 36
3 closes 36
6 opens 36
12 closes 36
36 opens 36

- lgbasallote

for 49
1 opens 49
7 closes 49
49 opens 49

- anonymous

I guess the number which has odd number of factors will be open

- shubhamsrg

in 36 you missed some in between but result remains open,,hmm
i see maybe you're correct..

- anonymous

And the room number which has even number of factors will be closed

- lgbasallote

for 64
1 opens 64
2 closes 64
4 opens 64
8 closes 64
16 opens 64
32 closes 64
64 opens 64

- lgbasallote

what comesafter 64 again?

- shubhamsrg

81
1o
3c
9o
27c
81o

- lgbasallote

yes that

- shubhamsrg

finally 100
1o
2c
4o
5c
10o
20c
25o
50c
100o
nice..
thanks..

- lgbasallote

welcome

- shubhamsrg

can there be some reason why its always sq. no ?
or its just a pattern ? ! ?

- lgbasallote

it's a tedious proof... lol

- lgbasallote

i just know the trial and error proof of it

- anonymous

I guess only sq. no have odd number of factors

- shubhamsrg

aha..yes it seems so..lol..might make up another good ques that only sq no. have odd no. of factors! :D

- shubhamsrg

damn i closed the quess,,wait i shall tag,,

- shubhamsrg

@mukushla
@eliassaab

- anonymous

Oh..... got it....... I think I can prove it

- anonymous

IF its not an sq. number.... for example 12
then the factors are 1,2,3,4,6,12

- anonymous

right?

- shubhamsrg

yes..

- anonymous

|dw:1345558048314:dw|Now,

- anonymous

Now, lets look a sq. number ... for example 25
its factors are 1,5,5,25|dw:1345558167578:dw|

- shubhamsrg

hmm..so every no. will have even no. of factors if we count same no. twice..really ? that quite amazing!

- anonymous

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

- anonymous

TO make it more clear......|dw:1345558421778:dw|

- anonymous

and sum of even numbers is always a even number

- shubhamsrg

cool..i get it! nice.. :)

- shubhamsrg

now lets return to my ques,,why only sq. nos should be there?

- anonymous

To...... write and make other understand is a very difficult......

- anonymous

without speaking

- anonymous

Am i right?

- shubhamsrg

lol..you did a pretty good job sir.,thanks..

- anonymous

I mean it is easy for u guys........... but for others its really hard

- shubhamsrg

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?

- anonymous

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