[Aaryan]

A very long hallway has 1000 doors numbered 1 to 1000; all doors are initially closed. One by one, 1000 people go down the hall: the first person opens each door, the second person closes all doors with even numbers, the third person closes door 3, opens door 6, closes door 9, opens door 12, etc. That is, the n th person changes all doors whose numbers are divisible by n . After all 1000 people have gone down the hall, which doors are open and which are closed?

[Peekay]

All squares (1, 4, 9, 16,...) will be open. All others will be closed.

[curr3nt]

31 doors open? All the squares.

[Aaryan]

that was quick.

[Guest]

All of the door numbers that are perfect squares. So it will be doors 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961.

31 doors total.

[Guest]

all square numbered doors will be open.

The reasoning behind this is you take each number and find it's divisors.All numbers out there have an even number of divisors, except square numbers.

Since 1 divides all integers, he will open ( "change") all the doors. after that, each person will change the state of all the doors with numbers multiple of the persons own number.

Since all numbers (except the squares) have an even number of divisors, the state of any door will be changed an even number of times and end up it its initial state, closed.

The squares are the only numbers with an odd number of divisors and hence they will be left open.

Edited April 11, 2011 by Fesoj