Question

There are 100 prison cells in a row. All cells are locked Jailer is given permission by the Prime Minister of India to release any number of prisoners. In Round 1, Jailer opens all the doors. In Round 2, he closes every alternate door (2, 4, 6...). In Round 3, every third door (3, 6, 9,....) if Door is Open,, he closes it. If Door is Closed, he opens it. In Round 4, every fourth door (4, 8, 12..), if Door is open,, he closes it. If Door is Closed, he opens it. He does this for 100 Rounds. At the end, who are the lucky prisoners ?

Answer 1

All doors are closed at the moment.

If he does closing and opening of doors for 100 round, all those doors that have been interfered with even numbers of times will remain closed.

Take door 10

It will open in round 1, get closed in round 2, open in round 5, get closed in round 10.

So most number of doors will remain closed.

Except perfect squares.

Take door 25

It will open in round 1, get closed in round 5, and will be opened in round 25.

So all doors having a perfect square will remain opened at the end of 100 rounds, because such doors will be interfered with only odd number times.

So the lucky prisoners are 1,4,9,16,25,36,49,64,81,100