Monkeys and doors  

Read Time: 1 min 00 sec. 

Problem:

There are 100 doors, all closed. In a nearby cage are 100 monkeys. The first monkey is let out, and runs along the doors opening every one. The second monkey is then let out, and runs along the doors closing the 2nd, 4th, 6th,… — all the even-numbered doors. The third monkey is let out. He attends only to the 3rd, 6th, 9th,… doors (every third door, in other words), closing any that is open and opening any that is closed, and so on. After all 100 monkeys have done their work in this way, what state are the doors in after the last pass? Answer with the number of open doors. Answer is a integer. Just put the number without any decimal places if its an integer. If the answer is Infinity, output Infinity.

Solution: 

The solution is pretty simple. Consider a number between the range. Let's say in our case it is 100 doors and 100 monkeys, So pick a number between 1-100. I considered number 56 where we find all the divisors of 56. Why divisors, lets understand if we take the divisors of 48 it is 1&48, 2&24, 3&16, 4&12, 6&8. Monkeys will visit all the divisor doors of 56 where on pass 1 they open the door and on 2nd they close, on 3th they open again and on 4th they close, 6th they open and 8th they close, 12th open, 16 they close, 24th they open and finally 48th its close which gives a conclusion that consider any number and you find at last it comes under the initial state, but in consider of any perfect square number say 16, the divisor are 1&16, 2&8, 4&4, here the 4 repeats and in similar cases every perfect square have a number repeated in divisor as they called perfect square numbers. Now on pass 1 they open, on 2nd they close, 4 they open, 8th they close but in 16 at end they open which is opposite of initial state. This shows the perfect squares doors will be in the opposite position of initial state and others are in initial state always. So the number of perfect squares in range of 1 to 100 is 10. So the answer is 10 and to get the number of perfect squares in any range here is my code.