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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, which doors are left open and which are closed?

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**Chaos in the bus**There is a bus with 100 labeled seats (labeled from 1 to 100). There are 100 persons standing in a queue. Persons are also labeled from 1 to 100.

People board on the bus in sequence from 1 to n. The rule is, if the person â€˜iâ€™ boards the bus, he checks if seat â€˜iâ€™ is empty. If it is empty, he sits there, else he randomly picks an empty seat and sits there. Given that 1st person picks seat randomly, find the probability that 100th person sits in his place i.e. 100th seat.

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**Madmen in a circle**N persons are standing in a circle. They are labeled from 1 to N in clockwise order. Every one of them is holding a gun and can shoot a person on his left. Starting from person 1, they starts shooting in order e.g for N=100, person 1 shoots person 2, then person 3 shoots person 4, then person 5 shoots person 6 then person 99 shoots person 100, then person 1 shoots person 3, then person 5 shoots person 7 and it continues till all are dead except one. What's the index of that last person?

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An innocent person is accused of a murder in a village. The head of the village knows that the man is innocent but he also knows that the truth may be uncovered one day that his own son is the murderer. Just to hinder the attention of people, he offers a chance to the man. In a jar, he places two chits of paper â€“ one reads innocent and the other reads guilty. Among all the other villagers, he asks the accused person to pick up a chit. He says that their almighty god is watching them and the person will meet his fate truthfully with the chit.

Of course, the village head is cheating. He has put both chits with Guilty written on them. The accused person somehow suspects the same. But no one will believe him if he tries to expose the head. Also, he knows no one will look at the other chit once his fate is decided by his selected chit.

**What should he do to prove that he is innocent?**

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