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How many eggs were there

Jack is a farmer and owns some chickens. One day, he took a basket of fresh eggs to the market to sell.

The first person bought half of all the eggs in the basket and half an egg.

Then, the second person bought half of the remaining eggs and half an egg.

Later, the third person bought half of the remaining eggs and half an egg.

Now Jack's basket has only 1 egg left.

If no eggs were broken in the process, how many eggs did he originally have in his basket?

3 switches puzzle

You are standing in front of a room with one light bulb inside of it. You cannot see if it is on or off. Outside the room there are 3 switches in the off positions. You may turn the switches any way you want to. You stop turning the switches, enter the room and know which switch controls the light bulb. How?

Password Riddle

one day an employee of a high-security institution, found that he could not log in to his computer terminal. His password wouldn't work. Then he remembered that the passwords are reset every month for security purposes. So he went to his boss and they had this conversation:

Man-"Hey boss, my password is out of date."
Boss-"Yes, that's right. The password is different, but if you listen carefully you should be able to figure out the new one: It has the same amount of letters as your old password, but only four of the letters are the same."
Man: "Thanks boss."
With that, he went and correctly logged into his station.
What was the new password?