In a long corridor in a strange hotel, there are 500 rooms numbered from 1 to 500. Someone walks along the corridor and opens all the doors. The next person to come along the corridor closes all doors with an even number. The third party examines doors 3, 6, 9, . . . and closes the door if it is open and opens it if it is closed. This is how parties come to blows, where party i only examines doors i, 2i, 3i, . . . and opens whatever is closed and closes whatever is open. He leaves all other doors alone. Which door is open after 500 people have walked past it? What about the general case when there are n doors and n parties follow the process described above. How many doors are open after such a process?