A hotel has rooms numbered 1 to 1000. One thousand visitors enter the hotel.

Question

1. The first visitor switches on the light switches of every room.
2. The second visitor switches off the light switches of every other room, starting with room 2.
3. The third visitor toggles the light switches of every third room, starting with room 3,
In general, visitor n toggles the switches of every nth room, starting with room n.
After the thousandth visitor, in which rooms are the light switches on?
Be sure to justify your answer!

Answer 1

After the thousandth visitor, the rooms that the light switches are still on are the ones that the numbers are a perfect square. For example, room number 4, 9, 16, 25, and 36 will all be on.