One Hundred Rooms
There are 100 rooms in a row in a building and inside each room there is a lamp that is turned off. One person enters each room and switches the lamp inside. Then, a second person enters every second room (2, 4, 6, etc.) and switches the lamp inside. A third person switches the lamp in every third room and so on and so far, until person #100 switches the lamp in room 100. How many lamps are turned on at the end?
We can see that the only switches that have been switched an odd number of times are the ones in rooms with perfect square numbers.
Indeed, if person N has switched the switch in room M, then person M/N has done that as well. Since person N and M/N coincide only when M=N², the claim above follows.
We conclude that the number of lamps that are turned at the end is equal to the number of perfect squares less than or equal to 100; that is exactly 10 rooms.
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