Santa has 100 reindeer in 100 numbered stalls and 100 elves to boss about. He picks reindeer by the following system: Initially all the stall doors are closed. Elf 1 opens every door, elf 2 then goes to every other door (i.e. multiples of 2) and changes its state (open to closed or vice versa), elf 3 does this for every multiple of 3 door and so on to elf 100. On Christmas Eve Santa chooses the reindeer in stalls with open doors. Which reindeer does Santa choose?
The square numbered doors will be left open and Santa will choose the reindeers in these stalls. Each door will be either opened or closed by every elf whose number is a factor of the door number. All numbers have an even number of factors except square numbers. The nonsquare numbered doors will be open/closed an even number of times and hence result in being closed. The square numbered doors will be open/closed an odd number of times and hence result in being open.