100 people walk down a hallway of 100 closed doors. Person 1 flips every door, person 2 flips every 2nd door, person 3 every 3rd, and so on up to person 100. Which doors end up open?
The answer comes down to pairs. Most doors have factors that come in pairs (like 1x12, 2x6, 3x4 for door 12), so they get flipped an even number of times and end up closed. Only perfect squares, like door 36, have a repeated factor, so they're flipped an odd number of times and stay open.
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Final answer: 10 doors (1, 4, 9, 16, 25, 36, 49, 64, 81, 100).
