# Puzzle 16 | (100 Doors)

There are 100 doors in a row, all doors are initially closed. A person walks through all doors multiple times and toggle (if open then close, if close then open) them in the following way:

In the first walk, the person toggles every door

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In the second walk, the person toggles every second door, i.e., 2^{nd}, 4^{th}, 6^{th}, 8^{th}, …

In the third walk, the person toggles every third door, i.e. 3^{rd}, 6^{th}, 9^{th}, …

Likewise,

In the 100^{th} walk, the person toggles the 100^{th} door.

**Which doors are open in the end?**

**Solution:**

A door is toggled in i^{th} walk if i divides door number. For example, door number 45 is toggled in 1^{st}, 3^{rd}, 5^{th}, 9^{th},15^{th}, and 45^{th} walk.

The door is switched back to an initial stage for every pair of divisors. For example, 45 is toggled 6 times for 3 pairs (5, 9), (15, 3), and (1, 45).

It looks like all doors would become closed at the end. But there are door numbers which would become open, for example, 16, the pair (4, 4) means only one walk. Similarly all other perfect squares like 4, 9,…

So the answer is 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

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