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1) First, let's consider what it means for a door to be toggled by a monkey:
- If a door is toggled an odd number of times, it will end up open.
- If a door is toggled an even number of times, it will end up closed.
2) Now, let's think about which monkeys will toggle each door:
- Door n will be toggled by every monkey whose number is a factor of n.
3) So, the question becomes: which doors have an odd number of factors?
4) A number has an odd number of factors if and only if it's a perfect square. Why?
- Factors usually come in pairs (if a is a factor of n, then n/a is also a factor of n)
- The only way to get an odd number of factors is if one factor pairs with itself, which happens in perfect squares
5) The perfect squares up to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Therefore, after all 100 monkeys have passed, only the doors numbered 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 will be open. All other doors will be closed.
To summarize: 10 doors will be open, and 90 doors will be closed.
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