Hi rajarshee,

This question has a 'visual component' to it that you can use to 'track' when each door is opened or closed. While the pattern that chetan2u describes is what defines the correct answer, most people can't make that deduction immediately; as such, you have to do enough work to PROVE that that pattern exists.

Rather than work through all 20 people and all 20 doors, I'm going to work through the first several so that we can define the pattern involved...

Remember: All the doors start off CLOSED...

Door 1: Only Person 1 touches this door. So it IS OPEN at the end.

Door 2: Person 1 and Person 2 touch this door. So it is closed at the end.

Door 3: Person 1 and Person 3 touch this door. So it is closed at the end.

Door 4: Person 1, 2 and 4 touch this door. So it IS OPEN at the end.

Now, stop and look at the work that we've done so far... Which doors do we know for sure will be open? Door 1 and Door 4. What do those two numbers have in common? They're both PERFECT SQUARES..... Let's see if that pattern continues...

Door 5: Person 1 and 5 touch this door. CLOSED.

Door 6: Person 1, 2, 3 and 6 touch this door. CLOSED.

Door 7: Person 1 and 7. CLOSED.

Door 8: Person 1, 2, 4 and 8. CLOSED

Door 9: Person 1, 3 and 9. OPEN.

Notice how the next door that we know will be open in the end is Door 9. It is ALSO a PERFECT SQUARE. Given the work we've done so far, this MUST be the pattern, so we're ultimately looking for the number of perfect squares from 1 to 20. They are 1, 4, 9 and 16.

Final Answer:

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Rich

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