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At a movie theatre, the manager announces that they will give a free ticket to the first person in line whose birthday is the same as someone who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday and that birthdays are distributed randomly throughout the year, what position in line gives you the greatest chance of being the first duplicate birthday?

 [Ref: ZBMH]

Hint: This is a very difficult puzzle.

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