Problem 29. Birth of Archimedes
Nick and Nikoo know that Archimedes’ birthday is one of the following 10 dates, but they forgot which one it is (Khordad, Shahrivar, Azar, and Dey are names of months in Persian calendar):
Khordad 4, Khordad 5, Khordad 8,
Shahrivar 4, Shahrivar 7,
Azar 1, Azar 5,
Dey 1, Dey 2, Dey 8
The soul of Archimedes comes back to life and whispers the month of his birthday in Nick’s ears and day of his birthday in Nikoo’s ear. After this, Nick says:
“I don’t know Archimedes’ birthday, but neither does Nikoo."
After hearing what Nick said, Nikoo responded:
“I didn’t know when his birthday was previously, but now I know."
Then Nick says:
“Now I know too."
When is Archimedes’ birthday?
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1694967860131729819
The correct answer is Azar 1.
Two dates have unique days: 2nd of Dey, 7th of Shahrivar. If what Nikoo heard was a 2 or a 7, she could have guessed the date of the birthday without knowing what the month is. Nick, who only knows the month of the birthday, is sure that Nikoo does not know the date of the birthday, (that is, he is sure that none of the above two dates are the answer). How can you be of that by just knowing the month of the birthday? Because the month he heard is neither Dey nor Shahrivar!
Thus, the birthday is one of the following:
Khordad 4, Khordad 5, Khordad 8,
Azar 1, Azar 5.
After hearing what Nick said, Nikoo tries to find out whether the month of the birthday is or Khordadeh or Azar. Keep in mind that she is already aware of the day of the birthday and at this point Nikoo confirmed that she knows the date. Thus, we imply that the day is unique in the above list, so 2 more dates are ruled out and the solution is narrowed down to the following:
Khordad 4, Khordad 8
Azar 1
Now Nick knows that only the above dates can be the birthday and at this point he is realizes what the birthday is. Therefore, we can imply that the month of the birthday is unique in the list above. Thus, the birthday is Azar 1.