Problem #WSP-000435

Problem

Pascal and Pierre are trying to work out Pythagoras. Being a bit obnoxious, Pythagoras puts a list of \(10\) possible dates on the table in front of them. In private, Pythag tells Pascal only the month of his birthday, and Pierre only the day of the month.

Pascal says “I don’t know Pythagoras’ birthday, but I also know that Pierre doesn’t know."
Pierre then says “Pascal was right that I didn’t know, but now I do know." Pascal replies “I know now Pythagoras’ birthday."

What is Pythagoras’ birthday?