What is Cheryls birthday?

About: can you solve the maths question for singapore schoolkids that went viral?

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Seth says
This is a very interesting puzzle.   I am almost sure now that Cheryl's birthday is June 18.  Why? ... well just because that is the only date listed for which: if Albert knew the Month and Bernard knew the day, would be unique.  What i don't know is how Albert and Bernard established that situation by what they said to each other. 

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