Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is.
Cheryl gives them a list of 10 possible dates:
May: 15 or 16 or 19.
June: 17 or 18.
July: 14 or 16 .
August: 14 or 15 or 17.
Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.
Albert: I don't know when Cheryl's birthday is, but I know that Bernard doesn't know too.
Bernard: At first I did not know when Cheryl's birthday is, but I know now.
Albert: Then I also know when Cheryl's birthday is.
So when is Cheryl's birthday?"
Cheryl's birthday is July 16.
Albert's first statement reveals that the Month is not May or June, because if it was then Bernard might know the date, if it was on the 18th or 19th (each of which uniquely determines the birthday).
Having narrowed the month down to July or August, Bernard's statement indicates that the day is either the 15, 16 or 17. If it was the 14th, he would still not know the Birthday.
Having narrowed the day down the 15, 16 or 17, Albert's 2nd statement indicates that the Month is July. If it was the August, he would still not know the Birthday.