OK, This puzzle is taken from Adam Spencer's book "World of Numbers". According to Adam, it was a problem "set for Singaporean Year 5 students, later upgraded to 14 year olds in an Olympiad test". That in itself suggests that not too many of the Year 5 students worked it out lol ...
Albert and Bernard have just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.
May15, May16, May19,
August14, August15, August17.
a) Cheryl then tells Albert the month she was born, (May June July or August), and Bernard the numerical day she was born (14 15 16 17 18 or 19).
b) Albert says "I don't know when her birthday is, but I know that Bernard doesn't either".
c) Bernard replies "At first I didn't know when her birthday is, but I do now.
d) So Albert says "Well now I also know when Cheryl's birthday is"
So when is her birthday?