Catherine and Belinda just became friends with Mary, and they want to know when her birthday is. Mary gives them a list of 10 possible dates:
January 15, January 16, January 19 February 17, February 18 March 14, March 16 April 14, April 15, April 17
Mary then tells Catherine and Belinda separately the month and day of her birthday, respectively.
Catherine: I don't know when Mary's birthday is, but I know that Belinda doesn't know. Belinda: At first I don't know when Mary's birthday is, but now I know. Catherine: Then, I also know when Mary's birthday is.
So when is Mary's birthday?