Okay...now please explain it for the mentally handicapped over here.
Cheryl tells Albert and Bernard separately the month and day of her birthday respectively.
This means that Albert knows the month, and Bernard knows the day.
So...they are looking at the list of dates Cheryl provided and Albert (who knows the month) says: "I don't know when Cheryl's birthday is, but I know that Bernard does not know too."
What month could Cheryl have told Albert that he would deduce the above statement from?
Well, if the only way Bernard would know the date is if the day that Cheryl told him was unique (never repeated). The only days not repeated are 18 and 19. The months with those days are May and June, and since Albert is sure Bernard does not know the day, we can assume that the month Cheryl gave Albert was NOT May or June.
So we rule out May and June dates.
Then Bernard (who knows the day) says: "At first I don't know when Cheryl's birthday is, but now I do."
If he is sure about the day of Cheryl's birthday that must rule out the number 14, because that day is repeated. So now we just have the options of July 16, August 15, & August 17.
Then Albert (who only knows the month) says: "Then I also know when Cheryl's birthday is".
Since Albert only knows the month, the only way he could be sure is if the month is July, since august has two options.
So Cheryl's birthday is July 16.
