For anyone wanting to know a... more simple way of understanding this that wasn't just copy and pasted by others, it goes like this...
Albert knows the month, meaning he knows it's either May, June, July or August.
Bernard knows the day, meaning he knowing it's either 14, 15, 16, 17, 18 or 19.
Now that we know what they potentially know, it's down to eliminating possibilities. May 19 and June 18 are singles, no other date corresponds to their day of the month, meaning they are possibilities. As for the rest in May and June, they share the same day of the month with other months. Since Bernard knew the day, but not the month, if he knew the day were 18 or 19, that would mean he also knows the month. Since he does not know her birthday, that means it cannot be May 19 or June 18. That's two dates down, 8 to go.
Albert knew Bernard didn't know the exact day of birth, but why? If Cheryl told Albert the months of her birthday were either May or June, then it's possible that the birthday would have been 19 and/or 18. However, since Bernard did not know the birthday, but knew after Albert made the statement, that would mean it cannot be May or June, so it must be July or August. 5 down, 5 to go.
Now we have July and August left. Bernard now knows the exact day of which Cheryl's birthday is. The day 14 pops up twice. Since Bernard now knows the date, that means he knows exact month, meaning he couldn't do a 50/50 guess on July 14 or August 14, eliminating those 2 dates from the pool. Now we have July 16, August 15 and August 17.
With 3 dates left, we now know both parties know the exact dates of which Cheryl's birthday is. If Bernard know the day number, he would know the correct day and month, as only 1 of each number remains. Albert too knows the answer, but how do we know which of the 3 it is? If Albert knows the day and month, then if Cheryl told Albert her birthday was in August, he would have a 50/50 chance of choosing the right day, August 15 or August 17. Since he doesn't know which is correct, but statement says he does know, August cannot be a remaining month.
That means, of the 10 dates given, the only possible date left is July 16th.
