If Bernard had 19 he'd know her birthday was May 19th and if he had 18 he'd know her birthday was June 18th as these are unique dates, all the other dates occur in pairs.
This means Albert had a month that doesn't have 18 or 19 in it, so rule out May And June.
Now Bernard knows it's not those months. And knows the answer. It can't be 14 because that occurs in both months and knowing it's 14 doesn't tell you what month. So throw 14 out.
That Leaves two possibilities for month:
July 16
August 15 or 17
If Albert had August he wouldn't be able to tell if Bernard has 15 or 17 and this would not know still.
But if Albert had July, he'd now know Bernard could only have 16 which as the only possibility that makes the initial statement true and is therefore her birthday.
Q.E.D.
