Brain Treaser

[Marcus von Cube]

Thanks for the nice riddle! Let me try:

A knows the month (one of May, June, July, August). He cannot know the exact date but he knows that B can't either. This rules out May and June because if it were one of the latter there is the possibility that B were told days 18 or 19 which are unambiguous. Therefore we have either July or August.

Now, B knows the exact date. He does the same reasoning as above and rules out May and June. He states that he now has enough information to know the exact date. So the day cannot be 14 which is ambiguous, so it must be one of 15, 16 or 17.

Now A knows that B knows the exact date. If the month were August, A still would have two choices, August 15 or August 17, but he states that he now knows the date, too. This leaves only July 16 as the solution.