[thedufer]

In the last line, A notes that B knowing means that A now knows. This is only possible if it is July, since knowing the month isn't enough to distinguish between the two possible dates in August.

[ramy_d]

This is correct. Although this is how we solve the problem, how did B know? that's where problems like these bug me.

[_bfhp]

B knows because B had the information "16th", and "16th" is unique among the July and August dates.

[cafard]

Quite, but B could have easily known "15th" or "17th" for all that A knows in this particular part of the story.

[rwbhn]

Nope, A knows July from the beginning - which rules those out. We, the readers, don't know that until A's final stmt.