[rr56]

Because A knows the month and says B doesn't know the birthday, it cannot be May or June since they have unique days (Which would allow B to know the birthday). So far so good.

After this statement B know it's one of the remaining two months, July or August and that's enough for him to know the correct date hence they day he was told cannot be the 14th. leaving only July 16th, Aug 15 and Aug 17.

Since this information is enough for B to know the exact date, it clearly cannot be August, leaving only July the 16th.

[mjfl]

Why couldn't it be in August? For example if B was told 15, he could still do the same analysis, eliminate May and June and come to that conclusion.

[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.

[gcb0]

only 19 is a unique day.

[mynegation]

No, 18 is also a unique day