After Albert's first sentence, it has to be July or August (since in May or June, there are days 18/19 that occur only once, in which case Bernard would know the date immediately).
Bernhard also infers from the same logic that it has to be July or August. Further, the day cannot be 14, since then Bernhard still wouldn't know in the second step (cannot decide between July 14 or August 14), but he claims to know.
That leaves July 16, August 15 or August 17. Albert, following up to here, then claims to know the date. If it were August, there would still be two choices and Albert wouldn't know. So it has to be July 16.
Disclaimer: I think I've seen a similar puzzle before, but I had to think about it again.
July 16.
After Albert's first sentence, it has to be July or August (since in May or June, there are days 18/19 that occur only once, in which case Bernard would know the date immediately).
Bernhard also infers from the same logic that it has to be July or August. Further, the day cannot be 14, since then Bernhard still wouldn't know in the second step (cannot decide between July 14 or August 14), but he claims to know.
That leaves July 16, August 15 or August 17. Albert, following up to here, then claims to know the date. If it were August, there would still be two choices and Albert wouldn't know. So it has to be July 16.
Disclaimer: I think I've seen a similar puzzle before, but I had to think about it again.