Bernard would know if Bernard had been told 18 or 19. Those only have 2 possibilities.
So Albert knows that Bernard does not know, because Albert had been told something other than May or June (i.e. July or August).
Once Bernard knows that July or August are the only possibilities, then any of 15, 16, or 17 would imply an exact date. So how can Albert’s knowledge advance from there, I am unsure. It seems like they are trying to imply that it is July 16, but more conversation would have to transpire.
To finish up, the fact that Bernard knows tells Albert that it’s the 15th (Aug), 16th (Jul), or 17th (Aug). Albert knows the month — if it were August, then he still wouldn’t know which day, but if he knows July then there’s only one possible day (the 16th, therefore July).
“Once Bernard knows that July or August are the only possibilities, then any of 15, 16, or 17 would imply an exact date. So how can Albert’s knowledge advance from there, I am unsure. “
I came up with July 16.
After May and June were eliminated by A being 100% sure that B (who knows the number) won’t be able to figure out the month, then B from that clue and the 5 remaining dates suddenly knows.
For him to know with the available information, it cannot be 14 since there are two of them, so at this point it has to be 15,16 or 17.
Then for A, who knows the month but not the day, to suddenly declare that he knows too, means that it has to be July 16, since if he had august there would still two possible days and he wouldn’t be able to know which one.