The dates range from 14 to 19 among the 10 that are given with only 18 and 19 occurring once. Albert, having seemingly been told the month rather than the day, first says he doesn’t know when her birthday is – eliminating both 18 and 19 as possible days. If Cheryl had told Albert that the month was May or June, then the day could have been May 19 or June 18, and Bernard may have known the right day. But, as the question says, Albert knows Bernard does not, meaning that Cheryl has said her birthday is in either July or August. Out of the five remaining days in July and August, the day ranges from 14 to 17, with 14 appearing twice. If Cheryl told Bernard her birthday was on the 14th, then he would not have known but, he does, meaning it can’t be on the 14th. That leaves only 3 possible days: July 16, August 15, and August 17. After Bernard speaks, saying he knows the birthday given that information, it eliminates August from being a contender since he still wouldn’t have known whether it was August 15 or 17. Therefore, Cheryl’s birthday is on July 16.
