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Here was the answer I posted on Facebook last night.

SPOILER - don’t read this comment if you don’t want to know the answer. You’ve been warned.

Okay, right off the bat, you can eliminate the entire month of May and the entire month of June. If Cheryl had told Albert that her birthday was in May or June, then the possibility would have existed that Bernard could know the birthdate simply by knowing the day, because if Cheryl had told Bernard either the 18th or the 19th, then Bernard would have immediately known that her birthday was either May 19th or June 18th. But, Bernard didn’t know immediately, so it couldn’t have been either of those days. However when Bernard heard that Albert knew that Bernard didn’t know immediately, that gave him all the information he needed to figure out what day and month it was.

Bernard’s initial problem was that the day he was told existed in two separate months on her list, and he couldn’t differentiate between them. However, when Albert said what he did and eliminated the months of May and June, that gave Bernard the information he needed to narrow his list down to one date. There was only one date that provided such ambiguity. That date was the 16th, which could have been in May or July. Bernard wasn’t sure initially which one, but when Albert helped him to eliminate May and June completely, then Bernard knew immediately that Cheryl’s birthday was July 16th.

When Bernard stated that he now knew what her birthday was, that gave Albert the information he needed to complete the logical analysis that I just did. That’s how Albert was able to say that he now knew what the date was, even though Bernard had not told him the date.