Problem - when is my birthday - ANSWER

How to Figure Out Cheryl's Birthday

We asked our Maverick Mathematician for the answer. He has given us a clearer version of the story to help make the assumptions clearer. Cheryl has given Albert the month and Bernard the day. Examine what Albert and Bernard say. 

Albert "I don't know when your birthday is, but I know Bernard doesn't know, either". The first half of the sentence is obvious in that Albert only knows the month, but the second half is a critical clue.

How could Bernard know, Cheryl only whispered to him the day? If Cheryl had whispered "19", then Bernard would know the exact date (May 19) because there is only one date with 19 in it. Similarly, if Cheryl had whispered "18", then Bernard would know Cheryl's birthday (June 18).

For this statement (by Albert) to be true means that Cheryl did not say to Albert, "May" or "June".

Bernard: "I didn't know originally, but now I do."

Bernard now also knows that Cheryl's birthday is not in May or June, eliminating half of the possibilities, leaving July 14, July 16,Aug. 14, Aug. 15 and Aug. 17. If Cheryl had told him "14" he would not know, because there would still be two possibilities: July 14 and Aug. 14. Thus we know the day is not the 14th.

Now there are only three possibilities left: July 16, Aug. 15 and Aug. 17. 

Albert again "Well, now I know too!"

For Albert to know, the month has to be July, because if Cheryl had told him, "August" then he would still have two possibilities:Aug. 15 and Aug. 17.

Therefore Maverick believes the only possible answer left is July 16.