The way to solve this problem is to look carefully at the question and then work through each statement in turn and see what we can deduce from it.

From the question we know that Albert has been told May, June, July or August.

Bernard has been told 14, 15, 16, 17, 18, or 19.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.

• Bernard has been told the day of Cheryl's birthday. There are only two days, 18 and 19 that appear once on Cheryl's list. This means that Cheryl's birthday cannot be May 19th or June 18th as if it was then Bernard would know the answer.
• Remember that Albert has been told a month and from the statement he knows that Bernard does not know the birthday. For Albert to be certain that Bernard does not know Cheryl's birthday, the month Albert had been told must not have been May or June.
• Therefore Cheryl's birthday must fall in July or August.

Bernard: At first I don't know when Cheryl's birthday is, but I know now.

• Based on the above Bernard has worked out that Cheryl's birthday is in July or August.
• If Bernard now knows Cheryl's birthday the day he must have been told would be 15, 16 or 17. It cannot be 14 as this falls in both July and August and as such he wouldn't know Cheryl's birthday for certain.

Albert: Then I also know when Cheryl's birthday is.

• Based on the above Albert has worked out that Cheryl's birthday is one of the following:
  • July 16
  • August 15
  • August 17
• If Albert now knows for certain when Cheryl's birthday is he must have been told the month of July, as if it was August there would be two possible options.

=> Therefore the answer is July 16.