Since she told Bernard the day, but May is the only month when 19 appears, and June is the only month in which the 18 appears, he would think it was May 19 or June 18. However, when Albert says he knows that Bernard does not know the full date, Bernard must realize that Albert knows it is not either of those dates. Therefore, it has to be July 16. Oh damn! Unicorn has beaten me to the answer before I could explain it!