> On the basis of Albert’s statement that Bernard cannot know the answer, you can rule out any month with a unique day; because if Bernard were holding [May] 19 or [June] 18 he could know the Birthday without any information from Albert.
Albert was given the month, Bernard was given the day.
Albert who only knows the month announces that he knows that Bernard who only knows the day can not know the date. This tells Bernard that the day is not unique. Which rules out unique days - NOT the month those days are in.
This was given to fifth graders?
You're not using part of what you know. You are not using the fact that Albert knows the month.
Albert knows that Bernard cannot know the answer without his help. That means he knows Bernard is not holding any date in May or June. How can that be? Only if Albert is Holding July or August.
Here's another way of looking at it:
Suppose, to the contrary, that Albert is holding June. Then he knows that Bernard has one of the days in June. Therefore, he CANNOT claim Bernard can't know the birthday without his help -- because Bernard might then be holding one of the unique days. We arrive at a contradiction. Therefore, Albert cannot be holding June.
The proof for May is identical. Assume, again to get a contradiction, that Albert is holding May. He can no longer make a definitive statement that Bernard cannot answer the question without him, because Bernard might very well be holding the unique day that occurs in May. Therefore Albert cannot be holding May, either.