Can You Figure Out the Singapore Birthday Problem?

[XLFD]

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates: 

May 15, May 16, May 19, 

June 17, June 18, 

July 14, July 16, 

August 14, August 15, August 17.

Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively.

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

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

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

So when is Cheryl’s birthday?

To find the answer and the reasoning behind it, check the link out at the end of this article, and try not to spoil the answer for others in any comments below.  This was a high school question in math posed originally in Singapore, that recently went viral, with many educated people perplexed about the answer.  With my math and puzzle background, I was able to solve it without too much effort, though I admit it is tricky.

If you are having a hard time solving this problem, here's a version of one of my favorite puzzles in the same vein, but simpler, with answer and reasoning just after the ensuing picture:

Three men are buried in the sand all facing forwards with their heads above ground. Each man has a hat placed on his head selected from a bag containing 3 red hats, and 2 black hats. The men cannot turn around to see the men behind them. The man at the back is asked what hat he is wearing. He replies 'I do not know'. The middle man is asked what hat he is wearing. He also replies 'I do not know'. The man at the front is then asked what hat he is wearing. He replies 'I am wearing a red hat'. How did he know?

Since the man at the back could not determine his own hat, this means that the front two men could not have been wearing black hats and that, therefore, there must be at least one red hat on the two front men. Therefore the middle man must not be able to see a black hat otherwise he would know he had a red one on. Therefore the front man must be wearing a red hat - which finally he deduces. Interestingly, the other two can never determine their own hats.

The same reasoning applies to the Singapore problem, Albert and Bernard rely on what the other knows and does not know to figure things out logically.  Still stumped?  Look here:

http://www.businessinsider.com/answer-to-singapore-high-school-math...

[Willy]

I've always love the hat riddle. Many years ago I finally solved it after much thought. The way I learned it is that each man was wearing a mask and one at a time was allowed to look at the other two and make a guess and was re-blind folded after guessing.  Most folks I knew never figured it out but a relative of mine who had never heard the riddle knew the answer immediately and explained it. He has one of the most logical minds of anyone I know.

As far as the birthday riddle is concerned I would like to know what the paragraph below means:

"Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively."

Did Cheryl tell Albert only the "day" and Bernard only the "month"? And were they aware she had revealed part of the answer to each of them.

[XLFD]

Yes on both counts.  Albert knew Bernard knew the day, and Bernard knew that Albert knew the month.  You also must presume in this case that both Albert and Bernard are using the information they are given to the best of their ability.

The hat riddle has a whole host of variants, most using the same line of reasoning that gets the answer to the birthday riddle.

[Willy]

X, I sent you a message with what I think is the answer. I didn't check to see if I was right because If I wasn't I still want to try and solve it.

[XLFD]

You made the proper inference from Albert's first statement, but using the information from both Bernard's and Albert's second statement is a bit more subtle.

If you get to a point where you have narrowed it down and get stumped, perhaps it would be a good idea to check each date remaining to see whether it would work or not work.  Only one should remain.