Cheryl's Birthday

Maybe you've heard of a math problem out of Singapore (meant for elementary students), which people in other countries have been going crazy about online. Some say it's impossible, but there's way to solve it, and understand it. The question is:

Albert and Bernard just met Cheryl, and they don't know her birthday. She gives them a list of ten dates, one of them being her real birthday;

• May 15
• May 16
• May 19
• June 17
• June 18
• July 14
• July 16
• August 14
• August 15
• August 17

Then, Cheryl tells Albert the month, and Bernard the date, and they don't share the information with each other. Albert says he doesn't know Cheryl's birthday, but he knows that Bernard doesn't know either. Bernard didn't know either at first, but because Albert can't figure out Cheryl's birthday, Bernard can. And when Albert finds out that Bernard knows, he too figures it out. So, when is Cheryl's birthday, based on their conversation?

It seems impossible to figure out the date and month, when you don't know either. If they both guess different dates, there's a 20% chance one of them will be right, but, there's a way to figure out for sure when Cheryl's birthday is:

Bernard, who was told the date, could not figure it out at first, which means that he was not told a date that only occurs once in the list. Using this, we can rule out the 19th (Of May) and the 18th (Of June). Albert knows that Bernard won't know at first, meaning that he definitely wasn't told a month containing one of the dates that only appear once, because or else. Bernard might have been told that date and have known for sure. This means that Albert wasn't told the months of May or June, leaving July and August.

So now, Bernard has figured out the date entirely out of August 14th, 15th or 17th, or July 14th and 16th. Since he knows the birthday, it isn't on the 14th. This is because the 14th is the only day that applies to both of the months, and if he had been told this, he would have been stuck between the two months.

Now, we're at August 15th or 17th, or July 16th. Keep in mind that Albert also knows this.

Since Albert eventually figures it out based on this and the month he is told, he wasn't told the month of August, or else he'd be stuck between the 15th and 17th.

So, Cheryl's birthday could only be July 16th.

It takes a few times, or twenty, to understand completely, but you'll get it eventually.