Last week, people around the world spent countless hours trying to figure out when Cheryl's birthday is, a strange priority considering we haven't yet cured cancer. Understandably, as experts in all things logical, we were asked about it by students and colleagues dying to know all about Cheryl. Is she a harbinger of Logic Games to come? We think not. Test makers are not terribly creative when it comes to stuff like that. Then again, keep giving them ideas...
The version of the Cheryl problem posted on Facebook was so poorly worded that, technically, we'd need to make a number of (unwarranted) assumptions in order to reach the correct answer. The problem would never pass LSAT muster. Did Albert and Bernard each know that Cheryl had told the other only the month, or only the day, of her birthday? We need to assume that each of them knew about the nature of Cheryl's answer to the other, otherwise the whole thing doesn't work. We also need to assume that neither Albert not Bernard are lying or confused in their responses. Here's the problem, when reworded:
Albert and Bernard just met Cheryl. "When is your birthday?" they asked her, making small talk of course. Cheryl responded, "I am not going to tell you, but I'll give you some clues so you can figure it out." She then wrote a list of 10 possible dates, and told them her birthday is one of these:
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Albert and Bernard were intrigued. "Fine, but which one is it?" Cheryl, reveling in the mystery she had suddenly created, decided that being coy is super fun. "Albert, I'll tell you the month, and only the month, of my birthday. Bernard - I'll tell you the day, and only the day. You boys ready to play?"
Neither of them had seen Saw III, so this line of questioning seemed perfectly normal. Cheryl then whispered in Albert's ear the month of her birthday; to Bernard, she whispered the day. Little did she know that Albert and Bernard are PowerScore LSAT instructors.
Albert: I don't know when your birthday is, but I know Bernard doesn't know either.
Bernard: I didn't know originally, but now I do.
Albert: Well, now I know, too!
Cheryl, stunned, let them live another day. So, when is her birthday?
First, let's think about how you can know the answer if you only knew the month, or the day, of her birthday. If you only knew the month (like Albert did), you cannot know her birthday because each month has multiple options. But, if you only knew the day (like Bernard did), you could: if Cheryl told you "18," you'd know that her birthday is June 18, because there is only one date with 18 in it. Similarly, if she told you "19," you'd know that it's May 19.
This line of reasoning explains why the second half of Albert's first comment is so critical: "I know Bernard doesn't know." How can Albert know this? If Cheryls had told him either "May" or "June," Albert wouldn't have been so sure, because if her birthday were May 19 or June 18, Bernard could have known it, which Albert is confident isn't the case. So, that rules out May and June since those are the two months where a day (18 or 19) might have been enough for Bernard to know, and for Albert to suspect that he might know.
Albert's confidence that Bernard doesn't know would alert Bernard to the fact that the month is neither May nor June. He is left with July and August. So, how can Bernard know the exact date, having only one number? If Cheryl had told him "14," he would not know, because there would still be two possibilities - July 14 and August 14. Thus, we know the day is not the 14th.
There are now three possibilities: July 16, August 15, and August 17. Any one of these three numbers would have been enough for Bernard to know the exact date, which explains the second half of his response, "... but now I do." But how can Albert know from this the exact day? If Cheryl had told him "August," he couldn't possibly know, because he would still have two possibilities: August 15 and 17. Thus, for Albert to know, the month has to be July.
Clearly, then, Cheryl told Bernard "16," which pointed him to the only available month, July. She told Albert, "July," which pointed him to the only remaining number, 16. And so, her birthday is July 16.
So, why was Cheryl so coy about the whole thing? Notice how conveniently she forgot to mention the year in which she was born. A deliberate act of omission? We think so. Botox can only do so much.