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By now I'm sure everyone has run into the math puzzle where Albert and Bernard try to deduce Cheryl's birthday, which is all over social media, and even traditional media! If you don't know what I'm talking about, see e.g. this Washington Post article, or the transcript below:

It's a nice logic puzzle, and I'm delighted to see any math fall into general public consumption, but to me it feels like hundreds of other logic puzzles. As someone who has spent lots of time unsuccessfully trying to get people into math problems, I wonder what the trick is with this one. Whatever it is, can we use it to design problems that students will be naturally puzzled by?

Here's a transcript of the puzzle:

Albert and Bernard just become 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?

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    $\begingroup$ not a full answer, but I think that after the first time reading the problem, most peoples' reaction is "....what??? but this doesn't tell me anything! is this a joke??" and then after reading it a few more times and thinking about it the slow realization that there is much more under the surface, might be an explanation for its attractiveness. Another riddle that gave me this exact same reaction is the Blue Eyes riddle which may also be useful to examine in relation to the puzzle in your question $\endgroup$ – celeriko Apr 16 '15 at 16:24
  • $\begingroup$ @celeriko Speaking of Blue Eyes, I felt like having seen that puzzle, the "Cheryl's birthday" puzzle was trivial. $\endgroup$ – Kyle Strand Apr 16 '15 at 20:16
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    $\begingroup$ Though I think there probably are aspects to the problem that illustrate pedagogic principles, my suspicion is that the puzzle "went viral" because its origin story plays into the "Asian students are better at math than American students" narrative, a narrative that presses a lot of people's "hot buttons". I think had the puzzle been presented absent that narrative it would be just another clever bit of recreational puzzling. $\endgroup$ – Eric Lippert Apr 16 '15 at 20:45
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I think I would personally dispute the claim that this "went viral". I didn't see any people in my social networks talking about this problem until well after the media started reporting that "Everybody is talking about this problem!" Even now I don't see anybody talking about the problem; they are only talking about the coverage of the problem.

Having said that, I do think there are two interesting features to this problem that contribute to the interest in it:

  1. There is something odd about the way in which the information is deployed over time. The fact that in Step 2 Bernard knows Cheryl's birthday tells Albert something entirely different from what he (Albert) could have inferred if Bernard had known Cheryl's birthday from the outset. One might say that the knowledge claims interact in a noncommutative matter; each piece of information is meaningful in a precise way because of the information that came before it.
  2. The second thing that I think is interesting here is the way in which meta-knowledge (knowledge about the state of others' knowledge) can be used to make inferences about the actual state of events. In this case, the fact that Albert knows that Bernard does not know is knowledge about somebody else's knowledge; it is rather surprising that Bernard can leverage that. And once Albert knows that Bernard does know, that is again a piece of meta-knowledge, and it is again surprising that Albert can leverage it. There is a strange "level-crossing" phenomenon here, in which knowledge of Cheryl's birthday gets entangled with knowledge of others' knowledge of Cheryl's birthday. That kind of level-crossing makes me think of the "strange loopiness" in Douglas Hofstadter's book Gödel, Escher, Bach, which is all about level-crossing phenomena and the ways in which they capture the imagination.
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One possible reason is that it is couched in "everyday terms". I am more familiar with a version that has products and sums, and it seems products and sums are less known than months, days, and birthdays.

Another is that it involves a social event. Disregarding gender preference or other possible norms, many are interested in birthdays and ages as a mode of comparison, and the choice of two (male-sounding) names considering something presented by a (female-sounding) name is a choice taken by many samples of popular and classic literature.

Finally, it is "simple" enough and "characteristic" enough that one can explain the unique solution in a way that the audience can understand and (here comes the viral part) explain the solution to someone else. Sudoku has a similar kind of simplicity and uniqueness about it.

It might be fruitful to discuss with people like Will Shortz what makes a good puzzle, and with people like Adrian Paenza (Leelavati prize winner) what would make it appropriately educational.

Gerhard "Two Cents Out Of Millions" Paseman, 2015.04.16

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In my mind there are two main reasons this became popular:

  • It is counter-intuitive.

  • It is slightly ambiguous.

The same could be said about other popular math-y puzzles and paradoxes, e.g., Monty Hall problem or "Tossing two coins one of them is heads what are the odds the other is heads too?"

The second point is quite important for the popularity, I think. This ambiguity, however, is somewhat at odds with class-room use.

A way to put this problem to good use in a class-room setting could be to discuss different types of "knowledge" such as "common knowledge".

To elaborate on the point on ambiguity: The New York Times gave the version below to remove such ambiguity (I put emphasis on the main change). In my opinion, it is a lot clearer this way but I am not sure it would have had that success in this form.

Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.

Cheryl thought a second and said, “I’m not going to tell you, but I’ll give you some clues.” She wrote down a list of 10 dates:

May 15, May 16, May 19

June 17, June 18

July 14, July 16

August 14, August 15, August 17

“My birthday is one of these,” she said.

Then Cheryl whispered in Albert’s ear the month — and only the month — of her birthday. To Bernard, she whispered the day, and only the day.

“Can you figure it out now?” she asked Albert.

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!

When is Cheryl’s birthday?

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    $\begingroup$ What's less "ambiguous" about the rephrasing? $\endgroup$ – Kyle Strand Apr 16 '15 at 20:17
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    $\begingroup$ Perhaps ambiguous is not the best word, but the sentence "Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively" is a lot harder to parse and to grasp in my opinion. I did not get on first reading that who got which information is in fact also given by the "respectively" construction. $\endgroup$ – quid♦ Apr 16 '15 at 20:37
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    $\begingroup$ The original version also used present tense ("At first I don't know but now I do") where the NYT paraphrase used past tense ("I didn't know originally, but now I do") and a more natural phrasing (to my ear). I think the awkwardness of the writing in the original contributed significantly to the sense (that some had) that this problem was "baffling". $\endgroup$ – mweiss Apr 16 '15 at 20:59
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    $\begingroup$ I got a different answer from this phrasing than the OP's. The question mark at the end of Albert's first line (in the OP) led me to believe that the answer was August 17 (because the "bernard doesn't know" appeared to be known initial conditions, rather than a deduction). $\endgroup$ – Brilliand Apr 16 '15 at 22:48
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    $\begingroup$ I too believe the problem just plays on a turn of phrase. Asking when would Bernard know? Makes the problem easy. There is perhaps something to be learned from the impact of literacy in a problem. $\endgroup$ – Karl Apr 18 '15 at 6:32

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