Since a math olympiad question from Singapore went viral a few days ago, there have been lots of explanations cropping up all over the net. I thought I’d chime in with a restatement of the scenario in which the participants, Albert and Bernard, think out loud as they arrive at the answer. Here goes.
Albert and Bernard ask their friend Cheryl when her birthday is. An onlooker observes the interaction.
Cheryl says, “I’m not going to tell you exactly when my birthday is, but I’ll give you all a clue. It’s one of these ten days:
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
Now let’s see if you can narrow it down.”
Albert and Bernard realize they have no way of deciding between the ten options, so they ask for another clue. Cheryl says, “OK, I’m going to whisper the month in Albert’s ear and then I’m going to whisper the day in Bernard’s ear. I’m going to ask Albert and Bernard to keep what I tell them secret and not share the information with each other, but maybe they’ll still be able to figure it out?” Cheryl goes ahead and whispers the day in Albert’s ear and the month in Bernard’s ear.
Albert says: “I still can’t tell when Cheryl’s birthday is. She did tell me the month, but within that month — in fact, within all the months provided — there’s more than one possible day it could be. The information Cheryl gave me does allow me to draw one conclusion though: I’m absolutely certain that Bernard doesn’t know when the birthday is either!”
Bernard: “So, Albert, you’re using your knowledge of the month to conclude that I couldn’t possibly know the answer? How can you be so sure of what I know and don’t know?”
Albert: “Well, Bernard, you were only told the day, not the month. The only way you could know the answer from the day alone is if Cheryl had told you 18 or 19. If she had told you 18 you’d know it’s June 18 because that’s the only one of the ten possible birthdays where the day is 18. Similarly if she had told you 19 you’d know it’s May 19. What I’m saying is that I know Cheryl didn’t tell you 18 or 19, therefore I know you have no way of determining the birthday. For any other number besides 18 and 19, you still have several options to choose from, with no way of narrowing it down further. For example if Cheryl had told you 14, you wouldn’t be able to decide whether it’s July 14 or August 14.”
Bernard: “But how do you know Cheryl didn’t tell me 18 or 19? Aha, the only way you could know Cheryl didn’t tell me 18 or 19 is if she told you a month where there are no possibilities on those days. So she must have told you a month other than May or June, which means her birthday must be in July or August. And with that smaller set of options, I now know exactly when it is!”
Albert: “Really, you’ve figured it out? Well then, I can gather that the day must not be 14, because if Cheryl had told you 14 you’d still have no way to decide between July 14 and August 14. Bernard, you yourself ruled out May and June based on what I said before, and now I’m ruling out July 14 and August 14, so the only remaining possibilities are July 16, August 15, or August 17. And considering those three options together with what Cheryl told me, I know the answer too!”
Onlooker: “Albert and Bernard, I’ve been listening to you talk through this whole thing, and now I also know the answer. If Albert narrowed the possibilities down to July 16, August 16, or August 17 and then declared that he knew the answer, I know Cheryl must have told him the month July. If Cheryl had told him August, then Albert would still not be able to decide between August 16 and August 17. So, now we all know Cheryl’s birthday is July 16!”