The Washington Post thinks the puzzle above is hideously difficult and says that it has become an “obsession”. The writer there certainly finds it sweat-inducing: “As the problem spread far and wide, it became a bit of a joke. Truthfully, we blame no one for choosing to laugh instead of crying out of frustration.”
Problem originates from Singapore, hence the dicey English. “The question is actually from a test given to students who are sophomores or juniors in high school”. Word has it that parents are “angry” over the question. I’m dubious. But somebody did mistakenly say that the puzzle was for younger kids. And you know how people like to check facts before commenting.
Anyway, the link to the Post story will direct you eventually to the answer. But don’t investigate until you’re sure of your effort. My guess is half of readers will solve it in under five minutes, and a quarter will take ten, and a quarter won’t get it and will peek. But the readers of this blog can hardly be considered the average “Internet.”
Now today’s lesson—and challenge. The order that you receive information matters in logic, hence in probability. Think of the Monty Hall problem. It was crucial for the solution that Albert said what he said when he said it. That’s the only hint I’ll give.
So our challenge is to make a superior probability puzzle out of this. I don’t want to post any of the ideas I have until later today (or tomorrow?) after you’ve had a chance to answer the puzzle. I’m betting we can come up with a good twist on this puzzle that makes it a true stumper—and not just something which fools reporters.
Update I see some wrong answers here and there. I’ll post a better-worded version of the solution here late this afternoon, after we’ve had more time to stew. Writing it took longer than thinking of it! (As usual.)
Both the problem and official solution are poorly worded, so I thought I’d clarify them here.
The setup would have been better written like this: “Cheryl wants to play cutesy and so tells Albert the month in which she was born but not the day, and she tells Bernard the day but not the month. She gives them 10 possible dates (month + day) and then lets the boys fight over her.”
If Cheryl told Bernard the day was 18 or 19, then because each of these days appears only once, Bernard would know the date. For instance, if she told Bernard the day was the 18th, Bernard would know the date was June 18.
Albert knows the month, and reasons that if the day were the 18th or 19th, Bernard would know the date, which means the month could only be May or June. But then Albert declares he knows Bernard doesn’t (or can’t) know the date. That must mean Albert knows the month is either July or August, since if it were May or June, it is then possible Bernard might know the date and Albert could not say with certainty Bernard is ignorant of the date.
Bernard hears Albert say that he, Albert, deduced Bernard could not possibility know the date based on the initial information provided by Cheryl. Bernard then reasons just as Albert did and deduces the month must be July or August.
Recall Bernard knows the day. If Bernard knew the day was the 14th, Bernard could only deduce, given Cheryl’s list, the month is July or August. So it can’t be the 14th because Bernard says he knows the date. If Bernard knew the day was the 15th or 17th, then he’d know the month was August, and if he knew the day was the 16th, he’d know the month was July. Whichever way, he deduces the date and tells Albert of his deduction.
And then, after hearing Bernard announce he knows the date, Albert, knowing the month, says he knows the date, too. The only way for this to be true is if the month were July, because if it were August, Albert would be left guessing the date is August 15th or August 17th.
Thus Cheryl’s birth date is July 16th. Don’t forget there are four perspectives here: Cheryl’s, Albert’s, Bernard’s, and yours. Each person has different information at different times, and the order matters.
We can turn this into probability by expanding the choices. If we do it cleverly, the probability Cheryl’s birth date is such-and-such a date would be different for the two boys. Seems to me Cheryl, the minx, would like it better that way.
I haven’t had the time to think of any solutions where both would have different probabilities, but it’s easy to imagine setups where they both have the same: i.e., simply by expanding the (different) days in July and August.