The Solution to that Annoying Logic Word Problem

A word problem that deals in logic was given to fifth-grade students in Singapore. It has swept the Internet, stumping people even more than the color of the dress.

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The New York Times translated it like this:

 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?

 

So what is the answer?

It has as much to do with seeing it through others’ eyes as it does logic.  You have to approach the problem based on what Albert and Bernard know, not based on what you know.

Cheryl tells Bernard the day and Albert the month. So Albert says that he doesn’t know (which is obvious because none of the possible months have just one possible day). He ALSO says that Bernard doesn’t know. So all you can logically conclude is that Albert knows Bernard doesn’t know because the month Cheryl told him has days in it that are also in other months. That rules out May and June because Albert wouldn’t KNOW that Bernard doesn’t know, for sure, since May and June both have days in them that are not in other months (18 and 19 don’t appear in any other month.

The fact that Albert doesn’t know will not help Bernard at all (because it’s obvious Albert wouldn’t know–none of the months have just one day as a possibility). However, the fact that Albert knows Bernard doesn’t know will help Bernard figure it out. We know Bernard didn’t know before, which means that it couldn’t have been the 18th or the 19th (these are the only days that are assigned to only one month). Bernard knows that Albert KNOWS Bernard doesn’t know. This means that Bernard can also rule out May and June. Since Bernard says he “knows now,” what it is, that means that the day isn’t 14, since that is the only number that appears in both July and August. If it was 14, Bernard still wouldn’t know. So it has to be 15, 16, or 17.

At this point, Albert knows Bernard has figured it out, so he too can rule out 14 as a possibility. Albert knows the month already, and he says that after he knows Bernard knows, that he now knows as well. The only way he could have figured that out is if there was only one other option than 14, which means the month was July (the only month left with only 2 options). So it has to be July 16.

And there it is!

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8 thoughts on “The Solution to that Annoying Logic Word Problem

    1. Thanks for your question. It can’t be the 17th because Albert says that after he knows Bernard knows, that he now knows as well. The only way he could have figured that out is if there was only one other option than 14, which means the month was July (the only month left with only 2 options). So it has to be July 16.

      If it was any date in August, since there are two remaining options after you rule out 14, then Albert couldn’t have figured it out.

      Does this help?

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  1. The July 16 solution to the puzzle is correct if one assumes that there is a possibility that Cheryl would tell Bernard either 18 or 19. The August 17 answer is correct if the reader assumes that Cheryl would not say 18 or 19 to Bernard.

    The rationale for the August 17-related assumption is based on the notion that Cheryl would not design the puzzle for Albert and Bernard and then just give the solution away. Is this not a reasonable assumption? If it is, then August 17 is a valid answer. If it is not reasonable, then why not?

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    1. But the rationale for the August 17-related assumption implies something about Cheryl’s psychology, while the July 16 solution is based solely on the facts of what we *know* that Bernard and Albert *know*. Not what we think Cheryl might or might not do. That’s why it’s not a reasonable assumption, because we don’t know anything about Cheryl’s psyche.

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  2. Now I have to ask this… How can they rule out the whole month of May just by knowing it is not May 19th? Both numbers 15 and 16 show up more than once, and if you then rule out 18th and 19th, and then also June 14th because that’s the last one in June, and then apply the whole logic with there only being one of the numbers left for them to figure it out, that would leave August 17th, and not June 16th.

    I am sorry, this will probably never make sense to me. I just don’t understand how knowing that ONE of the THREE options of May is wrong, will help a person figure out that the whole Month is wrong without actually knowing the month.

    The guy who knows the month will know it isn’t May 16th, but to the guy who knows only the number, May 16th should still seem like a valid option…

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    1. So in the beginning, Cheryl tells Albert the month, and Bernard the day. Albert says he does not know when Cheryl’s birthday is (of course he doesn’t know, since none of the months have just one day listed), but he also *knows* that Bernard does not know. How could he know that Bernard does not know? Because Cheryl’s birthday is in a month where all of the possible dates also appear in other months. Therefore, it can’t be in May, since if Cheryl had told Albert “May”, then it is possible that she told Bernard “19” — and Bernard would know, since 19 doesn’t appear anywhere else. Similarly, we know Cheryl didn’t tell Albert “June,” because if she had, then it’s possible that she could have told Bernard “18”, and Bernard would know, since 18 doesn’t appear anywhere else. So if Albert *knows* Bernard doesn’t know, Cheryl must have told Albert a month that has days that appear in other months. July or August.

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