William M. Briggs

Statistician to the Stars!

The Logic Puzzle That Is “Stumping The Internet.” Update: Now With Solution

logic

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.)

Solution

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.

Probability Challenge

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.

52 Comments

  1. I know the answer – am I s’posed to post it – not post it or suggest a NEW puzzle based on this puzzle (to post or not post)

  2. Briggs

    April 15, 2015 at 9:35 am

    John B(),

    Not yet. Wait until late in the day. Answer questions if you like, comment on the order of information, try your hand at new problems which make probability out of this, but don’t reveal the answer now.

  3. I saw this floating around yesterday and was wondering if you would address it. It kind of reminded me of the unexpected hanging paradox, too.

    I do not like Albert’s first statement, though. Somehow he knows that Bernard doesn’t know the answer. That’s implied from Bernard’s silence, but it struck me as odd (and I got hung up on it), since everyone else was very free with saying “I don’t know”.

    I would prefer it if the information was given as:

    Bernard: “I don’t know”
    Albert: “I don’t know as well”
    Bernard: “Now I know when the birthday is”
    Albert: “I also now know”

    As for improvements, I will cogitate.

  4. Rather than the Monty Hall problem, isn’t this more like the three logicians who walk in the bar and the waitress asks: “Does every one want a beer?” … !st logician : “I don’t know” … 2nd : “I don’t know” … 3rd : “Why yes!”

  5. John B(),

    That’s hilarious, I hadn’t heard that one. It reminds me of the three statisticians out hunting. They come upon a deer, and each try to kill it. The first misses to the left, the second misses to the right. The third exclaims “We hit it!”

  6. James

    See? You’ve made it exactly like the 3 logicians in a bar.

    Albert’s answer is precisely correct, because he knows the month, regardless of the actual day (number), Bernard can’t answer.

  7. James

    Now I have to cogitate your three “I don’t knows”.

    The logician joke is in a list of jokes that go by most people

  8. James

    YES!

    I like your approach! It might even make it just a bit more difficult?

    Then you when you post yours, you can post the logicians in a bar as an aid!

  9. Side note: for some reason, my brain has a hard time associating the names with their knowledge (month or day). I had to rename them “Moe” and “Dave” to help myself out with that. Hopefully it’s not a sign of low IQ… or a tumor. Does anyone else have this issue?

  10. Briggs

    April 15, 2015 at 10:39 am

    James,

    No. It’s time for you to step inside a bar with two other logicians.

  11. Briggs,

    I’m fine with being told I have to step into bar!

  12. James

    It’s NAUT a TOOMa (How do you spell a thick Austrian accent)

    I think a good variation to the logicians in a bar (freak out a lot a people)

    Third logician: Those two want a beer, I’ll have a whiskey neat!

  13. James

    I just noticed: ‘That’s implied from Bernard’s silence’

    Bernard’s silence IS NOT NECESSARY for Albert to know.
    Albert’s statement that Bernard CAN’T KNOW is then important to Bernard without Bernard announcing.

    Now I’ve got a headache … maybe it’s a tumor.

  14. I really miss Car Talk. They used to do great puzzles on that show. I know one of them died recently, though I don’t recall the date…

    JMJ

  15. James

    *possible spoilers*

    I don’t know if your new dialogue is helpful. Albert’s foreknowledge of what Bernard knew without being told was the key for me and for Bernard. The way yours is worded I can’t reason my way to the answer.
    Maybe if, in your version, Albert’s first response was changed to “I know you don’t!”

  16. Seth said: The way yours is worded I can’t reason my way to the answer.

    I agree that it makes it tougher – it can be reasoned, though

    Try it again – I know the previous reasoning gets in the way but try to unlearn it

  17. Seth,

    The thing I was proposing wasn’t supposed to be a replacement that provided the same information. I can see how it comes off like that, so my bad on the ambiguity!

    The replacement phrasing leads to a different solution of the problem, and the last sentence “I also now know” is probably not needed.

  18. Seth James – Apologies to everybody.
    (I knew I should’ve posted my first answer – or at least written it down)

    James repositing the problem CHANGES the answer! (But, it’s still avalid question in which there is only one solution – may be harder – may be not)

    But it is different and I just realized that now and I’m not sure now if I had the right answer in the first place

  19. John B(),

    Also my fault for not being clear. I think the way I reposited it makes it much easier. I’m worried/preparing for finding out that the new suggestion is either too easy, or that it’s impossible and I’m not all that bright, or that it’s possible but I phrased it poorly relative to my intent (leading back to the TOOMA).

    I think it’s just too easy, though. The hardness probably comes from residual thoughts from the original formulation. Below is a quick test to see if mouse-over spans will work for embedding spoilers.

    This is the text I want to have a mous-over

  20. Thought I had it in under 5. Then struggled. Then started to tell myself “see, I told you you were stupid.” Then went through it methodically and got it. :\

  21. Okay David Appell puts me in moderation

    I’ll leave it to Briggs to see if this is too big of a spoiler

    Solution to James’ reposited problem:

    Bernard: “I don’t know” (Albert knows it isn’t the 18th or 19th)
    Albert: “I don’t know as well” (Bernard now knows it isn’t June)
    Bernard: “Now I know when the birthday is” (Albert now knows it’s August 17th)
    Albert: “I also now know” (But this answer is different from the original)

    Not easier or harder – different

  22. Briggs

    I intentionally put myself in moderation, do you want to see if you’d want to post it assuming it isn’t a big spoiler?

  23. OK, I think I’ve got it worked out but there is, it seems to me, an ambiguity in the way the question is worded.
    Can I assume the wording means
    “Cheryl then tells Albert the month and Bernard the date”…
    if that’s correct I have it worked out.

  24. Bob,

    That is correct.

  25. I thought the answer rather obvious but I still went to the WP page to see what they had to say which wasn’t much (yes, the answer is sorta there). What WAS interesting was the link there to: How a middle school math teacher became Jordan Spieth’s caddie. While it may be usual for pros to avoid amateur caddies, why was it necessary to mention this caddie was a (former, I guess) middle school math teacher? I might want to hire a math teacher as, say, a CPA for creative accounting innovations. Was the math teacher hired for his math skills? He had creative mathematical insights concerning score reporting? Hmmm ….

  26. The more I look at the question the more I like it exactly as it is. The bad wording in the question just makes it better.

    Can we teach the right way to analyze the question though?

    Monty Hall is definitely involved in the question. Remembering that both Albert and Bernard ARE Monty is what I keep stumbling over. Trying to explain that Albert and Bernard are Monty to someone who doesn’t give a rat’s **** about Monty is really hard.

    The simple answer, taught to everyone attending a silver bullet seminar on business:

    Albert says out loud “She said ******* to me”

    Bernard says out loud “She said ****** to me”

    Suddenly they know the answer.

    (Silver bullet seminars often try to get the folks attending to just talk to one another! If you just tell your neighbor your goal, they will tell you theirs, and you often can all get together to get the best answer!)

  27. I have to admit that I wasn’t very good at the more difficult Diff E Q type logic where you had say five people in six locations doing seven different things and you had to tie people to location to thing.

    Sort of like the game of Clue except multiplied by five.

    The two examples already mentioned and the variations on the logicians in a bar are better because they’re so straightforward.

    Glad to see my self-moderation worked

  28. James,
    thank you. I had thought the answer was suppose to be the same with your adjusted dialogue and the original. Now that I know they’re different I reasoned the correct answer of your dialogue, as well. 🙂

  29. Applied logic:

    https://www.youtube.com/watch?v=jKF1u-TOXuE

    Leave Sheldon at home

  30. One of my fb commenters answered “on the day she was born”

  31. The answer is clear once you accept Albert’s last statement . I am having a problem with that though: i understand how Bernard gets the date but not how Albert can then choose the correct one from the options remaining to him.

  32. Albert can know because there is only one month/day combination which would cause Bernard to know.

    Note: Albert gets the month from Cheryl and Bernard gets the date from Cheryl (notice the “respectively”, it would be clearer and less wordy for the third sentence to read “Cheryl tells Albert the month and Bernard the day of her birthday”) , which might make it easier to understand their reasoning.

  33. Looking at the problem, reversing who has which information would lead to concluding that her birthday was June 17, which might cause the difficulty people have with it.

  34. Got it,, thanks Max

  35. It seems to me that Albert’s original declaration that he knows Bernard can’t know was incorrect since Bernard was able to say he did now know which is the same as saying Albert was quite mistaken. Given the sequence of information discovery, it’s quite easy to solve this but Albert seems to have jumped to a an unwarranted conclusion.

  36. DAV,

    Albert’s first statement was actually correct, but in a very specific way. Remember that Albert knows that the month is July. Therefore, Albert knows that Bernard has heard either 14 or 16. Since 14 and 16 are both repeated, Albert therefore knows that Bernand (who hasn’t yet heard anything except a 14 or a 16) doesn’t know the answer at that point in time.

    Briggs,

    I’ve been trying to also think of a probability application, where maybe Cheryl gives each of them probabilities of months and days each, and they have to somehow find a joint probability that gives them the most likely day. So far, no luck on making it interesting or challenging.

  37. Wrong!
    Your solution is correct up to the point where Bernard knows the date.
    But Albert does not “know the date too”. He may say so but he is lying. He only knows that it is limited to July 16 or Aug 15 or Aug 17.
    i.e. your solution assumes Albert is telling the truth when he is only guessing.

  38. James

    Albert MAY know it’s July and therefore the 14th or 16th but YOU don’t, so you have to realize that if it’s May or June, Albert can’t know that Bernard can’t know.

    May 19 or June 18 are knowable.

  39. Bill S,

    Albert has been told the month, so he either knows it’s July or August. If he was told July, he knows the date and the would say what he said. If he was told August, he couldn’t have said he knew the date, because there are two options.

    I went about this in a very explicit way, since I’ve been doing a lot of constraint programming. For Albert, we just look at each possible value of the month and determine if he could make the statement he could for that month. Doing so gives us the set of possible months.

    Then, assuming Bernard figured that out, we pick dates that allow Bernard to make that statement, given the set of dates Bernard has as feasible. Then, we repeat that one more time with Albert, once again looking at values of the months and saying “if Albert was told this month, would he have been able to make that statement”.

    It’s the same procedure for doing things like solving Sudoku with a computer. For people who are coding inclined (with Python), look at http://www.checkio.org for a bunch of programming puzzles that have problems like this! Lots and lots of nerdy fun. 🙂

  40. John B(),

    Yes, that rule has to be applied for each month to get the set of months that Albert could have been told in order to make the statement.

  41. James.
    Thanks.

  42. What’s missing is how did Albert deduce the answer given what he knew after Bernard spoke?

    Yes, I can deduce based on all of the statements, but it doesn’t follow how Albert “knew” the answer.

  43. Grr. I got it wrong because I didn’t think it through all the way and leaped to the wrong conclusion. When I was in high school many moons ago, I loved this kind of thing and used to get out library books full of questions you had to figure out by logic. I got very good at it but that was long ago…. too sure of myself to do the work, I guess! I remember a wonderful book based on Alice in Wonderland, I wonder whether it’s still available?

  44. James,

    I don’t think so. He only had to follow the same reasoning that Bernard did. In fact, to solve it, you did too. May and June are eliminated but after that JUL 16th would have still allowed Bernard to deduce the date just as he could have for the 18th and 19th which is how we finally conclude Bernard did it. Didn’t Albert see this possibility when declaring certainty that Bernard could not know?

    I’m fine with being told I have to step into bar!

    Man walks into a bar and says, “Ouch!”.

    Jim Fedako,

    Yes, I can deduce based on all of the statements, but it doesn’t follow how Albert “knew” the answer.

    So, then, how did you deduce the answer? Why couldn’t Albert do the same?

  45. DAV,

    I seem to have caught Brigg’s typo bug 🙂

    I think we’re haggling over exact states of information and the amount of foresight they have or are allowed to have. If you are very strict about the timing, and we peek into Albert’s mind, he knows for certain it is July. Therefore, Albert knows for certain that Bernard cannot know (prior to Albert making his statement!).

    Bernard confirms this ordering by agreeing that yes, he did not know the date before Albert said anything. Then Bernard goes on to say that, due to Albert, now he does know. Albert didn’t say “Bernard can now deduce it such that I can guess it later”. He just said “Bernard’s knowledge of the number is not enough information alone for him to know the birth date”. Albert might have been able to think ahead and modify that statement, but that would have been a decision for the writer of the problem.

  46. DAV,

    I assumed Albert was correct and based my answer on that assumption. But how did Albert “know” he was correct? That is the question.

  47. I quite like this version (which might or might not lead to a different answer — you will have to figure that out). Possibly a little easier than the other formulations:

    Albert: I don’t know the date, but Bernard might know it
    Bernard: I didn’t know the date before Albert spoke, but I do now
    Albert: I also now know the date

  48. James

    Cool – Sudoku

    After I wrote what I did, I DID consider that maybe logically you went through each month, but I went for the straightforward way to answer

  49. BB

    Not bad

    I was considering a series of statements where one statement is a lie.
    Because the liar eventually wants the answer, the lie is found out.

    Requires 5 or 6 statements

  50. Milton Hathaway

    April 16, 2015 at 7:50 pm

    What I find most interesting about the problem (and was the biggest mental hurdle for me trying to solve it) is Albert’s assertion “Bernard does not know”.

    This assertion is true in the split second before Albert makes it, but immediately turns false in the split second after he makes it.

    It’s almost as if some sort of spooky quantum logic is at play, only on a macroscopic scale.

  51. Milton

    What Albert is saying is that the month is not May nor June where the date could be 18 or 19 and Bernard could then figure out the birthday

  52. I got it! Maybe now I can get a job at the Wapo.

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