Cheryl’s Birthday Problem – It depends on your point of view


Hi, everyone! So there’s a maths problem that’s gone viral this week, and I’ve got a new blackboard! So I thought– you know… So, the problem–you may have seen it already, it’s called Cheryl’s Birthday Problem, but if you haven’t seen it, well, you know how it works: I’m going to give you the problem, you’ll have time to try and solve it for yourself, if you want, and then we’ll talk about the solutions. So the problem is this: So you’ve got Albert, Bernard, and Cheryl– Or A, B and C if you prefer. Now Cheryl–she sounds like a bit of a nightmare to me– Cheryl says “My birthday is one of these ten dates.” So what have we got here? We’ve got May 15th, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17. So Cheryl says “My birthday is one of these ten dates.” She gives Albert the month of her birthday, she gives Bernard the number of her birthday. And then there’s a following conversation between Albert and Bernard. Albert says, “I don’t know when the birthday is, but I know Bernard doesn’t know, too.” Bernard says, “At first, I didn’t know when the birthday is, but now I know.” And then Albert replies: “Then I know the birthday, too.” When is Cheryl’s birthday? You can use this information to work out when is Cheryl’s birthday. So, I am going to give you time to pause the video, or have a go for yourself, if you haven’t already. And then we’ll start talking about the solution after this flash. So this problem has caused a lot of controversy on the internet this week. There’s been a lot of debate about what the correct solution should be. Now, I can tell you what the expected solution was. The expected solution was July 16, but that might not be the answer you’ve got, and I want to talk about alternative solutions in a minute. But for now, let’s run through what the expected solution was. So, from statement one: Albert knows Bernard doesn’t know. So Albert is holding the month. He can’t have May or June. May and June have unique dates in them. The 18 and 19 are unique numbers. So Albert cannot be holding May or June, or he wouldn’t be able to say that statement. Statement two: Bernard now knows. Bernard has gone through that same logic we’ve just said. He knows the month must be July or August. He’s got the date–he’s got the number. It can’t be 14, because there’s two choices for 14. So it can’t be 14. It must be one of these three: the 15, the 16 or the 17. Statement three: Albert knows the birthday too. If Albert knows the birthday too– he’s gone through the same logic. He’s worked it down to these three answers: July 16th, August 15th, August 17th. He can’t be holding August, because he would have two choices, which doesn’t help him at all! So he must be holding July which means the answer is July 16th. Now, that’s the answer I got when I first saw this posted– it was by Alex Bellos on the Guardian blog. And underneath, there was a lot of comments, there was thousands of comments of people debating the answer. Now, the majority did get July 16th, but it wasn’t an overwhelming consensus. So let’s have a look at some of these alternative answers. It turns out, it might depend on your point of view! So the third most popular answer I saw was August 17. And it all depends on how you choose to interpret This first statement: “I know Bernard doesn’t know.” Now, if you choose to interpret that as fact, then you get this different answer. And let me just run through the process of that for you. So Albert says, “I know Bernard doesn’t know.” So, maybe he’s realized that Bernard hasn’t just jumped in with the answer. So he knows It’s not a unique number. So he can now eliminate May 19 and June 18. Now Bernard says: “Oh! I know the answer!” Well, Bernard has gone through the same thinking, he’s eliminated June 18, May 19 as well. He’s also noticed that Albert hasn’t jumped in with the answer. If Albert was holding June, then June 17 would be unique. So Albert can’t be holding June–he would know the answer. So we can now eliminate June 17 as well. Now, Bernard knows the answer. His number must be unique in the remaining solutions. What have we got? We’ve got two 14’s, so he wouldn’t know. We’ve got two 15’s, so he wouldn’t know. We’ve got two 16’s, and we’ve got two 17’s, But, we’ve just eliminated June 17, so it must be August 17, because Bernard knows that it’s August 17. Meanwhile. Albert–ah!– annoyed that Bernard has beaten him to it, Albert goes through the same process, and gets the same answer, August 17. Now, that is valid, and that all does work but it does depend on your interpretation of this first statement: “I know Bernard doesn’t know.” So what we’ve got here is an example of the difference between a statement, and knowledge of the statement. So the statement here is, “Bernard doesn’t know.” Now, one argument says, we assume this is true, this statement– we go through the logical process, so something, something, something– and–we get the answer, and it was August 17 in that case. The alternative argument is we assume Albert’s knowledge of the statement is true, we go through the process, something, something, something– and we get the alternative solution, which was July 16th. Now, the people who wrote this question originally– it was actually a Maths Olympiad question from Singapore– they have now rejected this alternative solution. They’ve said that this first statement is a statement of knowledge, not a statement of fact. But I can see why people have misinterpreted that first statement. Especially people who aren’t familiar with maths questions like this. I can completely sympathize why they would read that as a statement of fact. Now, the second-most popular solution I saw, I actually have less sympathy for this. The second-most popular solution I saw was June 17th. And let me run through the argument for that. So, from the first statement, “I know Bernard doesn’t know”– so like we’ve done before, we now eliminate the unique dates. So that’s the May 19th and the June 18th. Now Bernard knows the answer. And what people were arguing was Bernard has 17–he knows the answer, so he must have June 17, because it’s all on its own in the month of June. It’s unique in the month of June. It must be June 17–that’s how he knows! Which is–wrong. And I think what they’ve done there is they’ve not quite put themselves in the position of Albert and Bernard. They’re seeing themselves as themselves, like these are statements to you, the reader, and they haven’t put themselves in the position of Albert and Bernard, or the knowledge of Albert and Bernard. And I can see why they’ve done that– so that’s how many books are written– they are written to you, the reader. That’s how many of these maths questions are written– statements are written as information to you, the reader but I think that’s where that falls down, not looking at things from the point of view of Albert and Bernard. So Cheryl’s Birthday Problem is actually an example of something called Public Announcements Logic. It’s where knowledge, or truth, can be determined with new information. And logic really goes to the heart of mathematics, and it is the foundation on which we base all other mathematical results. Another example might be the old mathematician’s joke which you might have heard of before: “Three Logicians Walk Into A Bar.” So three logicians walk into a bar, The barman says, “Does everyone want a drink?” The first logician says, “I don’t know.” The second logician says, “I don’t know.” The third logician says, “Yes.” And I imagine that’s how Albert, Bernard and Cheryl like to spend their evenings. And that’s all from me, so if you have been, thanks for watching.

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Comments

  1. hrm… i initially thought it was july 16th. now i think it is august 17th.
    Albert does not specify why he knows that Bernard doesn't know.
    This causes Bernard to not know how Albert knows that Bernard Doesn't know. (this is important because Albert could know that the days are not 18 and 19 for another reason unknown to Bernard)
    So all Bernard can know is that Albert knows it is not june 18th or may 19th. (18 and 19 are the only numbers that could cause Bernard to know)
    Albert also mentions he doesn't know the answer, which he would know if it was jun 17.
    Therefore Bernard knows it is not june 17th.
    In the second statement Bernard knows the answer and the only unique solution is august 17th.

    The overall problem with july 16th is the fact that Bernard doesn't have Alberts knowledge. So I don't think that it depends on how you interpret the first statement (as fact or as knowledge) because Bernard never learns what Albert doesn't know.

  2. It doesn't depend on your point of view or interpretation, Jul 16 is the only correct solution. There's a straight-up, distinct error in logic or reading comprehension (usually during a bit which is difficult to grasp) whenever someone argues that one of the other dates is the solution. Don't give the 'alternative solutions' credit!

  3. I love maths and I have seen many mathematician who complicate simple shit. But I still love it videos and I always find myself fascinated after watching and understanding the beauty of maths.

  4. Here's a simple explanation. You have to assume that Bernard keeps it a secret if he knows or not. You cannot assume "Oh well Cheryl wouldn't have just given him the date".

    Bernard keeps a poker face and is silent.
    If Albert has JULY and AUGUST, it is the only way he can say for certain that Bernard doesn't know. If Albert had May or June, while Bernard is keeping silent, he cannot say with confidence that Bernard doesn't know.

  5. isn't the elimination of all dates in may and June based on the 1st statement being "fact".

    I ask this because later in the video we are told not to assume the 1st statement is fact when arriving at the alternate answer.

  6. To those who think that you're smart because you can solve the problem: This is one of the Math examination questions for primary five pupils in Singapore.

  7. The alternative does not make sense because of the world 'too'. I could replace that word with 'as well'. If the word 'too' wasn't there then you have an argument for the alternative way of reading it. The word 'too' implies what Albert is saying, which is that he does know and neither does Albert.

  8. This also depends on what happens eventually.
    If at the end of this story, both Albert and Bernard got the birthday correct, then certainly the answer is Jul 16. But what if at the end of the story, it states that Albert got the birthday wrong, and Bernard got it correct? In the sense that Albert thinks that he got the birthday, but was instead fooled by Bernard into thinking that way.

    Here's the interesting way to look at it:

    Suppose that in the 2nd statement, Bernard actually lies about his knowledge.
    He says "At first I didn't know, but now I know." but in actual fact he still doesn't know because he was told the day 14. However, Bernard knows that if he says "I still don't know," then Albert would certainly know since Albert would have known that he was told 14 and also the fact that Albert knows the month. Hence, Albert decides to lie about it and, he says "Now I know," to prevent giving that information to Albert and then he waits for Albert's reaction.

    Albert then thinks he knows and concludes that it is Jul 16. Based on Albert's confidence, Bernard then concludes that the month is July, and figures out that Cheryl's birthday is actually Jul 14. (on the alternative case where Albert was told the month of August, Albert would say he still doesn't know, which would still lead to Bernard being able to conclude Aug 14, in both cases, Bernard's lie will enable him to figure out the birthday, assuming Albert tells the truth).

    Albert then gets the birthday wrong and Bernard gets the girl.

  9. I actually tried it out with two different improvised 'notations' and got July 16th and August 17th. Just fucked about on notepad, they're probably rubbish, but maybe I'll formalise them and they'll have some unexpected application and I'll have a Fields medal on my wall in a few years

  10. Where do you get your blackboards from?
    I live in the UK and can't find anything.
    I have a whiteboard but I am still wanting to get a blackboard for my workshop

  11. Cheryl Was not born on any of them dates. She was born on 30th june .

    https://en.m.wikipedia.org/wiki/Cheryl_(entertainer)

  12. hm, I would argue that the statement, "your answer is wrong because you didn't read it as we intended it" is a really poor statement and one should instead acknowledge that if one have wanted a question with only one correct answer one should have written a question with only one interpretation. And also saying people probably read this wrong because they are not used to math questions are also wrong. Math don't own questions. Questions are just questions.

    So in this case, if math wanted just a math answer, math should have asked a question with just a math answer. If it can't handle a non mathematical solution. It's not a question.

  13. Maybe it is June 17. Reason: Albert is an ass, realized that it is not June 18 (cause Bernard does not say something), realizes that it must be June 17 (cause Cheryl told him it is June), and now he is planing to be the only one to show up with a nice present and take her out for a dinner! 😀

    This would be the social solution to the problem!

  14. July 16 is the answer. Albert has July so it could be 14 or 16. Either of these are in 2 months. He hinted that fact to Bernard. It can't be 18 or 19 so May and June are eliminated. Since Bernard has 16, it is not August. He knows it can only be July now. So it is July 16

  15. It could be stated unambiguously by rephrasing the second half of Albert's first statement as "… I know that Bernard CAN NOT know, either."

  16. Maybe my brain is a little fried after going through the different solutions, but I want to clear something up…..
    1. Albert: Statement true ( equals fact?)
    Ergo August 17
    2. Albert : knowledge of statement true
    Ergo Juli 16

    Then you say the '' alternative '' solution has been rejected.

    2. is rejected. 1. is true

    Later you say that Albert's statement is a statement of knowledge not statement of fact.

    2. true. 1. false

    Which solution has been rejected now? Maybe there was a mixup?
    And don't get me started regarding knowledge of statement and fact for the interpretation of the statement. That's kind of hard to define precisely , I think. But maybe it's just a language barrier…

    Ultimately I just want to know the intended solution.

  17. I got the July answer first, but then quickly saw the August one seemed to fit and had to think hard to see why it didn't.

    James, I'm a bit confused about the knowledge versus facts explanation for this. Sorry, I'm not strong on formal logic, and the way I thought it through I figured the fundamental reason that August doesn't work is that I couldn't see how Albert could come to the "I know" statement at the end if he knows the month is August.

    Statement 1: The fact Albert knows that Bernard doesn't know the answer means it must be a month that doesn't have a unique number. So from that statement we can eliminate May and June. Because if Bernard held a 18 or 19 he would know the month as well as they each appear in only one month
    Statement 2: Bernard now knows – thanks to Albert's information. Therefore Bernard's number must be unique to one of the two remaining months. That leaves either July 16, August 15 or August 17. If he held any of these numbers, he would now know as Albert's statement eliminates May and June.
    Statement 3: Albert now knows. This is clearly not true if he holds August, as 15 and 17 are still viable answers from the information given. Therefore it must be July.

    I don't know why I landed at August 17 but not 15. Hmmm. I'm doubting my logic

  18. There is another option.
    Cheryl does not want her birthday known, if she did she would have told them when asked and not have made up this problem.
    And as such would have either given them false information or have given information that would lead to a null answer situation.

    Apply the logic of that.

    Once a rule is set it must always apply.
    Eliminate the singular possibilities.
    By date.
    May 19 June 18 are singular.
    June 17 becomes singular.
    August 17 becomes singular.
    Remaining dates/months are not singular.
    Answer null.

    By month because of date.
    May 19 removes May.
    June 18 removes June.
    This makes July 16, August 15-17 singular and removes both of those months as well.
    All dates/months eliminated.
    Answer null.

    Because Cheryl does not want her birthday known.

  19. Was it painful writing all of those dates backwards? Is that because they were written in that format in the original problem, and you didn't want to change them to the way dates are written in the UK for the halibut?

  20. This makes no sense. Why couldn't it be any of the dates? Albert could know it is May, Bernard could know it is the 15th! Who says it can't be because there is also Aug 15? WHY? Where does this idea of it being unique come into it?

  21. It is a rubbish mathematical problem which is written in such a way as to allow more than one right answer, depending on linguistic interpretation.

  22. I still don't understand how you came up with July 16. If Albert knew that Bernard does not know then why does he eliminate the whole month instead if just the day?

  23. It is way too early in the morning right now. For some reason I tried to wake myself up by watching youtube videos. It doesn't work, not I'm just confused.

  24. Wait so why wouldn't they just know Cheryl's birthday? I'm assuming they know her. I know my friends' birthdays…

  25. Wrong. Nothing to do with fact or knowledge. The argument given for August 17 also works for August 15. And it is consistent and possible with statements 1 and 2. Nothing to do with fact vs. knowledge. The problem is that then Bernard has no way of deciding, so statement 3 is impossible. That's how we get to know that the month is not august, so the date is June 16

    Nothing to do with point of view, fact, knowledge, etc. Just basic logic and a matter of double checking.

  26. It’s a logic problem on a math test. If one is going to ignore statements made by the characters, then one should just ignore the whole problem and move on to the next one. Let’s say when Albert says “I know that Bernard doesn’t know” he hasn't determined this with his unique knowledge, but rather, he means Bernard would blurt out the answer as soon as he knows it. But then, Bernard DOESN'T blurt out the answer when he knows it. So, Albert would have to think, oops, I got that wrong, I better rethink this.

  27. great video, I only got a little confused at the part beginning at 05:40. Is he confusing the alternative answer and the "expected answer" or am I missing something?

  28. If someone were to actually witness this scenario unfolding in real life, it would be unwise for the observer to declare that they know the answer. People use language informally in real life, when Albert says he "knows" something, that may mean that he highly suspects something. If this were actually to take place in the real world, it is not at all unreasonable for this scenario to take place and for August 17 to be Cheryl's actual birthday.

    While I think that July 16 is a slightly better answer than August 17 (though I think this conclusion is not quite as simple as most people make it out to be!), I also recognize that it is only a better answer because I'm "meta-gaming" the question. I'm taking factors outside of the question into account – namely, that it is a logic problem that was contrived and placed on a mathematics exam in the first place! If we were inside the universe where it was taking place, we would have to take into account that people don't always say precisely what they mean and that non-verbal cues exist.

    As such, if I were to give students this question on an exam, I would accept both of these options as correct, provided that the students could give a reasonable explanation of the situation.

    Ha! This kind of reminds me of mathematical logic and independence of sentences. Are we outside of the system or inside?

  29. theres no information that indicates any of those date it doesn't make snese how your supposed to answer the question without a reference to the date it could be. now i don't know if you missed something out but it seems like there is something missing in order to the solve the problem

  30. when did you mention anything about not being able to use may because its unique not once before you started through that part did you mention anything about unique dates or number

  31. This was actually very educational. I got August 17 but now I worked through it to get July 16th and I can clearly see the difference between statements of fact and statements of knowledge. I was thinking as a computer programmer not as Albert and Bernard. In some ways… you've taught me empathy lol 😉

  32. Typical that they would pick names of old, white men.

    They might as well have changed Albert and Bernard to Adolf and Donald.

  33. "I know Bernard doesn't know" can be interpreted as "Based exclusively on my date information then I can prove Bernard doesn't know" or "Based on something that could be anything then I know Bernard doesn't know" (maybe he read his Diary, or know's his tells). Interesting how both give a unique answer. I have to say, assuming either of them is more than the problem tells you.

  34. I interpreted it the wrong way. I did not know that the Bernard and Albert were being literal; that they actually know whether they know the answer or not.

  35. Realistically theres no way for albert to know the answer but Bernard can which is the problem. After Albert states he knows Bernard doesnt know it confirms that it cant be in may or june because if he saw may or june on his card or whatever he wouldnt be sure that Bernard got the number 19 or 18. After this info bernard realizes what he said and knew the answer because the number he got was also in may or june so he confirmed it to be either july 16, aug 15, and aug 17 By saying he knows the answer. Heres the problem how could albert possibly know the answer after his statement? He could have had the number 15 and say the same thing? Its like they left information out in this problem. Albert cant know after what bernard says unless bernard gave more info.

  36. I got August 17. I don’t see why you’d erase the whole month of May and June because they each have one unique number

  37. This guy is wrong in his logic, it's July 16th, Albert does not just cross out May 19 and 18 but the whole months take it from there

  38. This is wrong, this hinges on the statement "bernard doesnt know" when in fact the statement is "I know bernard doesn't know". Since Albert has the months, consider if he had been told the month was May: knowing this he wouldn't be able to say "I know bernard doesn't know" because its possible that bernard has the number 19, with would give him the unique date of may 19, meaning bernard COULD actually know the birthday. The reason he says "I know bernard doesn't know" is because Albert does not have the months of may or june, therefore eliminating those months with unique dates entirely. Its not based on your point of view, its misconstruing Albert's first statement.

  39. The reading of Albert's statement "I don't know, but I know Bernhard doesn't know, too" as "I know without reasoning Bernhard doesn't know. But even given that fact, I can't deduce it", leading to Bernhard's: "Aha, then it can't be in June. Now I know", giving Albert the clue "Aha, than it's the only remaining unique day" leads correctly to Aug 17, but is a far stretch from the implied reading in the original wording. I would never have considered that 'creative' reading when I solved the problem with July 16. but it's interesting and the logic following from it flawlessly yielding Aug 17. Thank you, great video.

  40. OK this problem has no solution due to the first statement. Lets take steps;

    1) The first statement clearly states that they both study the given 10 dates, they both process the given data in a similar fashion but can not figure out the birthdate. Which takes us to step 2
    2) The date can not be May 19 or June 18. If it was one of them Bernhard would immediately know it. So we eliminate those dates. Which takes us to step 3
    3) Once May 19 and June 18 are eliminated June 17 stands out single and had Albert been given June, thinking about step 2, he would simply reason June 17. But he can not. Eliminate June 17 as well. Which takes us to step 4
    4) Now that June 17 is eliminated, August 17 stands out single and had Berhard been given 17, thinking about step 2 and step 3, he would simply reason August 17. But he can not. Eliminate August 17. Which takes us to step 5
    5) The remaining dates are May 15, May 16, July 14, July 16, August 14 and August 15. There is no way they can solve it according to the "first statement".

    Proof -> Assume Cheryl said May to Bernhard and 15 to Albert.

  41. If the date would have been July 14th the conversation could have gone something like this:
    A: I know Bernard can't know the birthday.
    B: That's right, I don't.
    A: But now I know.
    B: And I do too.

  42. Solution 1: know because of deduction from my own info
    Solution 2: "know" because of seeing other person looking like they don't know

  43. I got July 16.
    But can please someone explain what is the difference between an argument and the knowledge of an argument? It's mentioned but not explained in the video.

  44. I think the problem with this is that it is phrased like a conversation. It shouldn't be, because it is ambiguous. If they only want one answer, they should phrase it like an unambiguous logic question and say what they actually mean.

  45. Here is my interpretation to explain why the “regular” solution is not satisfying
    Of course comments are greatly welcome
    Let’s imagine that 16/7 is really C birthday
    It should lead in a deterministic way to a flawless logic
    If it doesn’t the solution is not acceptable, ok?

    If 16/7 is the bday Albert gets July as information
    If so Albert knows for sure that the possible days are 14 or 16
    Knowing it is July is not enough to know the bday so it is justified the statement “I do not know the bday”
    However the statement “but I know that also B des not know” does not convey ENOUGH information to exclude may and June because it just says that B does not know and in fact B could have heard 14 or 16 (the only June options)
    But both 14 and 16 are not unique (options are 14/7 and 14/8 or 16/5 and 16/7)
    From Albert point of view B simply does not know but not necessarily means that June and May can be eliminated
    If you cannot necessarily eliminate may and June with A statement the rest of the “regular” solution fails
    If Albert has stated “I do not know but I know that B cannot be sure” I would have judge the statement sufficiently conveying the usable information that unique days (and therefore June and May) could be eliminated and the “regular” solution approved

    By the way my try when I first heard the riddle was 17/6

  46. James, and people in YouTube. There is something that I still can’t figure out. I’ll be happy if any of you can help me to get it through!

    James, your first solution, the one from the guardian, sounds quite fine to me until (2:52).
    Basically after Albert made realise to Bernard that the months can’t be May or June, Bernard says that he knows the day, which means that the date are Aug 15, Aug17 or Jul 16. Right?

    BUT

    How can Albert affirm that “Now he knows”? Based on what???

    Let’s imagine for a second that the number given to Bernard was the 15.
    Bernard would have said the same thing “At first I don’t know when the birthday is, but now I know (and he would think Aug 15)
    And Albert could still say (for a reason I don’t get) the same thing “Then I also know” and think that the date would be Jul 16.

    I mean, they would come out with two different birthdays dates lol.
    Please can anyone explain me how Albert can affirm out of the blue that now he knows too if there where three possible dates from Bernard’s point of view???

  47. is nobody gonna talk about the fact that cheryl's birthday scenario is a bit fishy?

    it sounds like a distraction ya know?

  48. I thought X is the month "know" is the amount of times you are going to jump after the months/Days are arrenged. So you eliminate 6 and you end up with july 16th

  49. Aug 17 is stupid!! Albert knows that Bernard doesn’t know just by knowing the month. We can therefore eliminate May and Jun. I see no legit argument against this logic.

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