That Viral Math Problem (Cheryl’s Birthday) – Numberphile

That Viral Math Problem (Cheryl’s Birthday) – Numberphile

I’m so happy I’m here to be able to get to talk about a viral maths problem. I mean when does that happen? Come on. And this is not in the maths community. We’re talking about like in the big wide world. This is really exciting. Have you heard of it, Brady? It’s got a few names, at the moment. It’s like the Singapore maths problem it’s a competition maths problem originally from Singapore. it’s also “The Cheryl problem” Not saying anything against Cheryl Cheryl’s not a problem Cheryls are usually pretty nice So if you’ve been living under a rock I’ll quickly explain what it is It’s actually, from a mathematical point of view it’s a classic logic problem It involves 3 people Cheryl, Albert, and Bernard Ok So Cheryl is trying to get Albert and Bernard to guess her birthday For what reason, we don’t know Maybe she’s just trying to play Albert against Bernard Or maybe Albert and Bernard, y’know, are trying to scam Cheryl for like her secret question for a bank account Who knows? But Anyway She gives them a bunch of options Ok so, Cheryl has shared a possible pool of dates But now, what’s she’s done is she’s actually gonna give a piece of information to Albert and a piece of information to Bernard An incomplete piece of information about her birthday so this is how it goes Albert, I’m going to tell you the month, – whisper whisper whisper and Bernard, I’m going to tell you the day, – whisper whisper whisper So now all they know is Bernard knows that Albert knows the month and Albert knows that Bernard knows the day But they don’t know it together And so then, they have a little exchange So Albert states I don’t know when Cheryl’s birthday is But I know that Bernard does not know too then Berard replies at first I didn’t know when Cheyrl’s birthday is, but I know now. So now suddenly, Bernard knows But what happens? Is Albert left the in dark? No Somehow, Albert is then able to deduce then I also know when Cheryl’s birthday is. So now they both know. And presumably, Cheryl is either happy or upset Or her bank account has been broken into. Something like that. We don’t know exactly. The question is how can you find out the information Nothing is explicit, is it Brady? it just seems like they playing games Are they Jedis? What’s going on? Are they reading each other’s minds? and this is the nature of these logic problems Believe it or not, this isn’t the first time I’ve been faced with a problem like this As mathematicians, we get asked these problems all the time And so the way to answer this Is to actually break it down and try and attack it logically because otherwise you’re not going to get anywhere. I am now going to rearrange this information here Because this is really important we need to have something to focus on So what I’m going to do is I’m actually going to split out the dates at the top here I’m going to..I’m going to put 14 uh… 15 The dates there in columns and in rows I’m going to place the months So here is May So now, what I’m going to do there is I’m just going to have a look at this so May 15 A cross is just basically, this is an option. This could be it. 16 or 19 Yeah, these are also contenders. What I’m going to do is actually give us a bit of understanding here because once we get into the logic we could get lost Let’s look at the options for this column. For the day or the date 14 how many different options do we have? We’ve got 2 ok 15 we have 2 16 we have 2 17 we have 2 18 we have 1 and 19 we have 1 as well. Ok, now let’s have a look for months. So now for months how many options do we have for each month? So we have 1,2,3 for May 2 for June 2 for July and 3 for August See where I’m going with this, Brady? So now what’s going on here is that I’ve actually just, I’ve just spread out the information Before I start bringing in the weird conversation they had The weird terse conversation that they had so now Albert says I don’t know, I don’t know what the birthday is. OK. (Brady: Of course he doesn’t.) Yeah, but that’s a weird thing to say. Can I just say that? It is a weird thing to say because in a logic problem Obviously you don’t. We both know the game here. You’ve only been told months. There is 3 options, 2 options, 2 options, 3 options. The only way you’d know is if there was 1 month there with only 1 date. Just saying that. I’m just saying that. What’s interesting though is Albert says “But I also know that Bernard doesn’t know.” We’ve got the information in a way that, ya know, I’m gonna be able to work with, but just looking at this straight away there are some red-hot dates. Here June 18 Well if you have a look at it it’s only one 18 that applies to any of the months It happens to be June. In actual fact the same for May there’s 19 19’s not anywhere else Can you see that, Brady? There’s nothing around there What this means is if Cheryl went over and said to Bernard “18” or “19” *smooch* What that means is straight away Bernard would be in. It’s like a gimme. It’s like “Bang”. I know exactly what it is. You give me the information. It can only be 1 thing. But see, Albert in sort of a gloating way, I mean really Albert needs a bit of emotional intelligence I think because he’s basically gloating and he’s going, “I’ve got information that excludes the gimmes” “So I know, Bernard, that you don’t have gimmes.” But how can he not know? Because he doesn’t have the date information. He’s only got month information. Ok, what are the months that had potential for gimmes? Well they were June and they were May. And that’s the only information that Albert has. He only has month information, yeah, monthly information So. What can we deduce from this? We’re mathematicians here. We’re all getting mathematical… Albert says, “I don’t know the date”. Yes, ok, that’s a weird thing to say. “But I also know that Bernard doesn’t know the date.” So logically, what you’ve got to say is therefore Cheryl didn’t whisper May or June in his ear. So, obviously cheryl whispered something like “July” or “August” That’s what she did ‘Cause if she said May or June, Albert wouldn’t be saying that, with conviction. Well that means that this is… no longer an option. And this… is no longer an option. But because Albert has said “I know that 18 and 19 are not an option” the only way Albert would’ve known that is through monthly information. So, in actual fact, this is gone, this is gone, and this is gone. Basically, everything in May and everything in June is gone This is great because with my prized little chart here, I can actually reduce some numbers here. So now this goes down to 0. This goes down to 0. So now Bernard actually knows…. Oh! An actual fact, because all of May and all of June, all the dates that we had on the table are all gone, that affects the total number of options I have for each day. It doesn’t matter about his date. He’s actually just looking at all the dates So now for 14… Well, there’s still 2 there. That hasn’t changed. But for 15, we’ve just struck out May so now this is only 1 option. For the 16th that was in May and July. May is struck out. So now… there’s only 1 standing in this column. For 17 same thing, look at this. There’s June 17 and then there’s August 17 So this one’s getting struck out. And then for 18 and 19 and they turned down to 0 too So Albert’s given this information Now Bernard has been able to reduce the options. But to be clear, for Bernard to know the birthday he already has the date. He just needs the month. So now Bernard says… “At first I don’t know when Cheyl’s birthday is, but I know now.” So, from what Albert has said Bernard is now able to say what the month is. Ok, so now, let’s put ourselves in Bernard’s shoes. So now he’s actually discounted May and June. He’s reduced the number of options. Really you’re only talking about July and August. Now if he did have 14… right? There are still 2 options on the table So… He didn’t have 14. If he did have 14… he would have said something like… Uh… yeah, I still don’t know. OK? But instead he goes, “I didn’t know before”, which means he didn’t have the gimmes. But now he knows! How does he know? Because there are actually new gimmes on the table, Brady, you wouldn’t believe it. Look at this. The new gimmes are 15, 16, and 17 So now, if he’s holding onto 15, 16, or 17 in his head, or really, in his heart, because we’re dealing with Cheryl here… What’s happening here is he’s going “Wow, there’s only one 15 that applies to August, there’s only one 16 that applies to July and there’s only one 17 that applies to August.” “Look, if I have any of these dates, I will know exactly which month it applies to.” So, at this stage of the problem, it could have been 15, 16, or 17. So now, Bernard is going… “Sho sho sho sho sho…” “bup bup bup bup bup” And this one goes down to 0 But not only that because July and August have now lost a date. So that means over here, we need to go bang bang bang… 1. …and over here…. We need to go: “bang bang bang… 2” Right. So what’s the last statement? Albert says, “Then I also know when Cheryl’s birthday is.” Ok. So that means he’s got a gimme. So now if we have a look where the gimmes are… Let’s look at the options. He’s either got July or August. What options do we have here? Well August is no good because it’s got two options. That means it’s ambiguous. It means he doesn’t have gimmes. It could be either 15 or 17. But because he knows… it has to be July. Do you know what that means, Brady? It means that Albert is going… “Uh-uh, sorry.” and Albert’s going… “Uh-uh, sorry.” So this now goes down to 0. This goes down here. Well Albert’s doing it, so I’m doing it in blue. That one goes down to 0 too. So now, look what we end up with. We end up with 1 option for the month and 1 option for the date that they can both get. And that option is July 16. *sighs* Ok, what’s really interesting about this problem is the fact that because it’s using words it’s actually open to interpretation. If this was actually redone in just pure symbolic logic, it would be unambiguous. Because we’re using these sentences in actual fact, people have opened them up into interpretation.

100 thoughts to “That Viral Math Problem (Cheryl’s Birthday) – Numberphile”

  1. I got July 16th

    Haven't watched the solution yet

    Basically, because Albert knows that Bernard doesn't know, that means that it's not a unique day. This eliminates May and June because they both have a unique day (a day that only appears once in the choices). 19 is unique in May and 18 is unique in June.

    So all that's left is July and August.

    Bernard says he knows the answer, which means that it's a day that's in either August but not July, or in July but not August. Because of this, we can remove July 14 and August 14.

    Now, July 16th is left, August 15th is left, and August 17th is left.
    Since Bernard knows the day, he knows the date.

    Now, Albert knows the answer even though there are three options left.
    If the month he was given was August, he wouldn't know the answer yet because it could be August 15 or August 17. HOWEVER, he says he knows the answer! That means that the month isn't August. All that's left is July 16.


  2. So they're not gloating. They've just found a way to share the info with each other without Cheryl working it out. Subterfuge.

  3. would i be stretching it if i said right here in this Youtube channel us plebe's are actually treated to the literal best of the best? this channel has grown on me and i've only met the narrator of this video just in the last couple of days. i don't' know why i haven't subscribed yet but putting that in now 😉

  4. I thought it would be an interesting exercise to try to write this in few words. Albert isn't holding May because if he were he couldn't rule out Bernard holding 19 and already knowing everything. Similarly Albert isn't holding June because if he were he couldn't rule out Bernard holding 19 and knowing everything. So Albert holds either July or August. Once he speaks, Bernard knows that Albert holds either July or August. If Bernard were holding 14 then he couldn't differentiate between Albert holding July and Albert holding August, and Bernard wouldn't deduce the birthday at that time. But since he DOES deduce the birthday after Albert's speech, we know Bernard isn't holding 14. If Bernard is holding 15 or 17 he could rule out either July (no 15) or, respectively, June (no 17) after Albert's speech, and would know the date is (respectively) August 15 or August 17. But then Albert wouldn't be able to figure out whether Bernard held 15 and rule out July to get the date, or held 17 and ruled out June to get the date. Since Albert DOES figure it out after Bernard's speech, Bernard isn't holding 15 or 17. That means Bernard is holding 16 and Albert is holding July. Albert, holding July, knows that Bernard must be holding 14 or 16. But Albert knows that if Bernard is holding 14, Albert's speech wouldn't cause Bernard to deduce the entire birthday because Bernard would not be able to choose between July 14 and August 14, and Bernard says he CAN figure it out. So Bernard isn't holding 14 and must be holding 16. So as soon as Bernard says that Albert's speech enabled him to deduce the entire birthday when he couldn't before, it tipped of Albert to the entire birthday (July 16) as well.

  5. This problem really doesnt seem that complicated, the gimmes you first showed I got then I realized it was incorrect, I did a little more thinking wrote it out and got it pretty quick. Took me 20 mins tops

  6. There's a 50% chance that either Albert or Bernard shares the same birthday as Cheryl so they should just guess their own birthdays.

  7. Idk about you guys but that's my birthday so me and Cheryl boutta go gang out together.
    Edit: where my July 16 bois at

  8. Alberts initial statement is completely loaded and up to interpretation. I think there are different ways of looking at it.

    For this particular solution he should have really said “I don’t know the answer and I know that Bernard could have no possibility of knowing it.” This would better imply and lead to the solution presented in this video.

    Instead Albert says that he knows Bernard doesn’t know the answer. Well of course he doesn’t know the answer! He could have been whispered the 14th into his ear yet that very statement would still be valid. And this time there would be no reason to cancel out May.

  9. So, I have been able to deduce it is in July since if it was any month that had a unique day, Albert wouldn’t know that Bernard doesn’t know, and July is the only one with a non unique day, and once Albert said that, Bernard now knows the month, July, and the day, he was told, but I cannot for the life of me figure out which day it is, will finish video now, eager to find out

  10. but what if we include mind games. what if one of them lies, could he still beat the second guy and be the only one with the right info? and if so, what the lie should be?

  11. What is funny is that based on the interpretation of Albert's first phrase "i don't know when Cheryl's……… Bernard does not know too" you can get to different results. I thought he meant "i don't know, but since Bernard does not talk he propably does not know either" and from that point , based on the rest conversation, you can conclude that the date is August 17!

  12. im proud of myself for pauseing this opening up a spread sheet and solving it exactly how it ends up explained in the video XD

  13. This turned me on working this out; awesome braingasm! So much fun! Came here to confirm my logic was sound. It was! Love it. 🙂

  14. I have similar riddle: There are two camels. One is purple. How much does the dog weight when it's dark?

  15. on one hand, cool, I got the solution almost instantly, but on the other hand, I'm meant to be revising for a mechanics paper tomorrow and they're not going to be nearly as simple as this…

  16. That's also my birthday, although I never would've dreamed up such a screwed up way to tell people about it.

  17. But Bernard could still make the statement 'I now know the birthday' if it was August 15th or August 17th?

  18. The problem with the puzzle is Bernard can only work out the date if he has already heard Albert's second response. He doesn't have that information when he makes his deduction so he shouldn't be able to work it out.

  19. February first? Genuine guess from the beginning

    Oh never mind, I didn’t know those dates listed were the actual only choices

  20. Watching this video again after 4 years, I must say that I find it a bit annoying to watch.

    No offense intended for Mr. Pampena, but that presentation was a bit annoying. A bit too much repetition, a bit too much of saying the same things again and again, and with needless detail, the actual whispering was unnecessary and a bit annoying, and the whole thing just takes too much time to get to the point. I think this could have well been explained, clearly and understandably, in 5 minutes.

  21. When I read the problem, I understood it as meaning that Albert knew that Bertrand didn't know, not from logic, but from observation. After Cheryl gives each man a piece of the answer, Albert and Bertrand watch each other and then Albert says that he doesn't know, and Bertrand doesn't either. That eliminates only June 18 and May 19. But then Bertrand, seeing that Albert doesn't know, realizes that June 17 is off the table as well, since if Albert had been told June he'd have immediately known the full date. Since Cheryl told Bertrand "17" then he knows now that August 17 must be correct, since it is the only other date with a 17.

  22. It took me like, 2 minutes to solve it and i am still in school. Its not all that hard tbh… Its just the wordings that get most ppl, me included. At first i thought it was a joke but then somebody told me it was actually possible, and once i knew it was possible, it was a cake walk…

  23. Bernard never indicates that albert would know now, so in albert's head, it could have been either 15, 16 or 17. If it was 15, bernard would have known but it could have also been the 17th. I dont see how albert could have known just from the statement that bernard made.

  24. Lol i just used a little logic. The wording was bs so i just counted the words. Theoreticly saying that Bernard made a long pause in his sentence the total words up to the comma are 7. And he knows the month so it has to be July. And Alberts total words are 16 so its the 16th. And because they are mathmaticians and as we all know they count every word they both know the answer.

  25. Sorry dude , staring at you from under that proverbial rock, as I don't use social media much , but looking forward to your explanation .

  26. Without watching the video from 2:30 onward, i did get to a point where i had only these (3) as options.
    – July 16
    – Aug 15
    – Aug 17.

    when Bernerd says "at first i didnt know but NOW i do" …., that is applicable for any of those (3) options. I couldn't figure out, from that point, how could Albert know Bernard has 16???

    I'm lost in Simon's explanation on the final clue …. , how can albert "just know" it has to be july"??? how can Albert be 100% sure that Bernard wasn't given '15' or '17'?

    Let's assume Albert was given 'July'. Then since Benard said 'Now i know' …, then this WOULD BE the ansewr.

    But let's assume Albert was given 'August' and , yes, he would've had ambiguity between '15' and '17'. from Bernard's perspective, he knows for sure the birthday since he has the date. But from Albert's perspective, i don't see how he can know for sure what Bernard has (btwn 15 and 17) IF HE WAS GIVEN AUGUST….

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