Married Problem with solution (plus extra problem)


Hello, everyone! I’ve got a problem for you today, which you may have seen in my earlier video, but let me run through the problem again. What I’m going to do is go through the problem, we’re going to talk about the answer, and also, we’re going to talk about what we can learn from this problem, including how it applies to an actual real life maths problem, too. So the problem was this: Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person? There are three options for this. A) Yes B) No or C) Cannot be determined. Now the claim was that over 80% of people get this wrong. So if you haven’t tried this yet, pause the video, have a go. Otherwise, I assume you’ve got an answer in mind. So now, I can tell you that the correct answer is A) Yes. There is a married person looking at an unmarried person. Now I know some of you are not going to be happy with that answer. Why is it A? Why is it A, and not C–cannot be determined? So let’s look at this answer and also, what we can learn from this problem. I got this problem from a book called “The Science of Genius” by Scientific American, and in particular, this problem was described in a chapter by Keith Stanovic. Now, let’s look at the set-up for this problem. We’ve got three people. We’ve got Jack, Anne and George. Now we know that Jack was looking at Anne, which I’ll represent by drawing in an arrow, like that. So we had Jack looking at Anne. We had Anne looking at George, which I’ll draw in another arrow for that. Now, Jack was married, and George was unmarried. But we knew nothing about the marital status of Anne. Now, the way this question is written, it’s–I mean, it’s ridiculous. There’s a lot of opportunity to set up your own flirtatious back story between Jack and Anne and George, and married people stealing glances at unmarried people. But the question itself was used by a computer science lecturer at the University of Toronto, and when he tested it out, he found that over 80% of people were responding ‘cannot be determined.’ That was C in our three options. Now I can think of three reasons why people might say ‘cannot be determined.’ So let’s look at those. One reason might be that the question doesn’t say anything about where these people are standing. And I can see that might be a problem for people who are less experienced at these kind of puzzles, because they’re trying to apply it to the physical world. So they’re thinking “Are these people standing in a line?” “Are they standing in a triangle?” “Can they look at two people at the same time?” But people who are more experienced with these kinds of puzzles will know that when it says something like “Anne is looking at George,” then that is simply a relation between two objects. The other problem might be that the question doesn’t say anything about where George is looking. Now, again, people who are more experienced with puzzles might think, “Well, if that information is not included in the question, then I probably don’t need it to determine the answer.” The other point is, the question says “Is a married person looking at an unmarried person?” And here, George is unmarried, so it’s actually irrelevant where he’s looking. He could be staring off into space for as far as I’m concerned! The third problem is that the question doesn’t say anything about Anne’s marital status. Is she married, or is she not married? Now, this is the crux of the problem, because at first glance, it does appear that the question is not giving you enough information, and so you think that it cannot be determined. But if we take our thinking just one step further, it turns out that we actually can come up with an answer. If you think about it, there are only two possibilities here: either Anne is married, or Anne is not married. Those are the only two possibilities. So let’s consider each one. So let’s say Anne is married. If Anne was married, this would be the picture that we have. And in this case, the answer to the question would be ‘Yes.’ So a married person is looking at an unmarried person. In this case, we have Anne looking at George. On the other hand, if Anne was unmarried, then the answer to the question would still be ‘Yes.’ It’s still ‘Yes,’ because in this case we have Jack looking at Anne, which means we have a married person looking at an unmarried person. In this case it’s Anne who is the unmarried person. Either way, the conclusion is the same. Even though we don’t know anything about Anne’s marital status, we are still able to deduce that there is one married person looking at one unmarried person. In the book I got this from, Keith Stanovic explains that this sort of reasoning is called ‘fully disjunctive reasoning.’ That’s reasoning that considers all the possibilities. Now, this question does suggest that there isn’t enough information and so people take the easiest inference which is ‘cannot be determined,’ without considering the full range of possibilities. But people can do fully disjunctive reasoning if they know it’s necessary. For example, if we didn’t give you the option of ‘it cannot be determined.’ Now, this problem did remind me of another problem from mathematics, where the same sort of reasoning applies. Let’s look at that. So this second problem is far more mathematical in nature, so mileage might vary here. I know some of my viewers are hard-core maths enthusiasts, so for them, this is going to go down a storm, I’m sure. It’s about irrational numbers. Irrational numbers are those that cannot be written as fractions, so their decimals go on forever, like pi. Pi is an irrational number. And the question is this: Can an irrational number, to the power of an irrational number, be rational? Now, this is quite a hard question to answer, particularly if you try and construct an answer. If you took an irrational number and raised it to the power of an irrational number, you get something that is hard to determine if it’s rational or not. But, if we want to answer this question– can an irrational number to the power of an irrational number be rational– we’re going to use the same sort of reasoning that we used in the married problem, and the proof is kind of nice, as well. For a start, we’re going to take an irrational number to the power of an irrational number. In this case, we’re going to take the square root of two to the power the square root of two. The square root of two is known to be irrational. Now, I don’t know what the answer to this might be, but let’s call it ‘x.’ Now, I’m going to take the result here, x, and I”m going to raise that to an irrational number. I’m going to take x and I’m going to raise it to the square root of two, again. So what I’m getting here is the square root of two to the power of the square root of two, to the power of the square root of two, again. Now, if you’re happy with powers, and how they work, what this means is we’ve got the square root of two squared, which is actually two. Right. Just the number two. Now how does this help? Well, x here is an intermediate step. Now, the question is can an irrational number to the power of an irrational number be rational? Well, if x was rational, then the answer to the question is ‘Yes,’ by the first line here. If x was irrational, then the answer to the question is still ‘Yes,’ by the second line here. Now, I don’t know if x is rational or not, but it turns out it’s irrelevant because the conclusion is the same. It’s a lovely proof. Yes, we don’t have to construct x. We don’t have to know if it’s rational or not. For completeness, I can tell you that x, the square root of two to the power square root of two is irrational, but again for what we’re trying to do it’s actually irrelevant. It was just our intermediate step. By considering all the possibilities, we can deduce the correct answer to the problem. I teamed up with Alex Bellos, and he ran the married problem on the Guardian blog, which included a survey, where people could record what they thought. It’ll be interesting to see how well people do. The original claim with the problem was that over 80% of people get this wrong, so I’m going to include a link in the description to the results of the survey and it would be interesting to see, so you should check that out. But, from me, if you have been, thanks for watching.

About the author

Comments

  1. I enjoyed sinking my JAWS into the logic problem, but I was a bit lost (world) with the maths, sorry!

  2. But what if Anne is a horse, no one said that Anne was a person? (Edit) I have read the comments in the description box: It is not evident that the question is not a riddle, as it is improperly worded, and contra proferentum ambiguity (it is not even ambiguous in fact, it is simply not stating the necessary information) is construed against the person seeking to rely upon it, so if seeking to rely upon the assumption that Anne is a person, when it is not a necessary and unavoidable conclusion from the statement (and no assumption is permissible in interpreting the statement, as is your own case), then A is not the answer. A preferable wording would include the ending 'Of these three people, is a married person looking at an unmarried person?'.

  3. You can think of this as a sort of boundary problem. If in a series you have one element on one side of a boundary and then some later element is on the other side, then it doesn't matter how many elements are between them or where any of them are, at some point you must cross that boundary (assuming that you cannot "go around" it).

    The interesting part of the question is that when generalized to n number of people with unknown status, although that might make the problem seem more difficult, for me it actually makes it far more obvious. I think it's because by having only a single person between them and giving her a name it makes us focus on her and the fact that we don't know her status, so we assume we can't know the answer. Whereas a more generalize version of the problem keeps me from focusing so much on a single details and instead focus on the problem itself. At which point I might use some trial and error and simply assume one possible layout and decide if the question is yes or no in that case, and then try to construct an opposing solution.

    So I might say "if everyone between them is unmarried then the person behind Jack is unmarried so the answer is "yes", now can I create a situation where the answer is "no"? At which point I quickly see that there is no possible way to make the answer "no", therefore the answer is always yes.

  4. The framing of the question is defective, thus relegating this to mere 'riddle' status, as it does not state that Anne is a person. Furthermore, of 'unmarried', that is not unambiguous, as e.g.
    Queen Elizabeth died unmarried. She never married.
    King Edward VI died unmarried (before adulthood, although child marriages did occur). He never had the capacity to marry.
    Whereas Queen Victoria died a widow.
    If you said that 'Queen Victoria died unmarried', many people would regard that as a false statement, or at the very least, imprecise and misleading, as she had been married to Albert, who pre-deceased her.
    The problem lies in attempting to translate a logic problem into the English language without being sufficiently precise in meaning. If the set-up of the problem were to refer to three women, and replace 'married/unmarried' with 'pregnant/not pregnant', then it would be clear that Anne's ambiguous status is in fact implicit and unavoidable in the set up of the logic problem.

  5. What a waste of time. Btw sqr2^2 is not an irrational number to the power of an irrational number so the answer if 2 can't be valid. Is this a trick or did he genuinely not get that? Can someone explain this as well as the Jack solution?

  6. the answer cannot be a "no" because we don't know the marital status of Anne. It cannot be "cannot be determined" because 80% got them wrong. Hence, it's "yes" XD

  7. Hm, maybe somebody can explain this to me. Shouldn't the answer still be no, since the problem explanation doesn't say anything about the property of Anne? She could also have no property of marriage. We assume it, but the text doesn't postulate that everyone has a marrigal state. So the only married person looks at somebody that is neither married nor unmarried what means that the statement "A married person is looking at an unmarried person" is false.
    I understand, that if we assume everyone must be unmarried or married we come to the conclusion that the statement is true.
    Isn't this some kind of missing premise or where did I go wrong?

  8. If you're trying to figure out what the square root of 2 to the power of (the square root of 2) is, the square root of 2 is equivalent to 2 to the power of .5, which can be shown as 2^(.5). This means there's going to be a power to a power to a power.

    Consider a related problem, 2^3^2. Working from left to right, it becomes 8^2, which becomes 64, which is also showable as 2^6.
    2 x 2 x 2 x 2 x 2 x 2 = 4 x 2 x 2 x 2 x 2 = 8 x 2 x 2 x 2 = 16 x 2 x 2 = 32 x 2 = 64.
    So when you put a power to a power, they multiply.
    The answer may surprise you.

  9. Anne could be a dog and not a person at all. DISPROVEN.

    This illustrates a good point, and is very good to know… However it's vague and a bad example, in my opinion.

    The math example had a lot less room for debate. Abstract logical constructs (like math) work better for illustrations because the real world isn't determinant.

    Making assumptions will often result in errors. Funny enough, if computers made assumptions like this they'd be failing left and right. If you want something to work logically, you can't leave ANY holes or it'll fail. That's the whole point of logic, it's not random and open to crazy interpretations!

    Makes me think this is designed to deceive, like he said, by adding the third option. People as a whole aren't dumb.

    Deceptions lead to deceit! Big freaking surprise there ;p

  10. why is this even debatable ?, A is looking at B, B is not married, A is

    C doesn't matter

    I don't get why is this so big deal, TBH paused and stopped the video, don't want to get brainfucked lol

  11. Jack could possibly see George in his peripheral vision or if there are arranged in a straight line segment (or close to that arrangement), then it is likely Jack can see George in his direct line of sight. Since no information and no restrictions were given about the arrangement of the 3 people, we can put them in any arrangement we like and just qualify our answer with that. That part alone will make it so the answer is yes in some cases, thus ruling out answers not and cannot be determined.

  12. Sorry Mr Banana, I don't know how slack maths professors are about formal logic, saying perhaps, "hey guys, these are the assumptions that you make in puzzling, gimme a break!". But I think a decent logicien would require the answer "Indeterminate, for it is not stated that the names are those of persons, Anne could be a cat". Slack question! Did the survey include respondents' reasoning? Feel free to call me an over-rigorous partypooper, but my answer remains "indeterminate" despite what you said and another guy in a video that brought me here. Maybe some of the mathos in the survey were thinking similarly. And for a survey of the general public the answer "indeterminate" corresponds to "can't be arsed to think about it unless you're paying me, go away". Surveys aren't reliable.

  13. This video was a real hook in my duel with myself on whether I should subscribe to the channel or not. The turmoil was rather something evil. I finally decided to go with my heart, as always, and subscribed.

  14. I love how a single question mark could change the meaning of the title drastically

  15. I'm not a maths person, but I like to watch your videos from a pedestrian viewpoint.

    With the second problem, my immediate thought was that it had to be true, but my reasoning is that if you take an integer x and take it by the root of an irrational number, would not the answer be irrational?

    Is this proof wrong, just different, or am I missing something in the problem?

  16. It's not at all difficult to construct an answer for the irrational number problem. One good example is e^ln2 = 2

  17. i really think the biggest confusion here, is that people don't realise "married" and "unmarried" are contrary, so they don't think about anne state as if she could be something else than married or unmarried.

  18. The irrational problem carries some controversy considering the intuition logic and exclusion of middle part.
    Does rejecting the Exclusion of the middle part lead to unnoticeable situation?

  19. a is indefinite article so it means if at least Anne or jack is married then the answers is yes. don't get the "80%" figure its simple.

  20. You can solve this even if you consider an infinite number of people between Jack and George. This is just intuitive to me, so I'm not sure of the best way to explain it, but I consider the following. If a binary signal goes down a line as a 1, and comes out a 0, you know that the 0 turned to a 1 at least one time down the line.

  21. If there're only the two possibilities "married" and "unmarried", the solution is correct. But at least in some country exist legal statuses like "divorced" or "widowed". Then the solution is incorrect. That's the problem with such word problems. There's no proper definition of the used terms.

  22. This seems like the word puzzle version of the Intermediate Value Theorem. You start with married and end with unmarried, so you must cross from married to unmarried at some point in between. It seems fairly straightforward, so I'm surprised at how many people got it wrong.

  23. "80% are wrong" : and who says that ? The few retards that think they are right ?
    The only reason why you think you can answer A)Yes, is the psycholinguistic process that associate the fact that "Anne is not… defined" with "Anne is not… married" or "Anne is not… single". In the process, you chose the easier : she's not… single. It's easier because the idea that "we can't know for sure that she's not [something]" is working in the background, and crystallize around "single = she's not married" more than "married = she's not single".
    Indeed, married=1 and single=0, so "single" is closer to the negation (the absence), and the phrase "we can't know for sure that she's not [something]" is attracted by… "single". So, Anne becomes the contrary : married !

    This also transforms the feeling you have that Jack is "seeing throught Anne" in a positive conclusion, terminating the reflective process and giving you the feeling that you've discovered something logic… behind pure uncertainty.
    The rest, the mathematical blabla with a quantic sauce, is pure bullshit. We don't care that in an other dimension there is a Anne who is married ; We're talking about here, bunch of nuts ! And here, we DON'T KNOW, so we DON'T ANSWER YES NOR NO.

    You think you're less lazy than 80% of the population, when in fact… you are ! But it's easier for today's people to hide behind science. That's why Grottendieck exiled in his manor ; he was sick of this contemporary shit.

  24. I have a problem for you :
    A clever guys is near an hypocrit, wich is near a violent douchebag.
    Does the hypocrit defend the clever guy when this one say to the douchebag that he's a sucker ?
    A) YES, B)NO, C)MAYBE.

    You gonna switch perpetually between the two false answers, for sure !

  25. Just realised something J-> A, A-> G that means J-> G logical math syllogism thank you Mrs.tantri for teaching me basic high school math.

  26. I'm sorry, but there is no right answer to it, because the question lacks information. We do not know how these 3 people are positioned and looking at one another. They could form a triange, or two people looking at one, a straight line etc. What you did regarding the answer, was creating a scenario in which the answer would be yes. But yet your answer is only based on the assumption that they are standing in a straight line. If they are not standing in a straight line, for instance a triangle, then the answerer cannot be determined.

  27. I got it straight away. To me it seemed obvious. If Anne is unmarried, Jack is looking at her so it is true. If she is, then she is looking at George. Took me less than 5 seconds.

  28. Fuck that problem. I don't care, The answer is C. The location of George was never specified. It doesn't make sense to say "oh because you're trying to apply it to the real world". The problem IS a real world problem since we are talking about people and lines of sight. It matters where the fuck George was standing because Jack's line of sight only would have hit George if they were all standing directly in front of each other.

    THE PROBLEM SHOULD HAVE BEEN WRITTEN AS: Jack, Anne, and George were standing in a LINE. Jack was looking at Anne, while Anne was looking at George. Jack is married, while George was not. Is a married person looking at an unmarried person.

    Then the damn question would obviously be yes.

  29. e to the power of pi multiplied by i is a rational number. which means irrational and rational problem is yes

  30. Thanks!
    "Close Encounters of the Third Kind"
    ^^ My guess of Spielbergs movie title. Reasoning:
    1st kind of contact = looking
    2nd kind = looking and knowing some data about observable object ("I'm staring at married person, maybe i should stop…")
    3rd kind = looking and using some extra-meta-information to get even more information
    ("OMG, one of us is definately staring at married person, wish it isn't me…")

  31. I saw this once on an online journal. I solved it with truth tables to get (A), but I prefer (C). With a classical approach, you would get (A), but the excluded middle must be assumed (or double negation elimination). From a constructive approach, you cannot prove Anne is married or unmarried, so you can't determine an answer.

    For classical mathematicians, this might seem nonsensical; Anne has to be either married or not. But the constructive approach does provide real-world counter examples. For instance, maybe Anne's status is ambiguous. Maybe she is religiously married but not legally. Maybe Anne is actually the name of a potato plant, and the concept of being married does not apply.

  32. Oh, so you don't have to believe me, but I'm watching the video, and he hasn't explained the right answer yet, but I'm going to say what I think it is. So if Anne is married, she is looking at George. Married person looking at unmarried. If she is not married, jack who is married is looking at her, so when she's not married, it is a married person looking at an unmarried person.

  33. Surely if you have a quantity to be squared, put that in brackets and raise the whole thing to another power, when you remove the brackets, you add the two powers, not multiply them. So the sro two, to the sro two all raised to the sro two would be the sro two to the power of twice the square root of two. Not the sro two MULTIPLIED by the sro two.

  34. Did the married one , quite quickly,but your maths one ,your explanation
    confused me for a bit, if x was irrational ,then no,but the question was can it be,well as you shown,it can,therefore the answer is yes to the question.
    Oh s**t! just notice my left shoe is on the right foot,oh wait, that`s ok then.

  35. Extension Question!

    Jack was looking at Anne, and Anne was looking at George.
    Betty and Charlie both walk in on the scene. They walk in the middle of the "chain"- Jack is still at one end, George still at the other, and everyone is still looking at the next person along.

    Is the answer to: "Is a married person looking at an unmarried person", still "yes?"

    a) Yes, b) No, c) Cannot be Determined

  36. This is working on the assumption that there are only 2 possible states of a person's family status. Maybe it is my misunderstanding of the English language but what if Anne is divorced or widowed? Does that mean she's unmarried? If not, then the answer is C.

  37. The question is not specific enough to give a good answer. There are two possible interpretations of the question :

    1) the question is a general condition like for exemple "a married person, in the us, has a ring". Then the answer is C because not all married persons are looking at an unmarried person if Anne is married, and all married persons are looking at a unmarried person if Anne is unmarried.

    2) the question is a specific condition and only one married person need to be looking at a unmarried person. Then as explained in the video the answer is A.

    Since the question is not specific enough (it could have been "is at least one married person looking at an unmarried person") the answer is C

  38. I answered C, could not be determined when I watch the 'question' video simply because there was no detail about Anne. But the moment I got to you adding the arrows to your diagram, I kicked myself for not seeing it any sooner.

    This reminded me of how my mom always made me draw 'pie chart diagrams' for any question involving fractions back in grade school. It isn't that the questions are so hard that I couldn't solve them but I was apt to make mistakes that I would never have make had I seen them mapped out visually.

    Anyway, I've just found your channel. I've been casually watching videos on Numberphile as I do chores and stuff and from what I've seen, I think I would be watching more of yours soon. Congratulations on the eleven years on YouTube 🙂

  39. No wonder my maths teacher gave me 50% on the irrational question….

    I said "Yes it can, pi has a very rational explination"

  40. The supposed answer to this question is actually INCORRECT. Here's why. 

    There is nothing in the question that states or even implies that Anne is a person – Anne could be a Golden Retriever. If Jack is looking at Anne the dog and Anne the dog is looking at George then a married person ISN'T looking at an unmarried person. There is nothing in the question to say that Anne is a dog, it is merely a possibility within the terms of the question. If Anne was a person then the answer would be A) Yes. As Anne's personhood cannot be ascertained then the answer must necessarily be C) Cannot be determined. 🐶

  41. I think the difficulty also stems from having to identify the problem yourself and given how it's not placed in a context where logic is usually applied in this way.
    If you said "within the triple of numbers (1, x, 0) where x is either a 1 or a 0, is there a sequence of consecutive numbers (1, 0)?" the answer becomes immediately obvious.

  42. You overlook the MAIN psychological reason why 80% get this one wrong, and the reason is 100% social psychology. As soon as the question is concretized on people, even fictional, who might be committing a transgression against a social norm, subjects fixate on the "real" problem of identifying, naming and shaming the transgressor, rather than merely determining whether a transgression occurs. Since there is indeed not enough information to identify the transgressor, most subjects say "cannot be determined".

Leave a Reply

Your email address will not be published. Required fields are marked *