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.