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P10: Chivalry is Dead (Knights and Knaves #1)
471 Plays3 years ago
One tells a lie, the other the truth! Have fun with Sofía and Meryl as they investigate knight, knave, and spy problems!
Intro is "Breaking Math Theme" by Elliot Smith. Music in the ads were Plug Me In by Steve Combs and "Ding Dong" by Simon Panrucker. You can access their work at freemusicarchive.org.
[Featuring: Sofia Baca; Meryl Flaherty]
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Transcript
Introduction to 'Chivalry is Dead'
00:00:06
Speaker
I'm Sophia. I'm Meryl. And this is Chivalry is Dead, problem episode 10 of Breaking Math. This is on a series of episodes where we're going to talk about night and nave problems, right?
00:00:17
Speaker
So the idea of the knight and nave problem is that we have three characters we're working
Explaining Knights and Knaves Logic Problem
00:00:24
Speaker
with. We have the knight who always tells the truth, the nave who always lies, and the spy who doesn't always lie, but might at random. And so this is an example of a logic problem that you might see if you took, say, a computer science, discrete math class.
00:00:45
Speaker
So I've seen this problem before. I won't claim to be good at it, but we're going to work through some of them. Yeah. And, um, you usually have a limited number of questions you could ask, or you could only ask one person or you could ask them one at a time or something like that. Right. All right. So, uh, we, full disclosure, we recorded a little bit of this episode, like five minutes worth before we realized that I wasn't recording in the episode, but, um, Meryl, you're not, uh, you haven't done many of these problems, like you said, right?
00:01:15
Speaker
Right. And of course I've seen them, I have the exposure, these problems in particular, I have not. Yeah. And I found a whole slew of them on popularmechanics.com and we're going to be going with, we're going to be reviewing those problems. They're the riddles of the week. They have to do with nights and naves and they're really, they're really fun to work through. And we're going to be doing that on these series of problem episodes. So let's begin.
00:01:42
Speaker
So Meryl, you have a knight and a nave in front of you, right? You have a knight and nave in front of you and they're in front of two doors, right?
Presenting the Door Problem with Knight and Knave
00:01:51
Speaker
Okay. The doors are identical and you don't know who's the knight and who's the nave. One door leads to $50,000 cash. The other door leads to a spiky land of spikes. You don't want to go into it. So what question, you can ask one of them one question. What is the question that you would ask to figure out which door is which?
00:02:12
Speaker
Oh, and they're each in front of the door. Let's start off with something and say we want to figure out which one is a knight or a knave. That doesn't get us all the way. But what if we asked, what would the other person say that I would say I am?
00:02:33
Speaker
Right? So let's see if we asked that to the knight, right? You asked the knight, what would the other person say that you would say that you are? The knight, the knave would say that the knight would say that the knight is the knave, right? So knight would say knave? Yeah, so the knight would say, the other person would say that I would say that I'm a knave. Whereas the knave would say what?
00:02:59
Speaker
So the nave then would say that the other person who's the night would say, I'm the night. So yeah, let's say as the nave, what would the other person say that you say that you are, um, the night would say that the nave would say that, um, he's the, uh, night, right?
00:03:22
Speaker
The knight would say because he's telling the truth and the other person is the knave. So he would say the truth then that the knave would say that he's the knave or wait, this gets tricky. Yeah. Okay. So let's start, let's, let's start with the night. No, no, no, I got it. I got it. So the knight would say that the knave would say that the knight would say he's a knave.
Discussion on Identifying Characters with Logic
00:03:52
Speaker
Yep. Uh, because he's telling the truth about a liar, right? Right. So in the end night says Nate. Yeah. And if we asked the naval would happen. So he would say that the night would say that he would say he's a night and that would be the lie.
00:04:18
Speaker
Yeah. So, um, yeah, so, um, yeah, the, so the night, the name would say that he's the night. So the night would say that, uh, he would say that he's, uh, the night. So basically you're asking someone, what would the other guy say? What would the other guy say that I would say I am?
00:04:41
Speaker
Oh, I thought, wait, actually, let me, let me simplify the question. What would the other person say that they are? I think that's a similar one. So let's, let's consider the question. What would the other person say that they are? Okay. So let's start with the night. So if you ask the night, what would the other person say they are? Then the night would say that the other person would be the nave.
00:05:11
Speaker
Yeah, I would say that there would be remembered. The other person with lot would lie, right? So that the night is going to tell the truth about the lie and say that the other person is going to say that there are the night, right? Oh, I see. And if you asked the name of what would happen, the name would lie about the truth, right? Right. So the night would say that he's a night. So the name would lie about that and say that
00:05:42
Speaker
He'd say he's a nave. Exactly. Yeah. So that's a question they use to distinguish these two, right? Yeah. And so what we get then is that the knight would answer nave to that question and the nave would answer knight. Oh no, the nave would also answer a nave actually, but we are actually getting weirdly closer.
00:06:09
Speaker
Because think about it now, you're getting them to tell you the same answer to a question. That's progress, right? I suppose it is. Because we have this mechanism, right? Of the truth teller telling the truth about a lie, being the same as a lie about a truth teller, they're the same thing, right?
00:06:31
Speaker
Right. This actually is quite confusing to me. I'm not going to lie. All right. So let's, let's say, let's modify the question a little bit. What would the other person say?
00:06:48
Speaker
I feel like at this point you could just ask the person, do you exist? Oh no, but not even asking them if they, but yeah, but the, in the night, nave would say, uh, no to that. And the night would say yes. Right. So we know which one is which with that question, but we also don't know which door leads to the cash. Right. Right. We have still haven't solved that problem. So consider the question, would the other person say that the world exists?
00:07:16
Speaker
So the knight would say that the knave would lie about that. So the knight would say the knave would say no. But the knave, knowing the other person is the knight, knows that the knight would say yes. So the knave would also say no.
00:07:41
Speaker
Yeah. So do you notice that we're able to ask them a question now? No, it doesn't matter who we ask. We are able to ask them a question about the universe. So like we would say, so if we consider the question, because now we're getting nos to yeses, right? So would the other person say that nothing exists? So the night would say,
00:08:06
Speaker
that the nave would say yes to that. And the nave would also say that the knight would say yes to that. So, yep, you're right. So now how would you encode a question about the door into this? Because remember, if you could ask them just questions about the world now, you have a template, right? Right. So we could ask something that is constant about the doors then.
Determining the Safe Door with Truth and Lies
00:08:30
Speaker
How about just a direct question?
00:08:32
Speaker
So we could ask something along the lines of, so what if we asked what's behind the door? What's behind the door the night is standing behind?
00:08:44
Speaker
What's behind the door that the night is standing behind? Oh, you can actually absolutely ask that and then But think about this if you if the if the night is in front of the good door, right? Yeah The night would say that The night is standing in front of the good door, right? What whereas the nave would say that the night is not right makes sense
00:09:07
Speaker
Yeah. So we still don't have any more information about that because based on the yes or no, we still don't know exactly. But consider this, would the other person say that the world exists? If they say no, it's yes. Would the other person say that nothing exists? Would the other person say that a cheeseburger costs a million dollars? So we could ask something along the lines of would the other person say that, so what the other person say that they're standing in front of the good door?
00:09:37
Speaker
Alright, so if you ask the knight, would the other person say they're in front of the good door? If the other person is in front of the good door, they're gonna say no, right?
00:09:48
Speaker
Right. So we kind of have to build out our truth table here. So let's say the night is standing in front of the good door, then the night would say that the nave. So the night would say that the nave would say that he's standing in front of the good door.
00:10:09
Speaker
Yeah, exactly. And if you ask... And so if you ask the knave, so the knave knows that the knight's standing in front of the good door, so the knave would say that the other person who's the knight would say that he's not standing in front of the good door. Exactly. So whatever answer you get is going to be a lie, right? That sounds right.
00:10:37
Speaker
Yeah, so because if you swap it around and the knave is in front of the good door, the knight's gonna say the knave says he's not in front of it and the knave is gonna say that the knight is gonna say that he is in front of it, right?
00:10:51
Speaker
So can we conclude from this then that so the knight would say, wow, this trips me up for some reason. All right, so let's do a truth table. Let's say the knight is in front of the good door. Knight equals good, nave equals bad. And then the second scenario is a knight equals bad.
00:11:13
Speaker
Knave equals good. Alright, so if the knight is in front of the good door, if you ask the knight that question, what are they going to say that the knave is going to say he's in front of? So the knight would say the knave would say that he's in front of the good door. And what would the knave say? So the knave would say that the knight would say he's in front of the bad door.
00:11:33
Speaker
Right. So both of those questions are exactly the opposite of the truth. I mean the answers. Now let's do the second scenario. The knight is in front of the bad door and the knave is in front of the good door this time. So the knight would say that the knave would say he's in front of the bad door. Exactly. And the knave would say what? The knave would say that the knight would say he's in front of the good door.
00:11:58
Speaker
Yep. And, uh, yeah. So since every scenario that we could put in and no matter who we ask the question, we get exactly the opposite of the truth. How would, uh, we know how to get the truth from this, right?
00:12:11
Speaker
Right. So, and since you know which night and since you know that, um, and since you see them standing in front of the doors and right in front of you, all you have to do is say, would the other person say that you're in front of the bad door? If they say yes, um, then you go in. If you say no, then you go to the other door. I see. Yeah. It's a famous problem because there's so many little things that play, you know?
00:12:35
Speaker
Mm hmm. And I don't quite know what it is. I think it's the way that you're trying to find the question when you know the answers. That just makes this ever so tricky to me. It's a little bit. What's the word posteriori?
Solving a Three-Person Logic Puzzle
00:12:52
Speaker
Oh, yeah. Yeah, that's why I also really like these problems because they really like give you a sense of how to solve certain questions and logic.
00:13:02
Speaker
Ready to move on to our second and final problem You have three people in front of you one is dressed in red one in blue and one in green, right? So far so good the guy in the red says that the guy in the blue is a spy, right? right The guy in the blue says that the man in green is a spy Okay, and the guy in the green says that the blue dude is a spy. I
00:13:24
Speaker
Got it. All right. So, so red accuses blue, blue accuses green and green accuses blue. So in this, so in this case, we don't need to ask any questions. We just need to determine which person is which and see if it fits the logic and it should play out. Yeah. Cause you know that there's one, there's one spy, one nave and one knight.
00:13:55
Speaker
Now I'll give you a quick little hint. Both red and green say that blue is the spy, right? Right. So blue and green, red and green are both. So red and green are both saying that blue is the spy. Yep. So one of them has to be lying, right? Or at least not telling the truth. So I want to just plug in some stuff and test.
00:14:24
Speaker
Right. So basically they're both either both lying or they're both telling the truth. Or one of them is a spy. Okay. Oh yeah, exactly. Okay. So let's say, um, and if they're both lying, we know who the night is. If they're both telling the truth, we know who the nave is. So let's do exactly. So let's assume the blue dude is a spy. Oh, wait. If the case, if the blood, if the guy in blue is saying, if the guy in blue is a spy, right? Red and green are saying that blue is a spy, right?
00:14:55
Speaker
Right, so blue is the spy, then red and green are both tell telling the truth. But that doesn't work because one of them would have to be the nave. Yep, yep. And so we reach a contradiction. So let's just let's say, um, let's say, um, let's say the guy in green is a spy, right? The guy that let's say the guy in blue is telling the truth. And the guy or let's just say the guy in green is a spy, right?
00:15:24
Speaker
Okay. So if green is a spy, he could either be lying or telling the truth. Yeah. Which means that red, um, and blue must be the night nave. And so red is saying that blue is the spy and blue is saying that green is the spy. Yeah. So if red, red would be, so red would be lying.
00:15:55
Speaker
and blue would be telling the truth. And so red's the nave, blue's the knight. Awesome. So we know that that at least satisfies the problem, right? Right. There's no contradiction. There's no contradiction, but do we know that it's the only thing that solves the problem, right? Right. So let's keep trying. All right. So let's see if red is the spy,
00:16:23
Speaker
So red could be they're lying or telling the truth. So blue is saying that green is the spy and green is saying that blue is the spy, but both of those would be lies. So there we have a contradiction.
00:16:48
Speaker
Yeah, and so we've shown that green has to, that the only color for the spy is green, right? That seems to be the case. And if green is the spy, then we know what red and blue are, right? Right.
00:17:01
Speaker
So just to spell it all out, green is a spy, red is the nave, and blue is the night. And since that's the sole condition, since green is a spy is a condition upon which we know that red and blue are the nave and the night respectively. And we know that spy can't be anything but green, we know that this is the only solution, right? Right. So Meryl, I know this was a short episode, but did you learn anything?
00:17:32
Speaker
Um, I think I stretched my brain a little. So in that case, I learned a little bit more about trying to at least set up logic such that you can solve problems where you might. So at least I learned a little bit more about how you can, you know, set up problems such that the outcome will be what you want. And you can test our hypotheses like this.
00:18:01
Speaker
Exactly. I couldn't have, yeah, exactly. It's one of the awesome things about doing these little puzzles is that, like you said, they stretch your brain.
Audience Interaction and Feedback Invitation
00:18:09
Speaker
So I'm on Twitter, um, uh, at Saipod Sophia. Oh, Sophia with an F. And I'm on Twitter as Future Pod Meryl.
00:18:18
Speaker
And Breaking Math is on Twitter at BreakingMathPod. Don't forget to interact with us if you would like to send us questions, comments, say what you'd like, didn't like about an episode. And of course, our posters are on facebook.com slash BreakingMathPodcast. Just click on shop.