 Hi, I'm Zor. Welcome to Unisor Education. Today we will have a lecture about logic. This is number three in this topic. And it's all part of the course MES Plus and Problems presented at Unisor.com. The course MES Plus and Problems is about solving problems. And there is a prerequisite course which is called MES for teens. So the MES for teens is much more theoretical with like theorems and proofs, etc. Now this is the course which is concentrated only on solving different problems. Which are not exactly a theoretical kind. Sometimes it's maybe more difficult. Sometimes, like today for example, it's just a common sense problems. The common sense actually is called logic by mathematicians. So it's very important to solve problems. Actually my personal understanding is that the whole purpose of studying mathematics is basically to develop your ability to solve problems. And the more problems you solve, doesn't matter which ones. And mathematics presents the whole field of different problems. So the more problems you solve, the better you will be equipped to solve real problems of real life. So today's problems do not require a lot of theoretical knowledge. Basically, I think it's just nonsense at all. Just a common sense. And I think it's very useful in this particular situation. They're not difficult at all. And what I suggest to you is whenever you listen to the presentation of this problem, stop watching, pause the video, and try to solve the problem yourself. Spend some time. It's very important to spend some time thinking. Thinking is something, the process which I encourage you to get involved in. So you think about solutions to this problem. And even if you didn't solve it, listen to whatever I present as a solution, it's still very useful to think about this particular problem. Okay, so let's get to the problems. Okay, the first one is, okay, you have a tourist which comes to a fork on the route. One is called the city of truth, another city of lies. So in the city of truth, people, all people only tell the truth. In the city of lies, all people always tell lies. Now, he doesn't know where to go, but he wants to go to the city of truth, but he is on the fork and he doesn't know where to go. Well, luckily, there is a person here, right here, sitting, and the tourist can ask the person basically how to get to the city of truth. So the question is, since we don't really know whether this person lives here or there, so he doesn't live anywhere else, so either he is telling the truth or he is telling the lie. So the question is, what kind of a question should tourists ask this person to definitely know for sure how to get to the city of truth? So, what's the video? Think about this. Now, I will just tell the answer. The question is, show direction to a city where you live, where you live. So, let's just think about it. Let's assume this person is from the city of truth and he tells the truth. Then if we will ask him to show the direction to a city, he will show to the city of truth where he lives, right? So everything is fine. He will show direction to the city of truth. What if he lives in the city of lie? Well, then he is supposed to lie. Now, we are asking him to show direction to a city where he lives. So, he will not point to the city where he lives. He will lie, which means he will point to the city of truth. So in both cases, whether he lives in the city of truth and then he will show to the city of truth, or he lives in the city of lie and he lies about where he lives and shows direction again to the city of truth. We will have the direction to the city of truth. That's it. And the problem. So, let me just tell again. These are not really like requiring theoretical knowledge problems. These are just common sense problems and very easy to solve. All you have to do is just to think about it. Next problem. Okay. The next problem is a very old one because I remember myself actually. Somebody was asking me this problem when I was in school. That was like 60 years ago, maybe more. Okay. So, we have the man. We have the wolf. We have goat. And we have cabbage. Now, in the presence of man, nobody eats anything. But as soon as the animals, whatever they just bite themselves, then wolf can eat goat. Goat can eat cabbage. Now, we have a river with two banks. So, all these guys, man, wolf, goat and cabbage are on this side of the river. There is a boat here. But the boat can hold only a man who basically does the job to roll and to go to another side of the river. And one of these guys, either a wolf or a cabbage or a goat. So, he has to go to another side with all these animals and cabbage, whatever, so that nobody kind of eats anybody else. Everything on this side should be whole. So, question is, how can he do it? What's the sequence of actions? I mean, obviously, if he will take man and a wolf to another side, while he is on another side, goat will eat cabbage. Some obvious things which cannot be done. So, question is, how can he do it? Alright, so, again, pause the video, think about this, and I will just tell the solution. So, the first, obviously, wolf doesn't eat cabbage. So, if he will take the wolf with him to a boat, everything is safe. Okay, so first, from A, man and goat will go to B. Now, what's next? Well, obviously, it doesn't make any sense to get the goat back. So, the back man should go by himself. So, from B, man goes back to side A of the river. Okay, now we have a goat here, and man, wolf, and cabbage here. Okay, here actually we have two ways to do something, but I'm suggesting the one. So, the man should take the wolf and go to side B. Now, wolf is there, and wolf. So far, in the presence of man nobody eats anybody. So, he takes the goat and goes back. So, now, from B back to A, the man and the goat go back to A. So, now we have only wolf on the A side, and man and goat are here. Again, in the presence of man, goat doesn't eat cabbage. Wolf is just by himself. So, now, he takes the cabbage and goes this way. So, from A, man and cabbage goes to B. So, now, we have man and cabbage. Nobody eats anything. So, man can go back by himself. Wolf doesn't eat cabbage. So, man from B goes back to A. So, now, he's here. The wolf and the cabbage are remaining on side B. Wolf doesn't eat cabbage. Everything is fine. Now, he takes the goat and goes, and that's the end. So, these are steps to get all participants in this trip across the river. Okay, next. Okay, next problem. So, there is a revolver. Revolver has potential for six bullets, and chamber can be spun. Okay, now, there is only one bullet in the revolver. So, the chamber is spun, and the man is trying to hit the target. Well, he does a shot, and there is no bullet comes out. Okay, so it's only one bullet out of six, right? So, after they spun the chamber, he makes one shot, and there is no bullet. Now, the question is, should or should not he spun the chamber again to shoot the target? What is the better chance for him to make a real shot? Whether he's spinning or not spinning the chamber? Again, think about this, pause the video, now I will tell the answer. Now, let's just think about it. Whenever there is one bullet, you spawn the chamber. What's the probability of shooting the real bullet? If there is only one bullet out of six? Well, there is one out of six chance. So, it's the probability one-six. Now, if he does not spin the chamber, he knows that he has already made one particular shot, and the bullet was not there. So, there are only five remaining bullets. So, the probability that the real bullet is one of those five places, one-fifth, which means this is a better chance. So, he better not spin the chamber again, because that would increase the probability to hit the target. Okay, that's a very simple problem. You just have to realize what is a probability. The probability is basically a fraction, which gives you how many successful chances out of all chances are present. So, the number of all chances after the first shot, without spinning the chamber. So, the total chances is five, and the number of successful chances is only one, so it's one-fifth. So, if you spin, then you basically lose the information which you have. The information was that in that particular chamber which you just used, there is no bullet. So, now if you are spinning, you are basically disregarding this information, and the less information you have, the less chances to win, basically, you have. Okay, next. Okay, so you have two players, player A and player B. They are playing the game where either A wins and B loses, or B wins and A loses. They have agreed that if somebody wins, it pays one dollar to... I mean, if somebody is losing, sorry, then he is paying one dollar to the winner. So, the loser pays one dollar to a winner. Now, they both have initial capital of one hundred dollars. Both of them. Then they play at the end. Here is the information at the end. A won ten games. B has a hundred and twenty dollars. So, at the end, this is the result. A has won ten games. B has a hundred and twenty dollars in the pocket. The question is, how many games were played? What's the video and think about this? Now, here is the answer. Again, absolutely common sense answer. Let's just think about it. Obviously, the net result is, since this is twenty, that B has won twenty games more than A. That's obvious. But at the same time, A has won ten. So, how many games B should win to still have twenty dollars extra? Well, if he wins ten games, they should win thirty games. That would give him twenty dollars extra, which he has. And the total number is forty. So, they have played forty games. Ten games were won by A. Thirty games were won by B. And that's why he has a difference of twenty dollars. So, the net result is A has lost twenty dollars and ended up with eighty. And B has won thirty games. And that's why his total is a hundred and twenty dollars. So, total is forty games. That's the answer. Okay. As you see, all these problems which I'm presenting right now are, well, kind of real life, basically. I mean, obviously it can be presented in some other more real kind of conditions. This is still kind of logic, which is part of the mess. But again, the conditions of all these games do not require any kind of special mathematical knowledge. Just the common sense. Okay. My last problem is about three wise men. So, you have three wise men, A, B and C. Now, it is important that they are wise men. So, they have certain common sense, logic, whatever. And think about certain things and basically come to logical conclusion. So, they were discussing something very important, some very deep philosophical problems, tired and basically fell asleep. And it was actually happening somewhere in a square, let's say, in a park, whatever. So, these three wise men are asleep. And while they are asleep, some joker actually decided to have a joke, basically, on them. And he took the shoe wax and put some black wax on four heads of all of those three wise men. So, each of them has some black spot on the forehead. All right. Kind of funny, obviously, but in any case. So, they woke up at the same time. And each one saw the black spots on the four heads of the two other guys and basically started to laugh at them. Yes, how funny it is. The person has a shoe wax on his forehead, right? Okay. Then, all of a sudden, one of them, let's consider a who was just a little bit wiser than the others. He stopped laughing, realizing that the wax is on his forehead as well. Purely logical. Without, you know, touching anything, etc. He was just thinking and he came to a conclusion that the shoe wax is on his forehead as well. And he stopped laughing and went to the bathroom to wash his forehead, basically. And everybody then realized the same thing. So, the question is, what do you think his logical thinking might be to come up with this particular conclusion that his forehead is also with a wax spot? Okay. Now, pause the video and think about his logic. And here is what I think about this. A can think the following way. Alright, let's assume that I'm clean. Assumes is clean. So, I don't have anything on my forehead, but B and C do have because, you know, I see them, right? So, then, B is a wise man. Definitely not a fool. So, what does B see? B would then see that the spot is only C with a wax and A is clean. So, what does B think in this particular thing? So, what A thinks that B thinks in this particular case? Well, since B, assuming A is clean, since B sees only clean A and shu wax on the C's forehead, and he is laughing and C is laughing. So, B is thinking, okay, what is C laughing at? A is clean, so C must be laughing at me. And then B would immediately realize that the wax is on his forehead as well as on C, and he would stop laughing. But he is not stopping. He is still laughing. So, what is he laughing at? He is laughing because my assumption is wrong. That's why A has decided, and as I said, A was just a little bit wiser. So, he came to the same conclusion, to this conclusion a little bit earlier than others. But in theory, everybody else can think in exactly the same way. So, whoever is a little bit wiser came to this conclusion first. So, A came to this conclusion. Assume it's clean, then B would immediately see that the wax is only on C and not on A, and he would stop laughing because he understands that C laughs at him. Okay, so that's basically the logic. I think it's very important to come up with these solutions just by yourself. There are tons of problems of this type, and I will try to present more. But in any case, they are absolutely common sense. They do not require any special mathematical knowledge. Just good thinking. And that's exactly what I'm trying to encourage you to do. I think all other problems which do require some math knowledge are also very important that you are not just listening to the lecture, whatever I'm presenting, but you do it yourself. You stop wherever I finish presentation of any problem, not only logical problem, and then think about this yourself. You see, my initial problem for the whole course, which I'm presenting on Unisor.com, was to basically present a lot of very difficult problems. Not very difficult, but more or less non-standard problems. But then I realized that to present the problem which has a mathematical contents, you need the mathematical theory. So you need all these theoretical lectures, and that's why a big chunk of Unisor.com website is Math 14, which presents the pure mathematics with certain problems, but the problems are more with theoretical character. So that's why I had to really finish up that course, which is done basically, and you're welcome, obviously, to take any lessons from there. And then I started to do these type of non-standard problems which I basically call Math Plus and Problems. By the way, Unisor.com is a totally free website. There is no advertisement, no signing is optional, so you don't have to do it unless you're starting under somebody's supervision. So everything is for your consumption. Well, knowledge is power. So gain the knowledge and study mathematics. And by the way, Unisor.com also has Physics 14's course and even Relativity for All, so-called, where I present certain special Relativity concepts. Okay, that's it for today. Thank you very much and good luck.