it's time let's start welcome to another mad kangaroo webinar session i'm able to see that or try to see that you are here anyway today we're going to talk about logic problems and before you even start that one the warm-up question i would say you know like sometimes in life you have like some of the people that who always think about the worst case scenario you know they always look at the you know something bad happened or whatever so you can't think that way today when you approach this type of questions think about the worst case scenario what worse can happen when you choose those different color of balls let's see the least number of the balls, she has to take out her bag. Of course her eyes covered to be sure that she took out at least one ball of each color. Let's start timer guys. all right people keep joining let's just start over guys so i would say for this type of questions guys think about the worst case scenario please okay the two people who answered b okay we get b so far what about others i believe someone just chat direct message to me and say b i don't know if this is the same person guys can you please send your message to soham i might not necessarily able to check the text all the time you know so we can go faster i can kind of try to focus the teaching and having fun while you can send the chat so and he can check the you know chat box and tell us how many people says which one should be the right answer so far we have one vote for b what about others oh one of the b people changed their answer to a so far one for a one for b okay all right i just start timer we have 15 seconds right now okay the other person changed to a all right look guys we all know that we have more than 20 in this classroom so what about others i mean you can guess it's fine or type your reasoning to the chat box so we can interact and learn from each other it has been 42 seconds by the way Okay, please type your answers in the chat box and share with Soham. Then we can just see. I mean, you can just put some sort of shortcut or summary how it works. Okay, it has been one more than one minute anyway. Let's see. If A is correct, I wonder, are friends reasoning? Okay, great. A lot of people are joining today. All right. For our friends who just joined, guys, please think of the worst-case scenario here. What worst can happen while that kiddo Ada tries to just pick up the balls from the bag? So, do we have any answer or I'm gonna just start explaining anyway? It has been five minutes. Yeah, we don't have any more. Okay, I believe someone sent me a chat. Someone else says A now. Okay, guys, please make sure you direct message to Soham so he can check the chat box effectively. Okay, we have three A's so far. All right. So guys, here's the deal. Let's say, I told you before, it's going to be just, we are going to just work with the worst-case scenario. Let's say, she just starts picking up the balls and she always chooses the gray balls, all seven of them. You know, it might happen. Since we have four white balls and three black balls left over, let's just say she's so unlucky and keep picking other white balls. She just picked up every single one of them. Then, of course, we are going to have only three black balls in the bag. And they say, what's the least number she can take out from the bag? So she took at least one ball from each color. Well, if I just add one from that color, I should get the total. So when you add them, guys, you'd have 12. Okay? Some of you may think, you know what, I can just start with, you know, just get one from each, then it might be three. It's not that simple, guys. Remember, we have no idea if all of those colors are going to be different or not. Remember, you got to think the worst-case scenario, okay? Any questions about this one? I believe we are good. Otherwise, Suhan would tell me I'm going to move on, guys. As you see, we are almost done with our 10 sessions here. So what's logic? Logic is a branch of math which we use reasoning to obtain results. For example, we have deductive reasoning. Deductive reasoning is the process of drawing conclusions based on given premises. A premise is a previous statement from which another statement follows as a conclusion. So we get some conclusions. In terms of that conclusion, we're going to get some ideas. The pigeonhole principle or certainty. If n items are put into m amount of containers, which n is greater than m, then at least one container must contain more than one item. There is no other case, you know. So as you see, we have like four of those pigeons and we have only three of those cages. So if you're not going to leave any of those cages empty, at least one of them has to be more than one. Picking matching stocks is a common problem that can result within the pigeonhole principle. Think about the worst situation, remember I just mentioned from the warm-up question, that prevent the desirable event to happen. Then after all the bad situations have occurred, the desirable one must follow. There is no other case. All right. Cryptorhythms, an arithmetic problem in which letters have been substituted for numbers. Instead of letters, I mean instead of numbers, you are just using numbers, guys. And which is solved by finding all possible pairings of digits with letters that produce a numerically correct answer. As you see, we need to focus here that each different letter represents each, I mean, different letter basically represents different numbers here, guys. You cannot say s and e, they both represent number eight. It's not the case. Solving the type of questions requires strong number case, understanding of digit places, you know, ones, tens, hundreds, and so on. Understanding of carrying over or regrouping in addition, subtraction, and multiples, pattern and ending digit in multiplication. Using guessing and checking and elimination also another method you can use for this topic, guys. Knights and knaves. Knights always tell the truth while knaves always lie. There are problems in which you do not know who is telling the truth or lying, but are asked to determine the results or conclusion based on the given conditions. Because we are giving like four, five, or depends on whatever number of conditions, and you are going to just check each of them, maybe just try, then you are going to make some conclusion and find the answer. You can eliminate answer choices, or you can eliminate some of the items we have given. A common approach is using the casework by assuming certain statements are true, false, or assuming a certain result, then testing for contradiction. So we gotta talk about this one, guys, no worries. There are problems like that. Logic puzzles. In a logic puzzle, every item is matched to one and only one other item in each category. To solve the problem is to find the matches using the given clues, deduction, and eliminations. A chart can be used to track the deduction results. Yeah, for those questions, please create some sort of chart. Because I used to tell my students, when you have like even like huge paragraph of questions, you gotta make those given information some sort of organized, or you gotta organize given those information, guys. If you don't do that, it's gonna be really confusing. You will see that. Anyway, let's just move on. All right, first one from Metcalfe to 2021, and question number 10. Have fun, guys. I'm gonna start timer. Remember, I gave you a hint about those questions. If we have given a bunch of information, to be able to not get confused, please try to create a chart or diagram, you know, kind of some sort of visual, so you will see the relationship between, you know, ages of, no, heights of those people. We are two people for E. Okay. He has two so far. Third person says E. Look at that we have plus one. It has been two minutes guys. Can some of you please share what they get or I will just start. Just give us as a summary how you find that E if it is the right answer. Or you can write in the chat box. I would start with that method you know. At least we would know how you started the question. Or so I will if there is no one would like to share their ideas with us, I'm gonna share mine. Well, someone gave the reasoning with equations. Okay, how can we create equation? I mean, so suppose ABCD and E are the height of Aaron Byron, Karen, Darren and Aaron. So B equals A plus five, C equals B plus 10, D equals 10 plus C, E equals five plus D. Then do we just plug them into another equations to find something? And then we just plug them in and then find E in terms of A. And then it's E is 30 plus A. Oh, okay. That works. So I was kind of thinking different ways. I mean, of course, different. That works too, guys. If you want to work with that, you can do that too. So I would kind of create Y axis of number line, you know, since we are talking about their height, I just put it that way. So it says Brian is five centimeters taller than Aaron, but 10 centimeters shorter than Karen. We know that with that statement, Aaron is the shortest one. Okay. So, and also after the first sentence, it says Darren is 10 centimeters taller than Karen. Darren is even going up, but five centimeters shorter than Aaron. Aaron is the, you know, with the people with highest. So Brian is five centimeters taller than Aaron. I am going to just put here A. And since it says Brian is five centimeters taller, I will just put here B, and we have that five centimeters. Okay. Then it says, we still are talking about Brian, that person is 10 centimeters shorter than Karen. So maybe I should just put here C, remember, because in that case, that Brian is 10 centimeters shorter than Karen. Okay. Darren is 10 centimeters taller than Karen. Oh, look at that. Darren is even higher. I just put here 10, but Darren is five centimeters shorter than Aaron. That means Aaron is going to top place. And this is a five centimeter. So the question here is, I can ask you, does it make sense? As I told you, I just read the question and make the visual for you to understand easier. I believe there is no question. As you see from this chart, guys, you are not necessarily only comparing Aaron and Erin. So which is, you know, tallest and shortest people, you can also compare others' heights from only looking at that chart. Okay. And then, of course, you got to realize that five plus 10, plus 10, plus five would be difference between their heights of Aaron and Erin. So instead of putting in that order, you know, I will just make this five here to be able to add easily, because it's going to be a time test, remember. And you just get to it. Okay. Just make sure you count every single one of them. And the answer was A. Any questions so far? I believe you would say yes if there are any. I mean, in the chat box, not to us. All right. We are going to move on. Next, what about number two, which is from 2018? Question number 17. Remember, here, we are going to think of the worst case scenario. Okay. Pre-check the question and the start timer. Okay one person already said A. Oh okay. I was gonna say besides giving us answer please just summary your reasoning as well in the chat box guys. We have 30 seconds so far. We've got one more for a okay Guys little hint remember you need to think of the worst case scenario And now someone says II look at that, okay 17 through 35. Okay, right off different thinking it has been one minute and 20 seconds guys Please share your reasoning like in a summary Why do you think it should be a or e But what others no one wanna choose B C or D That's it. You can always change your answer. Think of that way I am saying if you want to change your answer, why not try that? A sound one says D. Okay, good. You have other answers to Again, whoever says a or e Do you think you may want to change your answer guys? It has been two minutes anyway Maybe I am literally telling if you think answer is a or e You need to think twice so d no another person changed their answer from a to d okay look at that cool all right guys let me just try to explain the possible of your reasoning whoever says a look we know that 2018 is an leap year so it's going to be there is going to be 365 days am i right i believe you all have to figure that out so but it says with the advertisement 350 days of sun every year well when we look at the difference we are going to have 15 days with no sun let's say if we think of all of the no sun days are consecutive i am trying to explain how you can get 17 then of course we can get at the end two more sunny days 15th year by no sun plus two sunny days it will be 17 but do you think with that logic we are thinking in worst case scenario no think about after starting with one night or no sun whatever if you had two sunny days we would just have the you know we've just sold it we can just have three days but what if i just keep pushing what if guys we have no sun sun no sun sun no sun oh my gosh look at that even if i start with sunny day to make even more worse so you are going to have those two groups sun and no sun together since i had 15 of those nights since they are group of two at the end guys i am gonna end with no sun but this is going to or 15 groups i should i should say with that case guys we are going to have 30 days remember these are a group of two well you know we just keep going we already spend all of those no no sun days in one month you know then of course we can have two sunny days because there is going to be no more no sunday so as you see we had 30 plus two days here answer was 32 guys i hope it does make sense if it is not please ask us again yeah when you just check the question first you would think it's 17 i got you but there's a reason we put that in a first answer choice for you to just choose 17 you know any question guys i guess no question i'm gonna move on that great i know you would ask me in the chat okay check number three please It has been 55 seconds, guys. And let me underline this first sentence. We have to work with one of the logic here. 1 minute and 40 seconds, guys. no answers that's all right it has been two minutes i can give you the full answer or instead let me just give you a hint guys look let's work with the first sentence it says there is always at least one green counter among 27 of them if there is at least one of them is green among those 27 can i say that at most 26 of those containers can be red plus blue plus yellow if you take out one of the green does it make sense you know we can have red plus blue plus yellow should be what should i put greater or less than or sign remember i said if there is always at least one green i took one of them but we have left over this might be the sum of red plus blue plus yellow should be at most 27 okay okay so i'm gonna say at most means what do you remember can i say less than or equal to no answer guys you gotta remember that too at most means less than or equal to remember i believe some of you remember then we should have 26 okay please go from there guys check the second sentence another one and try to find some sort of all of those system of inequalities then you gotta figure out something i hope Okay let's just check second sentence together. It says always at least one red among those 25. We're gonna get rid of red right now. We will say green plus blue plus yellow should be less than or equal to not 25, 24. Because we assume that one of them should be red anyway. With the same logic guys can we say always at least one blue among 22. So we gotta get rid of one blue that means we gotta have green plus red plus yellow should be less than or equal to 21. Remember one of them is already gone. Okay let's check the last sentence. Always at least one yellow among 17. Well one of them we just kick out from those group not no more one yellow we have left over 16. And since we take out yellow the remainder colors are green plus red plus blue. After that what we are supposed to do? By the way, I'm glad to see that no one says 87 or 91. Just add them all, you know, or take out one from here. What is the largest number of counters that could be in the box? You need to remember how to solve system of inequalities here. Remember, we can eliminate, we can substitute, we can combine the equations, add subtract. I don't recommend you to just try to graph. It wouldn't work. I mean, if you use space, that would be bad. From this case, it wouldn't work. Anyway. OK, guys, as you see, we have only one red, one blue, one yellow, or one green. Only one is a quantity. Do you think we can just add them all? I will just try that. Since all of those numbers before that is the same thing, just one, you know. OK. How many green do we have? One, two, three. As you see, how many red do we have? One, two, three. Among those, how many blue we have? One, two, three. And how many yellow? One, two, three. So can we say we have three times red? Let's just start. Order green, plus red, plus blue, plus yellow, which gives us less than equal to, when you add them, you should get 87, if I'm not mistaken. OK. Then what's next? I believe you should be able to find the last, you know. Oh, I just get that. We have one for b and one for c, OK. Thanks so much, Soham. I just realized that. Guys, as you see, to be able to solve an equation of inequality, if you have the positive number multiplied, we can just divide to cancel. So divide both sides by 3, OK? Well, that gives us green plus red plus blue plus yellow is less than or equal to. It should be 29, I believe, yeah. And that also says that what is the largest number we can have? Largest number we can have, 29 guys. Any question do we have for this? Look, it was number 24. You may feel it's a little bit difficult, but it's all right. We are just doing practice here. Well, I would recommend you guys, you just change some of the numbers and try this one by your own. And if you get stuck, you can always ask us, OK? It's fine. All right, I'm going to move on. I didn't hear anything from you, at least not yet. All right, number four, another logic question. Think about how multiplication works. If you cannot find it, I'm going to give you a hint. But before that, let's just start timer. All right, one minute, five seconds. Soham, do you have any answer? Some of the friends says. Two people say B. OK. All right, guys, let me remind you something. Look, as you see, the result is 300 something, because the 100th digit is 3. And second number is 20 something. Look, even if you multiply 20 by 20, you would get 400. Am I right? I mean, think about the multiplication. These are just zeros. It's not zero, but I'm just giving you the example. Since this number is less than 400, for the first little box here, represents 10th digit, I believe it's going to be 1. There is no other option, as you see. So if this is the case, then, guys, I really have to put here 1. Then I'm going to keep working. I will get 13 times 2 and box. And as you see, at the end, the last digit here is 2, guys. Let's see. I'm going to put here two digits here and two digits here. When you add them, remember the properties of adding or multiplying, we will get 3 and box and 2. OK. Think about, if this is 2, this 1 has to be 2 as well. 3 times what number give us 2 or 1th digit 2 or, you know, 20-something is going to be 2. At the end, we are going to have either 3 times something is 2 or 12 or 22. We cannot make it 32, because it's greater than 3 times 10 anyway. So as you see, 2 and 22 is not going to work. The only option is to multiply it by 2. The only option we have 12 here. We all know that, guys. Only 3 times 4 gives us 12. Well, since 3 times 4 is 12, we have 1 extra. 4 times 1 is 4, plus 1 is 5, as you see. Well, we can just keep multiplying. 2 times 3 is 6. 2 times 1 is just 2. So yeah, I can change the color anyway. We just keep the 2, guys. 5 plus 6 gives us 11, you know. But I am going to only put 1. And we're going to carry over extra 1. So we get 3 times 12. As you see, from the first box, guys, we found that it has to be 1. There is no other options. It's the first one. Then we find that the second number should be 4. And the last number should be 1. Then you add them. 1 plus 1 should be 6, guys. Thanks to two of our classmates, they already find 6. Any question about this one? OK, I believe we are good. We got to move on. All right. And this is going to be number 5 from 2009 and number 17. Let me tell you something about this question. The question is not necessarily difficult. But you just need to remember or think of only one property. If you get that, then you are good. OK. Let's see. Let's start the timer, by the way. It has been 12 seconds, come on. Just use regular multiplication ways, you've got to figure out something. Each letter represents different digit, guys. It's the same letters having the same value. How many different values can the product 3 have? All right, one minute so far. So do you see that we have any answers or not yet? None yet. That's sad, man. Come on. I'm going to give you more. This one is easy. I mean, I told you, it seems difficult, but not that difficult. You just need to remember one properties with multiplication. What works? Look, keep thinking numbers. OK, after one minute, I can just give you some hints. Look, we have A times I, G times H times T is equal to, you know, when we just do kind of cross multiplication, it's going to be F-O-U-R times T-W-O. Look, guys, we have a lot of different letters. I would say, first, you've got to count how many different letters we have. Then think about some of them might be like three, some of them might be seven, some of them might be five. And there is an equal sign here. How? How are we going to make that equal? Think about that way. Also, I can say, if you like, you can cancel these Ts. If you like, if you don't like, please don't. All right, because we have only multiplication and these are on both sides. What I am saying is, unless you remember one of the properties, it's not really easy to find that equation. Then what are we going to do? It is two minutes and 40 seconds, man. Man. Not yet. So I'm not reading. Should I tell them or no? Should we just keep it as a secret? Let them keep working. Oh, they are still working? Yeah, they're still working. OK, I can wait a little bit more. Because, guys, you can figure it out. Come on. Think about it. There has to be a way. So those products are going to be equal, but how? I might have three here, nine here, but still, I mean, three times three, nine. Maybe I can just, O is three and three, and E is nine. But what about seven? What about five? How do you cancel to make it equal? Or do you think we really need to think of finding equal or use some other rules? Well, one person says A. All right. We have A, what else? What I mean? I mean, what other answers? I meant who else, but... Okay, someone says A, that's great. Anyone else wanna say A? That's good, A is good. Guys, I gave you a lot of hints. Okay, so whoever says A, I really want that person to explain how they get A. Why do we have only one values for the product 3, half? I mean, T times H times R times E squared is half. Why do we have only one? Is T equals zero? Exactly, yes. Guys, look, there is no other way. I am so excited and at the same time sad. Come on, man, how come you couldn't figure it out? But at least one friend figured it out. Look, unless you give guys T is zero, not O, it's zero, you know, my way is zero. There is no way you can make that type of equation true. That's the reason I was just telling, think about seven, seven only cancels with another seven here. It's not possible. So think about other ways, you know. All right, anyway. I believe we are good. All right, next one, next one, come on. Don't give up, we are good. Okay, from 1999 to 28, this is one of the hardest questions. So when you just use your, you know, logic, please be careful. All right, I'm gonna shut up, then you can just work on that. Let's start timer. We had 31 seconds, guys. We will only add, remember, it should be relatively easy compared to multiplication. What is the largest possible of the results? Largest possible. All right. It has been 1 minute 20 seconds. To be able to get largest possible of D, R, E, I. Maybe you can start giving D, which is thousands value. Look at the answer choices. Oh, we ought to give you something. Look. D and R, 9 and 8. Anyway, look at those. You see? We are helping you. I told you. Okay. Then you gotta tell me it's number 20. It's not. You have only two digits left. I mean, we get 63, 73, 74, 75, 76. Come on. Many of them also include 7. Maybe we can, you know, vote for 7. Think about it, guys. It has been 2 minutes and 8 seconds. Come on. I'm gonna stop the timer. Let's see who gets something. Imagine you are solving one of the hardest questions from 1999. Was it 25 years ago? Wow. Quarter century ago. That's a lot, man. Wow. Many of you were not even here 25 years ago. That's sad, man. Anyway. All right, guys. Soham, do we have anyone says something? They say B. Okay. Someone says B. Two people says B, you are saying? Like that? Okay. That's good. Anyone else? Okay guys, if two of our friends says B, I would like to say, congratulations for those two friends, you get it right. But can you give us some explanation? It has been more than two minutes anyway, I'm gonna start explaining. That means we get nine, nine and eight, without even doing too much of, you know, working. Let's say guys, I would start with nine, eight and seven. Don't worry about answer choice, okay? Because at the end, our goal is to find largest possible, okay? For I, we don't know what's that. So it gives us what? We know that D is here and here, it's going to be nine. And we have O plus E gives us eight, am I right? So E is all of the same, I just put E as you see, I can just plug it here, like little tiny. But remember, I cannot make this letter of O more than two. If I just make this one as three, three plus seven is 10, then D is not gonna work. Remember, we just keep the D. Well, so we don't have too much of options for this one. And we got to get eight. Well, seven plus one is eight, you already know, I cannot make 18 anyway. So we found O is going to be one. What about other parts we cannot use? Oh, again, we have, let's just put E here as well, seven. Okay, seven plus something is going to be O. What else we can think of here, guys? I really don't wanna think, guys. Whoever found B, they just guess. I really wanna hear your opinion. So seven plus- Okay, X is six. How do we get X is six, guys? Because we don't have too much of numbers left, or what do you think, guys? You get the largest one left. You are right, exactly. We can put here six. So it's going to be 13, guys, as you see. Then I is going to be three. So we do not know about N and U, but we do not necessarily know that anyway, because think about that way. If N plus U is six, because we have extra one carry over here, as you know, if N plus U is six, N and U, either of them can be two and four, and these are the remaining numbers. Remember, I'm just also checking them, because after I assign them every letter, I mean, each letter represents different digits, basically. I need to assign them carefully. So that's the reason it should be nine, eight, seven, and three. If you try to choose other numbers, it's not going to work. I mean, you can try if you like. Then you have time. Any question about that one? I believe we are good. Okay. All right, guys. What about number seven? As you see, we have three different rooms with three different doors, obviously. So I would just try to organize information, create a chart or picture or whatever to be able to make it easy to solve. Okay, time starts. Have fun. One person says A, one person says E. Okay. Alright. One A, one E. And one B. Look at that. Okay, Soham, let's just make it. You say C and I say D. We have every single of them. No, I don't. I know you don't want to say C. I'm just messing with you. Okay. Alright, guys. Seven was from 2018 and number 14. Not necessarily too much hard, but it's in the middle. It has been one minute, 20 seconds, guys. I am going to start giving you some hints. Look, as I literally told you before, I would just say, guys, door one, you know, door two. Someone changed their answer from B to A. Okay. Door three. This one is zero and this one is plus one. Okay. Oh, I would put minus one. Anyway, you got the point. Okay, guys, these are three doors of the rooms. The line is here. I'm just going to put it here, guys. Another one says not here. We all know who I am talking about. That little line. We got another vote for A. One plus three is equal to three times. No, two times three. I can put three times three. It's the same thing anyway. You got the point anyway. Okay. Oh, you said one more person says A. Okay. Guys, what if I say answer is neither of them? Look at that. A, B, and E. So what? What if answer is C or D? Can you think of one more time? Only one of these sentences is true. Now someone says C. Look at you. All right, we get C. And whoever says C, that person got the big reward for nothing. I cannot give anything virtual, but you got the point, buddy. Okay. Someone else said C. What was that? Someone else said C. What was that? I'm sorry. Look at you guys. Two of you. Oops. You get cool logic here. Look, guys, let's just go with A. They said only one of these sentences is true. Let's just say answer is A. That means room one is true and line is here. Look, if this is true, that means two is also true because line is not here. The question, do you think we can get two of those true sentences at the same time? No. It says only one of these is true. That means we cannot have A. Three clue says A. How? Guys, you missed the point here. I am sad. Okay. What if we just assume it's room two, you know? If the line is in room two, that means not here. That statement is wrong. Here, that statement is also wrong, and that statement is also wrong. We both know that five is not equal to six. So all of them, every single of them are lying. Or these are all told by knights. I mean, names, not knights. So not B. What about C? Let's just try C. Let's say line is here, guys. Think about that. So line is not here. That's true, you know. I mean, I would say here. It might confuse you. Let's say line is here, guys. So not here is true, yes. And here is wrong. Also, even line is here. Whatever they put that equation, it's also wrong. As you see, we have only one true statement if the line is in the door. I mean, room three. Does that make sense, guys? Okay. It may be in any room. No. I mean, of course it's going to be in any room, but we are looking for a specific room, which is that. Anyway. That means it cannot be in any room, guys. It has to be. That line has to be in room three. All right. Almost there. I can just give you one more question. Maybe we can have more fun here. Come on. You ready? Number eight from 26. I mean, 2010, number 26. Come on. Quickly, please, guys. Please, just try to use the same methods. Assume that first person is a liar or first person tells truth. What's going to happen there? Go from there. Or second person interprets. Just keep working with that. It is 26. It might not be that easy. Anyway. But I know we can just solve it. One person answers C. All right, that's good. Guys, can that one person tell us, because I actually don't want to take too much of your time, guys. How do you find C, but can you just, you know, type really quick? Then we can go from there. Another man also answers C. Okay, all right. We have two people says answer should be C. It's right, guys, by the way. So the first, assume that the first person is a liar. Okay, so let's just check really quick guys. Let's say if there's only two option for each person, either liar or telling truth, you know. So if the person is telling truth, guys, think about that way. If this is true, there are no more than three of us in the room and each of us liar. Each of us liar statement should be true, but we are saying first person tells truth, it's a contradiction, it's not possible. So first person is a liar, look at them, okay. What about second one? Second one says, I should say L, not false. I meant to put L guys, sorry. Yeah, liar, okay. So second person, there are at most four people in the room, not every one of us is liar. What do you think about this one? If this is true or false? So do we have more answers? Not yet, that's sad. Think of that guys. Let's say second person tells the truth, so what? Okay, there are, that means it should be, there are at most four people in the room and not every one of us is liar. That means some of us liar, some of us tells truth. If it is true, that seems right statement, but if it was false, that means every one of us, I mean, if that person is lying, that means every one of us is a liar. It's also liar, it's kind of contradiction. Second person has to be telling truth, okay. That means also there are four people. Okay, third one says, there are five people in the room. We already know second people has to say truth. That's not right. That means this one is liar. This one is liar and that person says, all of us are liar. If it is not true, yes, you know, it doesn't make sense. So far, we have to have four people in the room and two of them has to be liars. Just check each of them with liar or telling truth, okay. I believe there is no question. Is there any, I can just check really quickly. None. All right guys, so far, I enjoy our class. I hope you enjoy that too. And I'm gonna see you next time. Take care, bye-bye.

logic problems

worst-case scenario

pigeonhole principle

cryptarithms

deductive reasoning

truth-tellers and liars

charts and diagrams

systems of inequalities

elimination strategies

interactive problem-solving