Hi, good afternoon. Hi, my name is Jing Qing. I'm really pleased to be able to spend an hour with you on a Saturday afternoon in New York. It might be early morning on the West Coast. And I'm joined here with Noah. So today we are going to do a special topic on logic problems. Now, I know you guys are a bunch of elementary school or early middle school students. So it must be interesting to you to see that we're working on a lot of problems. Because the concept of logic in math is usually not taught until high school. So we're actually doing this kind of early. But it's so much fun. And it's really helped us with problem solving and possibly doing well in math competition. So that's why we're doing here together. So logic problems. And we're going to start with just some housekeeping and just kind of tell you what is available for you. So even though this is a webinar that we cannot have students directly participating, but we are going to have interactive time with you guys. Because we created these polls on some of the problems. It's really fun. And you can do the problem. And you can vote on the correct answer as you see it. And then we're going to continue on to kind of review a bit on the four step problem solving method. I'm sure you guys are all familiar with it and have used it before. We're just going to try to see how we can use a problem solving method to help us to solve the problems more efficiently. And then we're together going to work on about 10 problems together. So we are going to go a little fast today, just because we want to make sure we cover all the 10 problems. And these 10 problems are carefully selected. And they're really fun. And they're really revealing and informative. And so hopefully, we can get to all of them. So let's just continue. So obviously, this is a webinar, which means that your microphone is going to be disabled. But you still have a chance to participate by putting comments in the chat to Noah, who's a gold medalist of math kangaroo competition. He's going to monitor the chat room and respond, if possible. And also, we can't get to all of your questions, obviously. But Noah will post some resources that answer common questions as they arise. So the poll feature is going to be really exciting, because you can vote on problems. Now, sometimes, depending on how your computer is set up, when the poll comes up, it kind of blocks the screen. So you may not be able to see the problem on the screen. So all you have to do is go into that box, the poll box, and go to the top left corner. And you will see that there's a feature called a move. So you kind of click on move. And then you can drag the box anywhere you want to put it. So that way, your view of the problem is not going to be blocked. And do your best. None of the poll answers are going to show your name. So have fun. So I am a math kangaroo teacher and really enjoy teaching problem solving with math kangaroo with elementary school kids and some middle school kids, as well. I have been able to contribute to the online course that Math Kangaroo USA has rolled out this past year. And I really look forward to working with more of you in the future. Noah, over to you. Hello, everyone. My name is Noah. I'm currently in 11th grade. And I've participated in Math Kangaroo since around third grade. And I generally enjoy any kind of math. Thank you, Noah. Maybe just quickly, when I do math kangaroo competition problems, sometimes I don't get perfect scores. So what is your secret? What is your secret recipe for being able to score perfectly in math kangaroo competitions? Do you have any quick advice for our students? Time management is very important. Make sure that you at least go through every single question. So just in case if there is harder questions near the beginning, you won't get stuck on them. And if there's easier questions near the end, you'll be able to solve them. Also, since I'm pretty sure there's no guessing penalty on this competition, if when you're low on time and you still have questions unanswered, just randomly choose one. Because in that case, you'll have a 1 5th chance of being correct rather than a 0% chance of being correct. Great, thank you. I can use some of that myself in my other activities. That's wonderful. OK, so let me get my touchscreen pen out so I can write on the screen. OK, I got it. All right, so let's just quickly review the four-step problem solving strategy. In fact, I'm going to model for you how you use the four-step method to help you solve problems. Obviously, the most important step is understand the problem. I tell my students, feel free to spend more than 50% on average the time you spend on doing a problem on step one, which is really understand the problem, understand what the question is asking you and what the question is not asking you, because sometimes people get tripped off. The second step is do not rush into calculations. Before you calculate, always think about, OK, so how do I approach this problem? What is a good way to approach this problem? What kind of strategies, like tools, I can use to help me organize the data, help me link up the different pieces of information? So this planning is very, very important. And again, we're going to model for you and you see why the second step is quite crucial. Only after you plan out your strategy, then you go ahead and implement your plan and then you do your math. This is where your math competition comes in. And I would suggest that the key thing to do here is after you do every step of a calculation, you double check, right? Because even I sometimes say, oh, 2 plus 8 equals 11. Hey, we're just human beings. We sometimes make sloppy mistakes, right? But when you check, you're like, oh, no, no, no. 2 plus 8 is 10, right? So check after every calculation. Finally, after you solve the problem, you go back and check and think about, okay, does this answer make sense? If the problem asks for you the smallest possible number, right, and then you look at the five answer choices and the answer you have is the largest value and you're like, well, wait a second. Did I get it wrong somehow? So go back to reflect and think about if the answer makes sense. So this is the four-step method and we're going to model for you, okay? So what is logic in math, right? So now, of course, if you go to a math textbook, right, the definition, it can be like several pages long, okay? And we're not doing that today, obviously, right? So I'm just giving you a very simple meaning, right? It's almost like a working definition, right? So you just use this to kind of help you to do this session with us, right? So in math, logic basically means a set of methods of reasoning that can help you determine whether an argument is true or not, okay? So one of these methods is the process of elimination. A lot of you actually probably heard about it, right? So here's a sort of a abstract version of the process of elimination. It basically says, okay, if I think something is A, then B should be the result of that, right? And then you check against the problem. Like, okay, if A means B and B does not meet the condition that given in the problem, then you know A can be eliminated, okay? And you can continue doing this elimination by checking against other conditions in the problem. So this is commonly known as the process elimination. Now, sometimes you do the logical reasoning and it helps you narrow it down to two answers, okay? And from there, you may use some other strategy to help you land on the correct answer, right? But by and large, the process elimination basically means that you try this if A, then B, and does it contradict with the condition or not type of method, okay? Go ahead. This is also called proof by contradiction. It basically states that if you start with a guess and then you make logical steps that, like let's say that you know you've done correctly and then you end with garbage, that means you've started with garbage. Very good. So Noah, being such a strong math student, he brought up the concept of proof by contradiction, which is one of the main method in mathematical proofs, right? So this is really quite advanced stuff, right? I mean, high school stuff, right? And some of the children at your age group may not even know the word contradiction. So I like to say the example of a contradiction is to say, I am an only child in my family and I love my sister. That is a contradiction, right? Because if I'm an only child, I won't have a sister. And if I said, I love my sister, that's a contradiction, okay? So what Noah says, proof by contradiction, in fact, indeed is one of the most important skill you can have in mathematical proof, okay? So let's look at an example of logic problem and why logic problems are important for us. Okay, so you hear, you're looking at a cartoon and this cartoon really cracks me up, right? So on the left of this cliff, right? I mean, this abyss, you have group of people holding the sign saying, be aware climate abyss. So these are the people who believe there's a climate crisis coming up and human race is about to falling a cliff, like a hit really bad outcome if we don't do something about the climate change, right? So they're saying, be aware of the climate abyss. And on the other side, there are people who don't really believe in the climate change thing and they basically, there is no abyss coming up, right? So we don't have to worry about the climate change. Now, so one statement contradicts with the other statement, right? So one of them is not correct, right? So if you do not have logical reasoning skills, you're gonna be that, like the person in the middle who basically fell into the abyss because he said, oh, I can't tell which one is telling the truth and therefore I decided that the truth must be somewhere in the middle. And where you see where that got him, he fell into the abyss, okay? So this is the kind of a funny way to show that why our ability to do logical reasoning and to be able to tell truth from not truth is fundamental, not just in math, but also in our everyday life, okay? And you don't wanna be like the person that fell into the abyss, okay? And now coming to the math kangaroo competition, of course, working with the logic problem, and there are a lot of them at a level five, level six competitions, like it feels like almost half of them, working with the logic problems really helps us to build a logical reasoning skills. And the logical reasoning skills in turn, help us solve all kinds of problems efficiently, okay? And we'll see shortly, you know, what kind of problems that logical reasoning skill can really be, really be helpful, okay? So today we're gonna cover five main tops of logic problems, right? I know we're ambitious, right? That's why I'm talking so fast. So if you don't, if you cannot follow me, feel free to put some questions in the chat room and Noah will help you out. So we're gonna cover five type. The first type is cryptorhythms, cryptorhythms, okay? And by the way, a rhythm is a Greek word, which means numbers, right? So crypto is like the cryptocurrency, and it's made up with the crypto and crypt and rhythms, which means some sort of puzzles having to do with numbers, okay? And the second type is the certainty problems, okay? And the third is the Venn diagram problems, okay? And the fourth one is whodunit problem. It's like, you know, it's like the detective, like, oh, who did this? You know, who stole this cat? Or, you know, who, you know, and that kind of problem is a lot of fun. And finally the truth and lie problems, which is probably the most classic logic problems. So for each type, we're gonna do two problems, okay? So let's do that. Okay, so the first problem is a classic cryptorhythms problem, okay? So now I'm modeling for you step one. Okay, we're gonna read the problem carefully, and together we can understand this problem really well, okay? So you see some numbers being added up in the image, right? So in this sum, the same letters represent the same digits and different letters represent different digits. Okay, where's my, okay. So I'm going to now underscore the important words, okay, the keywords, right? So same letters means same digits. Different letters represent different digits, right? So X, Y, and Z, they represent different numbers basically, okay? And they're not just any kind of number, they're digit, right? Digit means they're bounded, right? They cannot be greater than nine, and they cannot be smaller than zero, and even zero only, it's not, cannot be in the first digit, right? Because in the first digit, then that digit just disappear, right? So it's basically, it's between zero and nine for a digit, okay? So the question asks us, what digit does the letter X represent? Okay, so I'm gonna give you a little time to think about how to solve this problem, okay? So step two, we're doing planning now. What is a good strategy for this problem? Okay, shall we do logical reasoning? Shall we say, if Y is this, and then that means that, and that contradicts with the statement, shall we try with something like that? Okay, so Noah, do you think it's a good time to launch the poll? There is no poll for the first question. Oh, there's no. Okay, so there's a poll on the second question. Yeah, okay. Okay, so that's fine. Yeah, there's a poll on the second question. Yeah. So, what is a good way to approach this problem. Right. So I kind of already gave you guys a hint, right, it's a digit is no greater than nine. Okay, so what can we you how do we use this clue. Okay. Okay, I'm going to do this. Alright, so we're going to use the strategy of logical reasoning. So condition. We know the condition digit cannot be greater than nine. Okay, so if z is two or more than y must be greater than nine. Okay, so that will would have contradicted condition A, which is digit cannot be greater than nine. So z must be one. Okay, so that I eliminated the possibility that z is not one. Okay, z has to be one, because the greatest value y can take is a nine. Okay, so if you add a single digit to a number of 90 something, the biggest value you can get the sum you can get is 100 something right so z has to be one. Okay, so now we know z is one. So how do we go from there. Okay, so if z is one basically the sum is going to be one, one, one. Right. And what could y be. So again, in order to get a number that's greater than that is a little bit bigger than 100, you kind of want y to be fairly big digit. Right, so let's try nine. Okay, can y be nine. If y equals nine, then that means 2x plus nine equals 21. Right, because you need a two carried into the 10th digit. Okay, so that it becomes 21. Right, I mean, sorry, it becomes 11. Right, so you want a 20 to be carried over to the 10th digit. Okay, to get to get 11. Okay, so that means x has to be six. Okay, so now let's check forward now we're doing step four. Okay, let's look back and check. So if I have six here. Okay, and six here. And has 99 here. What do I get. I get 12 plus nine equals 21, I write down one and carry it over to two and two plus nine was 11. Bingo. So it worked out. Okay, so through step four, I went back to check. And I know I have the correct answer. So, the answer for this problem is x equals x equals six. Okay, so far so good right okay now you get a chance to sink your teeth into a logic problem. Okay, let me clear the annotation here. Okay, so again we're doing step one here, right. In the addition problem shown in the picture, every square stands for a certain digit. Every triangle stands for another specific digit and every circle donates denotes denotes yet another digit, what is the sum of the numbers represented by the square, and the circle, I'm going to slow it down on the question. I'm making sure I'm solving the right question. Okay, so this is a classic critter rhythm problem. Now, why don't you give it a try. How about for one minute, I give you one minute. Okay, go ahead. Now I do have any further hint to help our students. Well, it's. I guess you could notice how the fact that this is addition, which you've already covered before so I don't think I have anything new to add. Okay, fair enough. Our students are capable of tackling this I'm hoping. Okay, if you would like to launch the poll this might be a good time. All right. Okay, so the way we do this, I hope it's okay with you guys is that, you know, because we're, we also wanted to cover all the problems that we have here. So if we see more than 50% of the participants have put in their answer we're going to close the poll. Okay, I hope that's okay with you. Yeah, so Noah feel free to share the results when it's more than 50%. Okay. All right. Yeah, more than 50% students participated in the poll. Okay, we've hit 50%. Okay. So, sure results, pretty much. Wow, Noah, what's your impression. This question is hard. It's like, we don't even have a bell curve here right it's look like it's kind of a evenly divided amount. Well, you know, possible answer so we even have 9% of the participants saying that the correct answer is not here. It's the mother answer. All right, so let's, let's do, let's look at this together. Okay, so I'm going to do. Let's look at this. All right, something that will speed along this question really quickly. It's this column right here. Oops. I want to include the two as well. This part right here. So three squares, plus some carry over something here equals 20. This instantly means that the squares have to be six, because if the squares were more let's say there's seven they would add up to 21 which is too big. If they were less like five, it would add up to 15, and we would have to carry over five up here which is not possible when we're adding only three numbers. Excellent. This is why I love have Noah in my team because we actually teamed up to teach an entire semester of online courses together I love how we always help each other out like this that's exactly what I was going to say the way I think about it is similar to Noah, I basically say, Okay, look, there are three sets of numbers right there like in hundreds. Right, so three numbers in hundreds has to be equal to 2003, and that means each number will be around 600. This is kind of also what I what I did so I actually wrote it down. Oh, sorry, no answer yet. So this is exactly what I, what I was thinking. So I said, it has to be 600 something each number, right, and there's three of them right they add up to 2003, so I know the square has to be six. Okay, so now it gets really interesting right so if this guy is six. If this guy six right now I'm actually solving backwards right because I need to get something end up in three, and I have six here. So what can I do with these guys together. These the two of these together can be seven. Can it right because someone plus six plus seven is 13 so seven looks really good to me. Okay, so if this is seven and I get 13 here, and I know the six and six here so I already have 1212 plus one is 13. And what number do I need to get me to get me 20 right because this is zero here. Right, so seven fits right so I do I put in seven here. Okay, so I say seven is the triangle. Okay, and six plus six plus seven gives me 20 okay so I have to hear and a carrot over to. And these are three six, so that means 1818 plus two equals 20 bingo. We got it. All right, now let's do step four. Let's look back to check. Okay, so these kind of problem, it's always easy to check because you just add them up right 666 plus 660 and 677. Now, I'm sure your addition is solid. And I get to 003 check. Okay. In fact, I don't even need to wait until the results from the competition to tell me I got this problem. Okay, it's a 27 number 27 problem that means I have just secured five points solid. No doubt. Okay, congratulations team. We did this together. Okay. All right, let me clear the screen. Let's move on to the next time. What is the next time. We didn't actually solve the problem. The problem asked for what is the sum of the numbers represented by the square and the circle. Yes, the answer is the square is six and the circle is zero. So the answer is six. Okay, thank you. Thank you, Noah. You see Noah put me on my toes, right, I cannot, I have to be super, super duper on the task. All right, the next problem is the certainty problem, the certainty problem is quite popular in the logic problems as well. Essentially, it means that you are 100% sure This question has a poll, just saying, ahead of time. I love it. Yeah, I think that would be a good problem for the poll. So, step one, let's read the problem carefully. In the back there's three green apples, five yellow apples, seven green pears, two yellow pears. Okay, you notice there are two different colors, right, yellow and green, and they each, and there are also two type of fruits, apple and pears, right. Simon randomly, which means he just, he can't see, right, he just randomly takes out fruit out of the bag, one by one. That bag probably is solid, right, so he can't see through it. How many pieces of fruit must he take out in order to be sure? You see what I mean? In order to be sure, it's a certainty problem, that he has at least one apple and one pear of the same color. You know, the question did not say which color. The question only asks you that, you know, that he has to have at least one apple and one pear of the same color, and how many pieces he needs to draw out in order to be sure of that. Okay, so this is a certainty problem, it's an orthological reasoning problem, so you guys wanted to tackle it and have fun with it. Okay, you have one minute to work on it, and then vote on your correct, what you think is the correct answer. Okay, Noah, if you, anytime, if you'd like to launch the poll, it's your choice. So this problem, actually, I need to probably lead you on a little bit, okay, so whenever you have a problem like, you know, to be sure, right, and then there's a bit of a randomness to it, essentially, when you draw, you cannot really see which fruit you draw, only you see the color after you draw it out, right, is you wanted to think about the worst case scenario. Think about that, you can be very unlucky, okay, so you put your hand in the bag, you draw a piece of fruit, it's a green pear, okay, fine, you put your hand in the bag, you draw out, you get another green pear, right, so if you're really unlucky, you can be drawn out for the first seven times, all green pears, no match. Okay, so think about the worst case scenario, like you're being very unlucky when you draw out the fruit, and they happen to be the same color, and the same type, okay, and then think about in that situation, like you're in a very unlucky situation, how many pieces of fruit do you need to draw out in order to be sure you have one apple and one pear of the same color, okay? All right, I'll be opening up the poll now. Oh, it's already 50%. No, I meant opening, not closing. Oh, okay. You know, I really enjoy seeing that some students seems to be changing their minds by changing their choices. It's interesting. I don't think they can change their choices. It's just that the percentages are changing because more students are answering. Oh, they are. Okay, I see. So they don't have the option of going back to to reclick. Okay. Yeah. Okay. I wasn't aware of that. Okay, it's well 67 participated already. Okay, Noah, maybe we can share now, this is like 84, 85%. All right. All right. So, congratulations. It looks like this time the majority students have picked the correct answer. The answer is indeed, 13. Okay, and let's look at the solution together. Okay, I'm sure you have great solutions, and I'm just sharing mine. I mean any event will give you some ideas, right. So, um, so what happened here as I was saying, be sure you kind of wanted to making sure the condition is not even under the worst case scenario. Okay, so if a total of seven piece of fruit is taken out. And the worst case scenario is that he took out all seven green pears. Okay, so you, the condition is not being met. And then you continue on. Right, you after the seven, the first seven you draw another five pieces fruit. And guess what, and this time you're so unlucky you get all five additional five fruits are yellow apples. And yet, you have not met the condition, right, because there's no match of color in the seven pieces of green pears and five yellow apples. Okay, so now what happened when you draw the 13th piece of fruit. Okay, so now in this case, you will draw out either the green apple or the yellow pears. Right, so this time, you for sure has at least one apple and one pear of the same color. Okay, although that same color depending on what fruit you actually drawn on the 13th time. Okay, so if you, if the 13th fruit that you took out is happens to be the green apple, then the same color is green. Okay, if the 13th fruit you happen to take out is the yellow pear, the same color is yellow. Okay, but you don't care about that, right, because the problem just say the same color. So the 13th fruit would definitely give you that match and meet that condition. So the correct answer is 13. Okay, fun, right? All right, let's look at the next certainty problem. This problem is actually quite hard. I don't blame you if you need some time to think. So let's read the problem carefully. Okay, at least we understand what the problem says. So when it is raining, the cat stays in the room or in the basement. You notice that this first statement essentially gave you all the possibilities, right? So it's not possible to cast in the playground, for example. Okay, so the cat is either in the room or in the basement if and when it is raining, right? When the cat stays in the room, the mouse is in the foyer and cheese is in the refrigerator. Okay, and the third statement, you notice I numbered all the statements because I needed later on to show the reasoning. Okay, when the cheese is on the table and the cat stays in the basement, the mouse is in the room. And finally, right now it is raining and the cheese is on the table. So for sure, which of the following statement is correct? For sure, not maybe. Okay, so the cat is in the room, if the cat is in the room and the mouse is in the foyer, the mouse is in the foyer, the cat is in the basement and the mouse is in the room. And finally, the situation is impossible. Okay. So think about what is a good strategy you can solve this problem. Now, here is a bit of a hint for you. Okay. You notice there are four statements made in the problem, four statements. Normally, people tend to look at the statement one by one, right, they will look at the first statement first and look at the second statement second, and they try to use the statement in that order. And I'm telling you, you can look for a statement that would give you the most possible clue. Okay. So in other words, I have decided after look at the problem, I've decided I wanted to start with the statement number four. You notice at the IV, that's the Roman letter four, right? I decided it's better, it's more efficient for me to start with the statement number four. Okay, so you guys have one minute to tackle this problem. Do your best, don't worry about getting it wrong. And it's part of the learning. If we don't struggle, we don't make errors, we do not learn the most. Okay, so have fun. You have one minute. There is no poll on this problem, right? Yes, you can. For problems that doesn't have polls, you can just put the answer in the chat box and Noah monitors it closely, okay? He may not answer you, but he monitors it. The chat gets really messy. Oh, is it? But I am currently generally seeing one answer a lot. Yeah, and that's the correct answer, is it not? Let me check. Okay, so we're given this fact. Yeah, I saw two common answers. Yeah. Now for those students who selected E, I would suggest that you think a little harder, okay? You think a little harder. Good try. I just looked through the question and I arrived at the same answer as a lot of them. Okay, good, good. All right, so I think we're doing well. It looks like a congratulation is ordered. Most of you have gotten the correct answer. Okay, let me share my solution, okay? So as I was saying, for this type of problem, because there are quite a few statements, right? So you kind of get the feeling like, whoa, where do I start? So it's very important that you spend some time and effort thinking about where best to start, okay? So I decided the statement number four is a good place to start. So if the cheese, okay, because the problem tells you that it's raining and the cheese is on the table, right? So the cheese is on the table means that the cheese is definitely not in the fridge, right? Why? How do I know that? Because I know this from the statement number two, okay? Because statement number two says, when the cat stays in the room, the mouse is in the foyer and the cheese is in the refrigerator, right? Wait, okay, so hold on. Number two, okay, which means the cat, yeah, so that means that the cat is not in the room, right? Because the cheese is not in the refrigerator, so that means the cat cannot be in the room, right? And that reasoning is tight, right? It's tight because you can see from a contradiction, because if the cat's in the room, then the cheese would definitely be in the fridge. So the statement four cannot be right, so that's a contradiction right there. So therefore, we know that the cat is not in the room, okay? And then I also went to the statement number four, because it says when it is raining, okay? And we know when it's raining, the cat stays either in the room or in the basement, okay? So the fact that the cat is not in the room, okay, which means he's in the basement, right? And then that it is raining now, right? From the fourth statement, which it tells you it's raining now, it tells me that by elimination, the cat is in the basement, okay? And that is from the statement one and tells me by elimination, the cat is in the basement. Now, in the past, when I solved this problem with my students, oftentimes, they'd say, oh, we don't really need the condition, it is raining now. It is raining now is actually important condition, because if it doesn't say it is raining now, you cannot be sure the cat is in the basement, okay? It could be some other weather, and then we don't know where the cat is, frankly. We only know he's either in the basement or in the room if it's raining, all right? Okay. So finally, since the cat is in the basement, and the cheese is on the table, okay, the mouse is in the room, and we know that for sure from the statement number three, okay? And therefore, the answer is D, the cat is in the basement, and the mouse is in the room, okay? So when you use all these statements in the precise way, then the conclusion is 100% certain. There's no uncertainty here, okay? So this problem is using a different strategy from the one before, because the one before used the worst case scenario, and this one is by following the logical statement extremely closely. You notice each statement serves a specific purpose, okay? So that is how you do this kind of problem. It's hard, but it's really good training, and it strengthens our problem-solving ability, okay? Okay, we're going to go to the next problem. The next problem is the Venn diagram problem, okay? So the Venn diagram problem is not actually commonly known as a logic problem, but it is, it uses a lot of logical reasoning skills, so therefore, the Venn diagram is another type of logic problems. So let's do this problem together. This one has a poll. Awesome. 66 cats signed up for the contest Miss Cat 2013. After the first round, 21 cats were eliminated because they didn't catch mice, okay? Fine. So how many cats are still left at that point, right? So this is something you want to figure out. So now, 27 cats out of those that remained in the contest had stripes, and the 32 of those cats remaining had one black ear, and then it says all the striped cats with one black ear got to the final. What is the minimum number of finalists, okay? So let me underscore for you quickly, step one, right, the keywords, right? So 27 cats out of those remain, so you need to figure out how many cats remained in the competition at that point. And 32 had one black ear, and 27 had the stripes. Actually, I don't really need to underscore the black ear, okay? All right. And then all the striped cats with black ear got to the final. What is the minimum number of finalists, okay? So that's what the question is asking us, okay? So go ahead, try your hand at this problem, and Noah is going to launch the poll, okay? All right. All right. And I will end the poll and so quickly get. If you want to put in an answer, put it in really quickly. No, it's such a soft, soft hearted. In the past five seconds about 30% of the answers so funny. All right. The ending the polo. Wow. Congratulations. Again, the majority of you has gotten the correct answer. Okay, so let me share my solution. Again, my solution may or may not be more efficient than yours. Yeah, it's not supposed to be an A here. Oh, at the first at the first round. Okay, cool. All right, so let's, let's look at my solution together. Um, so it's as I was saying this is a kind of Venn diagram problem right. Okay. Um, so, first of all, we need to figure out how many paths for me in the competition, right so I have 66 minus 2145 cats for me. Right. So, and then what happened. Right, so let me draw the Venn diagram. All right, so we know the 27 cats has a stripes. Okay. And then maybe I use a different color. And then these many cats, 32 of them had one black, black ear. Okay. And I know that I'm going to use my favorite purple color, and that together is 45. Sorry, I would like to add that we know that this is 45 because since it asks us for the minimum, minimum number of finalists, and the only thing it tells us about the people, the cats that went to the finalists are striped cats with one black ear so we can define the minimum number we can just assume everyone else who does not meet these conditions did not go to the finals. Beautiful, beautiful I wasn't going to point that out because it says all that the striped cat with the one back black ear, I'll say, only the striped cat with one black ear got to the final, right, awesome. Yes, but since it says the minimum number we can change that to only to get the smallest number. Indeed, indeed, indeed, beautiful, beautiful. So, essentially the way we solve this type of problem in the Venn diagram problem is 27 plus 32. Notice this guy, right, the one that is overlapped in between these two circle get counted twice right they get added twice. Right. So you know, when you add them up in minus 45 which is total, and the 14 is the difference right because this guy in the middle. It's counted, it's added twice. Okay, so you wanted to take off one to get to 45, and that is the value of 14. Right, so 27 plus 32 minus 45 equals 14. Now, we're a little bit tight in time. These are this is to be expected. These are really kind of intellectually heavy problems. And we don't want to go too fast. Right. So what I would suggest that we do know at this point we're going to cover the other type of problems first, and I'll come back. If we have time to look at the other. The second type, the second Venn diagram problem. Okay. All right. Yeah. Okay. So that is, um, let me clear that. So the next problem we're skipping for now. This is actually a visual Venn diagram problem, it's absolutely brilliant. But if we have time we'll come back for it. Okay, so let's go to problem seven. Okay, so this problem is the whodunit problem. Okay. In fact, this kind of problem that there's a very good strategy to go with it. And so it's actually quite helpful. If this kind of problem comes up in math competition you know exactly what to do. Okay, so three friends, Miss Roberts and Pharaoh, each has one of these professions, doctor, engineer and musician. And each one has a different profession. So that's kind of important, right, they have a different profession. The doctor does not have a sister, nor brother. He's the youngest of the three friends, the doctor, right. I'm Pharaoh is older than the engineer and is married to Smith sister. What are the names of the doctor, the engineer and the musician in that order. So this is kind of like who's who, you know who's done it, you know, and each one has a different profession is kind of important feature here. So, you have one minute to think about this problem. There's a poll on this. Okay. And I would suggest you consider using a table to organize information, making a table is really a great strategy for this type of problem. Okay. Okay, go, go ahead. One minute. Okay. No, anytime if you like launchable. So elimination works really well in this type of problem. Okay. So for example, in the statement number three, it says Pharaoh is older than the engineer. That means Pharaoh cannot be the engineer, right so you can eliminate Pharaoh being an engineer got option. All right. All right. I'll be ending the polling. Three seconds. So quickly get an answer and if you want to. All right, I'll be ending the poll now. Wow. Congratulations, yet again, three times in a row on this class has the majority of students are getting the correct answer. And that's really quite impressive. So let's quickly share my solution. Okay. Um, so, as I was saying, making a table is a very easy strategy for this type of problem. Okay. So, you can already say that you can already cross out Pharaoh being the engineer because it says Pharaoh is older than the engineer right that's from the third statement statement number three. And you can also. So you can cross out Pharaoh being an engineer. And you can also cross out Smith, being the doctor, because it says the doctor does not have a sister. Okay, so if Smith has a sister then Smith cannot be the doctor. Okay. And so, because so because, and then, and Pharaoh is older than the engineer. That means Pharaoh cannot be the youngest of the three friends so that means Pharaoh cannot be the doctor, either. Okay, so you can see just from the statement three, the third statement alone. You can eliminate Pharaoh being the engineer or being the doctor. Right. And that by elimination you know Pharaoh has to be the musician. Okay, so once that Pharaoh is established to be the musician and Smith cannot be the doctor, and obviously he cannot be that musician because the musician is already taken by Pharaoh so Smith has to be the engineer by elimination. So at that point, you know Pharaoh is a musician Smith is an engineer then the only person left is Roberts and Roberts has to be the doctor by elimination. Bingo. You got it right so the answer is Robert Smith. Okay, great. So, um, let me get to the last type of problem, which is truth and lie problem. And if we have time we come back to some of the other who done that problem, and this one is actually quite hard. So let's skip forward to the truth and lie problem. Okay. So this is a problem that we actually brought over from last webinar when we did the number patterns and more. This is an interest me a logic pattern problem right think about pattern problems that pot problem has patterns built in. And this actually is a logic problem has patterns in it right isn't that kind of fun. Right. So, it says 14 people are seated at a wrong table. Each person that either a liar or tells the truth. It says both my neighbors are liars. Okay, what is the maximum number of liars at the table now here, the key word is maximum. Okay, number of liars at the table. Okay. The other key word I want to highlight is both my neighbors liars. Those are important clues. Okay, so you have one minute to work on this, and there's a poll on this problem. So obviously I gave you the hint of there's a clue, I mean there's a pattern to this right so you see if you can use the pattern to help you solve the problem. This question appears to be quite hard. Yes. And, and, and some students listed two answers, um, which is a great effort to narrow down the possibilities but see if we can go all the way. Okay, there's only one correct answer to this. Should I end the poll. You can end the poll. Yeah. So, would you like to tackle this one, would you like. Sure. Yeah, have fun. All right. I'm ending the poll now. This question appeared to be really hard. Okay. Everybody says both my neighbors are liars, so there only needs to be one person telling the truth for like you only a liar only has to sit next to one other liar to be considered a liar. If that makes sense. That's the explanation is kind of hard to understand so let me just try it out. All right, let's begin with a person telling the truth. All right, so we know that the people next to him must both be liars since he's telling the truth. This liar has one person next to him that's telling the truth so immediately he's already lying by saying both my neighbors are liars, which means it doesn't matter what we put for this next slot, we can put another liar here, and then truth. We can follow this pattern. So there's actually a lot more liars than many of you think. All right, if that makes sense. Like, let's have a look at. Let's just have a look at one of these guys. Let's specifically look at this one he's sitting next to someone who's telling the truth and someone who's telling a lie by saying both of my neighbors are liars, he is lying, which means that this space is checks out with what we're given. Now let's move on to this one. Both my neighbors are liars. Well that's obviously a lie since this guy to the. The guy who is more clockwise. Okay, that's confusing the guy who is on this side is telling the truth, which means by saying that both my neighbors are lying. He is lying. So, that makes that confirms what we've put down here now let's look at the person who's telling the truth well both of the people who are sitting next to him are liars. So, he's also telling the truth. And since we have the exact same pattern going out through an entire place. We can conclude that this does in fact work. This arrangement, which has nine liars. So, as Noah was saying, and the pattern is LT or TLL right so two liars, plus a truth teller. That's the pattern. Okay. In order to maximize the number of liars at the table, right. So when you count all the liars the answer is the maximum number of liars at the table is nine, nine. Okay. But if you would like to challenge yourself on the bonus problem after this webinar is what is the minimum number of liars at the table. What's the minimum number of the liars at the table, not changing the rest of the problem, only the question. Okay, go home and think about it. Okay, great. So we run out of time. But we covered a lot of meaty problems. Okay, and you guys did phenomenally well. So, and we know this from the poll. Right. So, just to recap. So we, we tackled a logic problems of five different types. These are substantial types and congratulations. You seem to have really, you know, mastered these problems, and the strategies we've used today that helped us to tackle them included drawing a diagram, making a table and logical reasoning is a strategy on its own, of course, and then we also use a bit of a And we also used working backwards as a strategy. Right. So, remember, working with the larger problems help us improve our logical reasoning ability. And this is such an important ability for solving problem in not only math, but everyday life, right, because we don't want to be that person who makes, you know, a poor logic reasoning conclusions and an ability into the abyss. Okay, so on that note, yes. Do you have the link to the survey? Yes, when I close the webinar, the link is supposed to come up. Let's see if it works. Right, because I added it. Right. Okay. Also, Do you want to post in the chat anyway? Yeah. Okay. So there's a survey guys. So if you like this webinar and would like us to improve further for future webinars, fill out the survey. I don't have the link to it. Oh, you don't have the link to it. Okay, let me close it and to see if it works. Okay. Before you close it. Yeah. You know where they can find the recording since I remember someone asked me that. Yeah, the recording is in the page where your parents registered the webinar for you. And there's a tab called the recording. Okay. Okay. Have fun guys. Go and spend enjoy the rest of your Saturday. Okay, bye. Bye. All right. Bye bye.

math webinar

elementary students

logic problems

math competitions

interactive education

logical reasoning

problem-solving skills

cryptarithms

truth-and-lie puzzles

climate change beliefs