topic is logical reasoning. Let's start with a warm-up question. Some students were solving an interesting problem from the kangaroo. The number of the boys who had solved the problem turned out to be the same as the number of the girls who hadn't solved the problem. Which number is larger, the number of students who had solved the problem or the number of all girls? So we have number of boys who solved the problem is the same as the number of girls who hadn't solved the problem, and we have to compare the number of students who had solved the problem or the number of all girls. Okay, so for this question, and same as a lot of large-scale questions, it's important to sort of figure out the right strategy and how to organize information, right? So, for example, over here we would have the boys, we have the, let me put it here, so we have the boys, the girls, and then we have a number of students who, you can sort of make a table, so the student who solved the problem and the student who didn't solve the problem, right? And another strategy we can use is that we can either pick like a specific number or we can use a variable, right? So in this question they say the number of boys who have solved the problem, which is this column over here, so I can make sort of a table like this, and I can call the number of boys who solved the problem is n, and the number of girls who hadn't solved the problem, those are the same, so either you can pick specific numbers or we can just put the same variable for these two quantities, right? And then we have to figure out, and these we don't know, so, you know, we can put like x and y because we have no information about these two numbers, and the number of girls would be this, this column is going to be y plus n, right? And the number of students who solved the problem would be this row, so that's also y plus n. And I mean some students can just get away, for example, you can assume this one is 10, this one is 10, you can pick any numbers for this and we still get the same answer, right? So that's what we're trying to do today, we see like, you know, some way to organize information or some strategies that are going to help us to solve logic questions like this. In case we need to review later. No, I don't know, I think it's fine. So these are the common types of logic questions. Since they all come in different shapes and sizes, it's hard to sort of have a, to categorize everything. But usually these are the things that we see in math current exam. The first type is the Piction-Hole principle, a very famous principle that you've seen a lot in lower grades, right? If there are n Piction-Holes and n plus one Pictions, then at least two Pictions must share the same hole, right? And then the problem using the Piction-Hole principle usually asks for a minimum quantity of something to make sure that a certain event will happen, like you have a subset of different colors in the drawers, and without looking into the drawer, what the minimum number of subjects you have to take such that you don't have a certain pairs of certain colors. So questions like that is fall under this category. And then we have a logic grid puzzles or correspondence between sets, right? In questions like this, we are asked to match two lists of items based on a set of clues. And these usually are sort of the most popular types of logic puzzles. And the way to solve this kind of question also kind of straightforward, right? We just draw a table with the list of item in one row and the list of item in the other set on the column, and then we use the clues and it saves time if we know what clues to use first, right? And the order of clues we use matters, and then we just finish the table. And then we have problems that have a bunch of statements and it describes a specific situation, and then we have to, based on what the question described, we have to determine whether certain statements are true or false. And then another very popular types of logic questions are nice and liars. You know, the nice always tell the truth while the liars always lie. And also there are problems in which the characters tell both lies and the truth, or they tell lies on certain days and truth on certain days. And based on what they say, we have to find out who are the nice and who are the liars. And in terms of problem solving strategies, proof by contradiction is a very popular strategy to use in logic and just in math in general, right? It's a very powerful way of proof. So assume, we first assume a particular statement is true, and then we see if that leads to contradiction, then we conclude that that statement is false. And then another thing is, another method is casework. So we just consider case by case, and we've seen this a lot in other topics, right? So for example, an island of nice and liars, we have to prove that no one will say that he's a liar, right? So we can just consider all the cases, and in this case, there are only two, right? If the person is a knight, then a knight cannot say I'm a liar. So that's a contradiction. So he must not be a knight. In case two, if the person is a liar, then the statement, I am a liar, becomes a true statement. But that's also a contradiction because the person is a liar. So the conclusion is that none of them can say that he is a liar. And other strategies that are useful are solve a small version of the same problem. We've seen this before, number parity odd and even. We've seen it before, and we also have one question like this in today's class. And draw a table or a graph to organize information that, this is sort of important in any question, in any types of problem, but especially in logic question when you have a lot of information and some of them are really lengthy, and then draw a table or graph to organize information or even to find problems, that's something very useful. And sometimes we just have to construct a solution and show that it is the only one that satisfy all the given conditions. And of course, we always have, with the aid of multiple choices, then we can eliminate these incorrect choices. So in this lesson, we probably have time to go through six, seven questions. And then in each question, we try to identify what strategies are used to solve that pattern in questions. Okay, so the first one, the inhabitants of a certain city always speak by means of questions. There are two types of inhabitants, the positives who always ask questions for which the answer is yes, and the negatives who always ask questions for which the answer is no. I met Albert and Berta. Berta asked me, are Albert and I both negative? What type of inhabitants are Albert and Berta? So yeah, lengthy questions. But I usually find that students are actually doing pretty well with... And this one, remember, we have a strategy, like assume one statement is true, right? And then see what happens to other statements. Or we can use casework, right? So that was what we were just talking about, casework. Because each of the characters can either be positive or negative. So a very common strategy we use for these kind of questions on liar and liars is to assume that character is either a positive or a negative and see what happens, whether we're gonna find any contradiction or everything is inconsistent with each other. And also, we just mentioned that we can use table, right, to represent information. So let me just use all of this. We have a border, right, and output, or maybe the other one is sort of more intuitive. I think that's fine, yeah, we can use border, and we have an output, and there has to be either positive, right, positive. It's sort of a table, like the warm-up questions. There are not many cases, so we can just consider the cases and see what we assume of the character is consistent or in contradiction with what the character says. And in this case, because Bota is the one who asked the question, so if we do case what we want to consider Bota first, right, instead of Albert, then we can compare, you know, her true identity with what she says. So, for example, if I say that, assume that Boddha is positive, right, so if we assume that Boddha is positive, in this case, then the answer to her question has to be yes, but that means she will be negative, right, because that's what, you know, the yes to this question means, and so we assume her as positive, and then it leads to the fact that she is negative, right, and that was a contradiction, so that should not be the case, and we can, you know, eliminate this case. And then if we assume the other case, I mean, in this case, only two cases, so that means we can conclude that Boddha is negative, so that was the definite answer, and when Boddha is negative, the answer to her question has to be no, and this question has basically, you know, two clauses, Albert is negative, and I am negative, but because Boddha is already negative, that means the other statement has to be wrong, that means Albert is not negative, right, and that means Albert has to be positive, so the answer is Albert is positive, and yeah, okay, so that is the answer. Is it correct? Yeah, it's easy to make mistakes, so if I make mistakes, it's just correct me, Andrew. And thank you, so that is sort of a standard method, so that's what we do in this class, and for each question, we try to identify the strategies, so in this case, remember, drawing a table to organize information, and also to sort of keep track of our reasoning, and doing casework, that's what, and fighting contradiction or consistency. Question number two. In a village, there are no two people with the same number of hair. Nobody has exactly 2,007 hairs. Joe has the highest number of hairs in the village. The number of the villagers is more than the number of Joe's hairs. Joe is the maximum number of villagers. You know, we didn't talk about that strategy, but I think trial and error in this case, or sort of goes through each choice, right, Andrew, is also a good strategy. And since the questioner is asking for the maximum numbers, then you can try 2009 and down to 2008 first and, you know, 2007 in that order to see how, you know, if you can figure out anything, to try from the largest to the smallest. So maybe Andrew can go to 2009 first to see if he can figure out anything. Yeah, so like Vance said, one way to solve this problem is to look at the answer choices. Because math theory is a multiple-choice test, we can just try to eliminate each answer choice, right? So let's say that we have 2009 people in the village. That means that Joe has to have less than 2009 hairs. So let's just assume that he has 2008, because that's the highest number less than 2009. So if Joe has 2008 hairs, that means that you must have 2009 villagers with hairs between 0 and 2008. But we know that from 0 to 2008, there's only 2009 numbers. But we can't have anybody with 2007 hairs, so we'll only have 2008 villagers. So that will work. OK, so let me do 2006. And we have to skip 2007, right? And we go to 2008, that would be all the possible numbers of different number of hairs, right? And that would be how many numbers in total here? That would be 2008. Yeah, so it's only 2008 different numbers. So that's not enough, because we have 2009 villagers. OK, so should we do another one, Joe, Andrew? Yeah, we can do 2008 as well. So when we do 2008, we get the same problem, where we can't have 2007, so we're missing a villager. OK, so the same thing with 2008, right? Because in 2008, Joe would have less than 2008, so maximum 2006. Yeah, because you're not going to have 2007. OK, so if we go like this, we only have 2007 different numbers. That's also not good. And that means, Andrew? Yeah, so then you can see, because if we remove one villager, we'll get the same maximum. That means that when we remove the villager, we keep the same number of numbers. So we can get 2007 villagers and 2007 numbers. So we know that that's going to be the answer. And then it's going to be, the number is going to be 0, 1, 2, all the way to 2006. In this case, we are good, right? Because we have exactly, yeah, 2007 numbers. So again, I mean, larger question, because it comes in all different shapes and sizes. So it's kind of scary when we want to do it. So again, I mean, larger question, because it comes in all different shapes and sizes. So it's kind of scary when we first look at it. But we're trying to convince you that if you apply the standard problem-solving strategy, usually after a while, a few minutes, you would be able to figure things out, or at least know where to get started, right? And in this case, really go through each choice once by once. Have to go highest one, try and narrow. So that's what we're trying to highlight in this class, in this lesson, the different strategies that we can use. Thank you, Manchu. Let's go to the next one. So this is a question with liars and truth-tellers. A certain island is inhabited by liars and truth-tellers, and as always, the liars always lie, and the truth-tellers always tell the truth. One day, twelve islanders, both liars and truth-tellers, gathered together and issued a few statements. Two people said, exactly two people among us twelve are liars. Four other people said, exactly four people among us twelve are liars. The other six people said, exactly six people among us twelve are liars. How many liars were there? So this is also a question that I just want to demonstrate that we can use table to organize information. So two people say that exactly two are liars, four people say that exactly four are liars, and six people say that six are liars. So everyone, two plus four, six, 12, so everyone made statement, because sometimes I have question in which only a few people, right, make some statement and the rest are silent. But in this case, it's important to know that everyone makes a statement. And from here, because there are 12 people, then we know that in this case, it means that 10, I always use, yeah, L and T, I think that's a good notation, 10, and this is going to be eight, this is six, right? So yeah, it's good to sort of summarize information this way. So give me a few minutes to see if we can figure it out. Okay, very good. I feel that for students, like for the logic, once you get a hang of it, actually, it's not that hard. It's always hard when you first read the questions, but, you know, there's a way sort of we can get started somewhere. So another strategy that I find super helpful in many liars and truth-tellers questions is to assume a certain number of people are truth-tellers or liars. You may assume a certain number of truth-tellers because these are the people whose statements are correct, right? We cannot trust the liar. So this is actually, I find that it works for many, many liars and truth-tellers problems. Sometimes, like in the room, only one person says something or everyone in the room says something. So just assume a certain number of people are truth-tellers. So in this case, for example, I can assume that... If I assume that one person is a truth-teller, then that would have 11 people are liars, something like that. Or if I assume two people are truth-tellers and 10 people are liars, then I have to basically figure out what is the exact number based on this table, right? So if I assume that one person is truth-teller, that means there's one statement that says that there are 11 liars, right? And I look at the table, there's no such statement. Sometimes someone is silent, but in this case, everyone made a statement, so that must not be true, right? And then I look at this case, that means exactly two people are going to say that there are 10 liars. But in this case, there's no such thing, so that's sort of also not the case. And after I do a few questions like that, for example, now I can jump to four, and this is eight. Or maybe I can do 10 or two to match this, but after I do a few cases, I can generalize. So for example, if I say that there are N person who are truth-tellers, then we would have 12 minus N who are liars, right? And then I would have exactly N statement in which the person say that there are 12 minus N liars. So that means this is a number of statement, this is number of liars. Then the two numbers have to add up to 12, right? And over here, this two number has to add up to 12. And that is the only case, six and six, and once you figure it out, we put it in. And after we do that, there's nothing special about two and four, so it's six. And once we figure out the number, always do a quick check whether that satisfy everything. Yes, so some of you got it quickly, that is six. So that is another type sort of related to the first question. Let's go for another type here. This one has numbers on it. So three boys play a game of word in which they each wrote out ten words. Each boy scored three points if neither of the other boys had the same word. Each boy scored one point if only one of the other boys had the same word, and no points were awarded for words which all three boys had. When they added their scores, they found that they each had a different score. Sam had 19 points, which was the smallest score, and James had the highest score. So the question is, how many points did James score? So this is also lengthy questions, and same as a question when I have a transformation. So our first step is to really try to really understand the question statement, right? So I think that's what I would do. I would spend time understand that. So for example, I have three boys, right? It's going to be A, B, C. A, B, C. Yes, that would be the first step. Suppose that we see what happened to A. If A score three points, right? Then B and C, if they have a distinct words, then they also have a 3-3, right? But it could be that B and C, they have the same words. So each of them have a 1.1 points, and that's the two cases for A, right? And if A share one word with other boys, then he would have a 1, and he could share with B, right? And in this case, C has a distinct word, so he would have three points. Or A could share with C one, and then that means B has three. And finally, if A shares the same word with both B and C, then all of them would have three. So that would be the first step I would do, really try to really understand the question by listing out all different properties, I mean, possibilities. And our next thing is that we have to find out, we have 10 words in total, and Sam scored 19 points. So basically, from here, we have to find out a combination of this, right? Three and one and zero such that with 10 words, he would make 19, right? So I will give you a few minutes to try to figure that out. And I would put here the same. He has 19, and he's smaller than, I don't know, maybe we would bend here in the middle, because they saw also different score, right? So it's also important to underline a few words, and the last person is James. So that's what we have to do, we have to find different combinations of 10 words to make up the 19. And so a combination of constructing the questions and, okay, some of you already started figuring things out, so I would try the strategy here with kind of, the strategy here would be constructing, right? Constructing the solution and then also sort of trial and error. Casework as well, yeah. Okay, good, so let's get started because I see the answer coming in. So now I know that Sam has 19, that's definitely to start from there, right? 19, and then it's easier to consider the number of questions that he has for three, so it's gonna be three times six. And then I would have one question that's worth one point, but because I have 10 questions in total, that means three questions, right, must worth zero points. So that would be the first case. And then I make sure that I list out all different possibility. If you have five questions that worth three points, then I need four questions that worth one point and then I would need one question that worth zero point. I'm sorry, one word that's worth zero point. And then after that, I think it's too small, right? Say three times four, and then you're gonna have one times seven, so it doesn't work. So we actually have only two cases, yeah. Okay, so in the first case, if he shares six distinct words, right? And then he actually shared three zeros worth with other boys, that means what? That means the boys that he shared, for example, the boys that he shared, for example, same here. If he share, if he has a three times six, and then he have a zero, zero, zero, with three times with Ben and James, right? And then we have, so this three times six is, yeah, six, this one. And then he have an one, and that one he had to share with either Ben and James. And because this, all of them, yeah, all is zero. So if he share one with Ben, and then even Ben get all the six, all the three point for the other six questions, then Ben would at most has any, as many points as Sam, right? Because he's limited by these three zeros. So that means Ben cannot has more points than Sam. So we can eliminate the first case. I think so, that would be, and sometimes I find it sort of easier just listing does like this. So that was done. And then we had to look at the second one. This is five and four and one. So Sam, he has three, three, three, three, five of them, right? Yes, and then he have a zero, and that zero we know. I mean, this three, we don't know because most likely these also have the three because they have more points than Sam, but at least I know for sure that in this case it's gonna be zero, zero, zero. And then I have a four ones. It's gonna be one, one, one, one over here, right? Now I want to distribute this one with Ben and Jim because I know that Sam, in each case, he had to share a word with either Ben and James. And how we do, we distribute it such that Jim has more than Ben, right? You have a four and it's split between two different numbers so it's kind of obvious that Ben would need to share one. He would need to share only one with Ben. I think one with James, right, Andrew? And then three with Ben, I think the other way around. So you want to maximize these. So in this case, all of them gonna have three, three here, right, you just have to fill in everything, three, three. Three, three, three, three. And in this case, I want Sam to share only one with Ben so that James would have a three here. And then, no, it's one here, three. Three. And then one here, have a three. You cannot share all four, right? Because then Ben would be equal to Sam and we want Ben bigger than Sam so we will put a three here and one here. Okay, and then we can sum up everything which is three times six plus three, so that's 21. Three, so that's 21. And James is three times eight plus one, so that's 25. Yeah, so this is also a question that doesn't really fall in any category, but I think the first step is really understand the question and then like making, organize the information in the tables and, you know, start from trial and error and K-squared. So that's question number four. Okay, so let's go to this one. This one is a good one. This one has a good strategies that we can learn from. At a conference, the 2016 participants are registered with the numbers from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as their registration number. How many hands did the 2016 participants shake? And this is, you know, if you see here's a handshake problems. And we talked about it before in the first lesson about problem solving strategy. And the second strategy I would like to talk about is solve a smaller version of the same problem. Because apparently 2016 is a very large number, right? So see if you can use the smaller numbers and, you know, by the pattern. Remember for the handshake problems, you have a certain number in the room and they shake hands with each other. And like even an odd number of people, each of them shake hands an odd number of times, something like that, right? So a very common strategy when you use it handshake is that we actually draw a graph. I think we talked about it in first problems. Sort of draw a connection between different objects and maybe try a smaller number. You can try, I don't know, you can, I think n equal to 4 is maybe, sometimes if you pick too small a number, you can't quite find the patterns. So in this case, maybe try with n equal to 6, for example. So these are the concrete strategies that we can use for other problems. I chose anacristic because it's even just like this right you you can use um art but then you probably realize that they they don't really match the pattern so also I sort of think what is good number to use you know it doesn't hurt to use a few of them. And also, like, when we draw the connection in the graph, you have to start with the connection that you 100% sure know how to connect, right? Don't start with something that you're not sure how to connect it to other vertices in the graph. I think it was handshake problems and has a one is that you can use the phone. And we draw, try to draw, it's like in the vertices of a regular polygon so that you can, you know, you have a space to connect every vertex to every other vertices, don't draw them in a line. Okay, very good, so you go, you figure it out, so that's the thing here. And I think you, if you look at past exam or you look at problems like that, sometimes the handshake is a sort of basic questions and then phone call or sport match, right, sometimes you have like, you know, different teams and how many times they play against each other. Problems that involve connection like that between like two objects and graphs usually a very, very useful strategy. So in this case, we consider any good as six, and then that means we have to go from P1 to P5, all of these five people, they shake hands with exactly the same number of participants as their number, right, so P1 shakes hands once, P2 takes a hand twice, P3 three, P4 four, and P5 five, but now the trick here is just like the correspondence between set or logic group puzzles, we have to use which clue first. We don't want to use P1, right, because we don't know who P1 shakes hands with. So we bypass one, same for P2, P3, so for among these five people, actually P5 is the one that we can start drawing, right, because there are exactly five other participants, that means P5 has to shake hands with all of them, right, so that would be, and I think that is a very common strategy in problems like this. So we connect this one, and once you're done, maybe you can just put a tick mark that P5 is all set, I don't need to consider it anymore. In the graph, these also correlate, you know, vertexes, and then P1 also, we could already shake hands with P5, so P1 is also all set, and now which person I need to look next, P2 also has so many choices, so maybe I look at P4 again. P4, right, so already shake hands with one, okay, so the good thing is that because P4 needs to shake hands four times, but P4 cannot shake hands with P1, because P1 is already all set, already has one handshake, right, so that means P4 has to shake hands with two, three, and six, okay, so I will draw the next set of connection, maybe use different colors. P4 has to go with this, and this, and this, yeah, so that is also something very commonly used. And after that, I make sure I mark P4, this person is all set, and now I move in this direction, so P3, oh, I also have a P2 all set because the two connections done, so I move to P3 is the only one, and P3 has a one connection, two connections, so P3 must shake hands with P6, right, that would be my last connections, P6, okay, yeah, right, and that means P6 gonna shake hand with, you know, sort of the lower half of the whole, the lower half of the, we can think of it as a regular hexagon, so P6 shake hand with, you know, sort of this half vertices, and from this, right, so smaller versus the same problem, and it has exactly the same structure as the number 2016, so if you imagine we draw a 2016 regular polygon, then 2016 gonna shake hand with this lower half, and that's exactly half of 2016, right, so the answer is D. I put it here, and then if you have a question, we can go back. Yeah, but just go back to pass the exam for level 9, 10, even 7, 8, you would see, you know, quite a number of questions that make use graph and use exactly the same strategy. Okay, so let's see what's next for this question. There are 30 dancers standing in a circle and facing the center. After the command left, some dancer turned to the center, and the other left, some dancer turned to the left, and all the others to the right. The dancers who were facing each other said hello. It turned out that there were 10 such dancers. Then after the command around, all the dancers turned around, made a half turn, and again the dancers were facing each other, said hello, and the question said how many, how many dancers said hello this time? Okay, so initially all of them facing the center, so we should start with this, right? The initial configuration is that some of them to the left, some of them to the right. We don't know how many, but that would be the initial configuration. And then when they turn around, in the final configuration, again, 10 of them said hello, and we have to, I don't know. Actually, if there is an N left, and this one is going to be 13 minus N, and final one is going to be 13 minus left and N right. So this is 10 and 1. So just like we used, you know, tables and graphs before, this one definitely a draw a diagram, like a picture of a circle, definitely very helpful. Another thing is that, again, 30 dancers is a big number, so maybe in this case, you can use maybe 4, again, 4 is too small, I would use 6, because too small, you can't really see the pattern. You can do 6, 8, or 10, but I think 6 might be already enough. Or you can, sometimes you need to use more than one number to find the pattern. you can draw a few cases of course here I already I mean draw before so I sort of more like draft but I also sort of know kind of what are the difference between these cases right. So in this case for example if I look at the first configuration right this is the initial configuration so in this case there's only one pair right this person is turning to the left so this is the only person they're going to say hello right hello in the initial configuration so I put like hello one and everyone else here this one is not because this person turns right into left so they were actually facing away from each other and these two you know you know lie so this not was h1 but then in this graph I also say that after the command around then this pair we actually this gonna face right and this person gonna face left so this pair we actually say hello to each other after the command around right so actually I would put h2 you know say hello the second time okay it's good some of you already figured out but you know for questions like this I always notice that some logic actually student a lot students are pretty good at logic you could figure things out very quickly but I was just trying to you know provide the strategies that hopefully is kind of standard that have you so many many different questions but a lot of the time if you notice something special some students sort of really a click and already figure out the answer so here left and right right this actually gonna have a h1 over here and left and right is one over here this is when they said hello initially and I have a right left so after we turn around this pair is gonna say hello and same for this pair gonna say hello and this one left and right this pair is gonna say hello the first time and this pair gonna say also hello initially while this pair gonna say oh actually this pair also say hello and after the turn around this pair is gonna say hello this pair gonna say hello and this pair gonna say hello so I do six but you can also do eight right and the point the point is here doing that you know here I have a one number of hello initially here I have a two numbers of hello and here I have a three numbers of hello that that that's sort of like my intention when I draw three different graphs so after you do this what would be h1 compared to h2 um in these three cases okay you notice that actually h1 equal to h2 right so it initially we have a 10 hellos it looks like least after the around command there will be exactly 10 hellos and um that we came back to the common session in which we sort of have a feeling that is the correct answer but we need to prove that in the exam I think you can just get away with that and to circle 10 that's what I would do but at home I think it's good to try to understand why and this is also a technique that I think that's very nice that I would like to present to you so we talked about like odd and even before so in this case for example if I can label l as 1 1 1 and r is 0 0 0 that's a sort of one technique but I think it's it's very powerful and you can use it for for other questions um and then I would label the if you have a connection between l and l I would label it zero l and l is also zero because I take the left number and subtract away the right number in this case because l minus one so I have a one here this is going to be zero this is going to be zero and this is minus one but the reason that if I subtract one number to this the right to the left the right to the left and the rest of that if I go around a circle the sum of the total number is exactly equal to zero right you have a two minus a one this is also like I know maybe telescoping sequence if you continue doing algebra something like that right and you go all the way to a n minus one minus a n and then plus a n minus a zero that sum is exactly zero if you go in a circle right so if we follow that pattern this one is going to be minus one it's going to be zero this is going to be one this is zero this is one and this is minus one the point is that because we go in a circle this time is zero so the zero zero these two people because they are in the same direction so they never say hello to each other but we would have exactly the same number one right if you have n one then we would have n minus one and the first um n one couples they say hello to each other and after they turn around the minus one couple is going to say hello to each other and the number exactly the same and that's why we are we also have a 10 hellos after they turn around okay um does it make sense um Andrew is it like does it make sense the the proof or you have a easier way to prove this pattern okay unfortunately we don't have a sound but if you have any like better way to um prove that pattern um feel free to share through the uh to the chat and I can incorporate in the next class because that's what I thought about you know these questions you can go back here and okay so that was um pretty much what we covered today so we went through um you know the first thing is read the question carefully and use table graph to organize information and develop the solution a common approach assume certain statements are true and false and look for contradictions and then determine which clues to use first um in a in a set of statement we can use algebra use diagrams use tables use graphs together with reasoning and then um other strategies are casework proof by contradiction fighting a pattern um you know so a smaller version of the same problems that would be um you know useful a strategy for solving logic questions and uh next week um so we finish with all of this and then we have a three remaining questions um lessons that are dedicated to geometry um I will send out please look out for your emails I will send out handouts for the next three classes um it will be helpful if you print out these before class or you can quickly join class but I I would

logical reasoning

problem-solving

mathematical logic

Pigeonhole Principle

Knights and Liars puzzles

proof by contradiction

casework

graphing tools

critical paths

algebra simplification