afternoon everybody. Welcome to webinar number nine in our level 3-4 fall series. We'll be starting in just one minute. If you have questions at any time during the webinar, you can chat to me or you can chat to our teaching assistant Shourya Vyas. It's good to see you Sharia, we have a lot of students joining. So, today's topic will be logical reasoning. If you like logic puzzles, this is the lesson for you. If you like those little brain teasers that are like, you know, these people are lying and these people are telling the truth, so figure out what's really happening, we're going to do some of that today. If you like those logic puzzles that are called one-to-one correspondence, where each person has a favorite color and you have to figure out what everybody's favorite color is, that would be, this would be a great place for you today. Okay. I think it is time to begin, so good afternoon and welcome. We will start, as usual, with a warm-up problem, so I will put that up on the screen right now and I'll read that to you. I do have a poll for this warm-up question, so after I've read through it and give you a few seconds, I will launch the poll for you to submit your answers anonymously, right? No one can see, so don't be afraid of making a mistake, just keep trying. Luke gave the bouquets of flowers shown below to his mother, grandmother, aunt and two sisters. Which of the bouquets did his mother receive? If we know that the flowers his aunt and sisters received were the same color and that his grandmother did not receive roses. Alright, I see students responding. Let me launch the poll. You'll see the question appears in the top of the poll as well. So you should be able to answer there. If you don't see the whole thing, just go ahead and try to scroll up or down. If it blocks your view of other things, you don't like it, you can always close it and go back to seeing the shared screen. Okay, I have a lot of students responding in the chat and in the poll and I like that like lets us know Shuri And I know that you're playing along with us. You're doing your best Here are the results About three-fourths of you 73 percent is pretty close to 75 percent have said that you believe the correct response is pink Roses, so we're trying to figure out which flowers his mother gets And let's see if that looks right when we go through it together So remember this was sort of a process of elimination there are a couple of different ways we can play this game of Logical thinking and this one is Elimination we know that his aunt and sisters and we know that there are two sisters. So aunt and sisters are three people Right, so his aunt and sisters received the same color So we need three bouquets that are the same color that go to the aunt and sisters So those will be the yellow bouquets of flowers. So we know mother did not get that We know that his grandmother Did not receive roses. So the two choices that are left for grandmother and it has to be the not roses choice That would make grandmother. I'm just going to use GM for grandmother the red carnations and that means that mother Gets the only flowers left the pink roses being so you did very well on that question. Good job All right, we are coming down to the end of our webinar series We have a meeting today and we have one next week And then I hope everyone will enjoy a nice winter break and we will have another webinar series starting in January So if you have been enjoying this webinar series if you're learning a lot Don't forget you can sign up for one with different problems all different sets of problems starting in January Okay, so logical reasoning we are going to use a lot of facts in our problem statements So remember our five our four-step problem-solving process was to make sure we read and understand the problem I have a feeling today We will be doing a lot of reading and rereading and possibly even reading a third time to make sure we identify all the clues in the problem statements and What we need to do may not be obvious because we might have to do different things in different problems We may have to even try an if and then what will that happen? What will happen? So maybe if you've been filling in Sudoku's or things like that You might try a number might try it small and pencil and see if the rest of the puzzle works and if it doesn't then you might have to Back up and erase that's kind of an if then if I try it this way Will I be able to make a correct conclusion? If not, you'll have to back up and try again So don't be surprised when that happens today Okay, so we call that a contradiction. What could be your contradiction in Sudoku? It would be having two fives in the same row. You can't have that right? Sometimes we might just have lots and lots of conditions so that we require very careful and we might want to make lists To list out what we have we might want to make tables to organize the information together So we will have some problems with truth and some problems with lies We will have we just sought an elimination You We might have problems with what we call one-to-one Correspondence each person only has one of the items being listed and we have to figure out where they go So be prepared and I hope you really enjoy today's lesson. It's not going to be so many calculations You won't need to be doing a lot of multiplying today But you will need to be doing some really good critical thinking and these critical thinking skills help develop your overall ability To analyze and do creative math problems All right, so here's a little kind of animal Sudoku Albert fills the grid with the five figures. It's a rhino shark frog a ghost and a flower Each figure appears exactly once in every column and every row Which figure must Albert put in the cell with the question mark We've discussed columns and rows before columns are up and down and rows are side to side across, right? So we you'll notice in the first Column, we have one of each animal So we need every column and every row to have one of each just like Sudoku with numbers There is no poll because I couldn't put these little figures and everything in the poll and have everything appear for you So, please use the chat if you'd like to send an answer or a question to me or to Shoria Nice. Sharia, hopefully the students are communicating and getting some encouragement from you as well. I'm seeing that. These webinars are supposed to be just me and Sharia talking to you, but we like it better when we can see how you're thinking and we can help you, right? So, I see a lot of correct answers, which is a ghost. I am seeing some answers for the rhino, and that makes sense, because if I look at the column where that question mark is, I am missing the ghost and the rhino from that column, correct? So, I'm going to try, since ghost is first, it's A, I'm going to try to place the ghost. If I try, I guess I could try to draw a little ghost. If I try to put the ghost here, you'll notice that I have two ghosts in that row. So, that is not going to be good work, and I also could not put the ghost here because there's a ghost in this column. The same reason I can't put the ghost here, because that puts multiple ghosts in the same column. So, let me erase all those incorrect ghosts. That means the only place I can put the ghost is here where the question mark is, and it is the correct answer. Let's double check. So, that was a process of elimination, right? Because I found all the places I could not put the ghost, and then I found out the ghost goes where the question mark is. But we also know we need to be able to place a rhino. So, can I put a rhino here? I'm not going to try to draw a rhino. I can put the rhino here. Where does that make me put a rhino? I need, in this column, I need a rhino, and I need a shark. There's no shark there. So, I can't put the rhinos next to each other. I would have to put the other rhino here, and then I would put a shark because that's missing here. Does that work? There are no other sharks in the vertical column, and there are no other sharks in the horizontal row. So, everything works out when I put the rhino in the column above the question mark. So, I can kind of check myself with this one and make sure I've done a good job in filling in the blanks. So, there are a couple of approaches. Now, I focused on the columns. Could you have started by focusing on rows? What's missing in each row? Yes, that's completely correct. There's no difference. There's two missing in rows. There's two missing in columns. So, start whichever way you feel like it, and it will work out fine for you. Okay, Shoria, you wanted to lead problem number two, so. Yeah. Okay. There is a poll. Seven children are standing in a circle. No two boys are standing next to each other, and no three girls are standing next to each other. Which of the statements below about the number of girls standing in the circle is true? So, out of these five, four of them are going to be false, one is true, and remember, we're trying to arrange them in a circle. So, it's going to be sort of like one, two, three, four, five, six, seven people. It's not, it's different from if they were in a line. All right, cool. So, I'll give you all some time to chat me the answer, and once I get quite a few answers, then we'll launch the poll. Okay guys, the poll has been launched so make sure to put your answers there. You can keep working because the answer on the question is still at the top. All right, so here are the results from the poll. No one picked four and five or five, so that's good. We were able to narrow it down to three choices, but there's quite a bit of a spread in those three. So let me work through this. And, okay. So, so from here, just looking at the answer choices, one of the tricks I can see is that four is one of the common answer choices. So four is an answer B, D, and C. So I feel like that means that four might be important when we're doing this, because it's in all three of these answers. So if we're able to prove that four isn't in one of these, then we can really narrow down our choices. Okay. So let me just try doing four. So if we have four girls in a circle, let's just say G, G, G, G, let's leave quite a bit of space between them. So remember, no three girls can stand next to each other. We can have two girls next to each other, but not three. Now, if we want seven people and we have four girls, that means there has to be three boys. And we could do like boy, boy, boy, because remember, two boys cannot stand next to each other. So there must be at least one girl between each two boys. Now, if you look at it, there is one girl between two boys. So this condition is true. And then if we look at the amount of girls, well, this is one, this is one, and this is two girls next to each other, but that's okay. Because, uh-oh, where did my Zoom go? Sorry, it kind of disappeared. Sorry, my thingy just like, okay, yeah, so looking at this, we can see that we have two girls together, but again, that's fine because our condition is going to be about three. So we can see that four works for our problem. So that means we haven't eliminated A or E, but we know that four is going to work. So that means that C, B, or D are possible, but A and E are not going to be possible because A says that three is only possible, E says five is the only possible. So now we have three, and now we're just going to have to like figure out does three work or does five work? So let's say we have three girls in a circle. Again, let's give it some space in between. And remember, each girl, like there can be a group of two girls, but not three. So let's say these two girls are going to be in a group. So then we can have a boy like here, here, but this is only five and we need two more boys. So we can't put two boys together. So we're going to have to put a boy over here between the two girls. Now the problem is we've already actually gotten six people, but we still need one more boy. And the problem is that there's no more girls to put the boy in between. So that means that three isn't going to work out because the boy can't fit anywhere since he's going to be standing to at least one other boy. So that's why three isn't going to work. And now we only have to try D, which means we have to try if there's five girls. So let's say there's five girls in our circle. There we go. That means there's only going to be two boys. Now, remember we can't have three girls standing next to each other. So that means we have to have them in groups of two, two and one, or really that's the only possible group we could have them in. But we only have two boys. So the most we can do is split them into groups of let's say two and three. Sorry, that shouldn't be two and three, but that doesn't work because they're always going to be in groups of three and two. So either it'll be like three and two girls or it'll be four and one or five and none. So it's not really going to work. So that means that five doesn't work. And so the answer that is the only possible answer is going to be C, four. So in this one, make sure that look at the conditions carefully. If there's two conditions, they both have to be satisfied by themselves. And again, just to make the problem go quicker, you can try the answer choice that's like in most of them. So the answer choice four wasn't C, B and E. So just by trying that, we were able to eliminate two answers, A and E. So that would be a good trick to use in these kinds of problems. Okay. Yay. Thanks, Shurya. He'll explain another problem in a little while. Let's go to the next problem. Okay. Each boy, Mike, Nate, Oliver and Paul has exactly one of the following animals. A cat, a dog, a goldfish and a canary. Nate has a pet with fur. Oliver has a pet with four legs. Paul has a bird. And Mike and Nate don't like cats. Which of the following sentences is not true? A, Oliver has a dog. B, Paul has a canary. C, Mike has a goldfish. D, Oliver has a cat. Or E, Nate has a dog. So a couple of things to think about in this question. There is a poll. I'll get the poll launched in a few minutes. But think about, we're trying to find a sentence that is not true. So if one is not true, four are true. And just to clarify, not all students know, but Math Kangaroo thinks you should know, a canary is a yellow song bird. So a canary, a canary is a bird. It's like saying a golden retriever and a dog, right? A golden retriever is a dog. All right, I'll give you a minute and then I'll launch the poll. All right, I'll launch the poll again. The problem is still there in the top of the poll. Hopefully, we'll get a lot of correct responses for this one. Yeah, it looks like most students have participated in the poll. Thank you. Remember, we're just, it's anonymous, so always try. On a math kangaroo contest, it's multiple choice and we do not deduct any points for incorrect guesses. But if you guess correctly, you might score some extra points just by chance. This is correct. The correct answer that the one that is not true that is false is Oliver has a dog. So very good work with that. Let's see. There is more than one way to solve this problem. I'm going to show you two ways. One is to figure out which animal each person has, which pet. So it says, Nate has a pet with fur. And over here, it says Nate does not like cats. So you can assume that Nate has a pet with fur that isn't a cat. So that would be a dog. So Nate has the dog is true. And Oliver has the dog is false. Right? You can continue with this. It says Oliver has a pet with four legs. The only four legged peck here that is not the dog would be the cat. And it says that Paul has the bird. That's the canary. And that would mean that Mike has the goldfish. So you can figure out the one-to-one correspondence and figure out what each person has. And that will definitely get you to the Oliver has a dog being false. The correct, not true answer here was A. The other option is to look at the answer choices themselves. Remember, we told you that we also might be able to find contradictions. We have here that Nate has a dog and also Oliver has a dog. They can't both have a dog in this scenario because they each have one of the pets. And we have here Oliver has a dog and Oliver has a cat. That is also a contradiction because he does not have two pets in our problem. So by eliminating A as a falsehood, we can remove the conflict between both answers D and E. So if A is false, then D can be true and E can be true. So that would be another way of telling that A does not work with the other answers. Okay. So more than one way, that's fine. Whichever way you like to do it, go ahead and practice, right? Okay. This one was one for you, Shorya. Yay. Okay. Let me zoom in real quick. Okay. Paul, Darius, Mika, and Jeff met at a concert in Chicago, but they came from different cities, Pittsburgh, Dallas, New York, and Washington. So we have a person who starts with P, a city starts with P, you know, they're trying to confuse us with all these big names. Paul and the boy from Washington met in Chicago early in the morning on the day of the concert. They had never been to Pittsburgh or to New York. Mika is not from Washington, and he came to Chicago later than the boy from Pittsburgh. Jeff liked the concert more than the boy from Pittsburgh. So we want to find what city is, what city is Jeff from? Okay. So I'll just give you all a moment to think about that. I feel like one of the good strategies just while you're solving this to write it out instead of trying to do it in your head, because there's a lot of clues. And you can give me the answer in your chat. I think there is a poll for this one also, right? Okay. So I'll just launch the poll once I get a lot of answers. Yeah. Oh, and as a side note, when you're doing this, not only should you write, but I feel that, remember the lesson on tables? You should try to make a table out of this. It might be helpful with all the clues. Just a hint. I'm seeing a lot of really good responses. Shall we launch the poll for you? Sure. Here, I'll do it. OK, there it is. The question's still on it. But of course, you can move it off to the side if you can't see all the answers. Okay, guys, make sure to put your answers in the polls so that I know what everyone wants to answer And then we'll just go over it Okay, I guess I'll just end the poll now and share the results It just disappeared off my screen Okay, there it is All right, so let me just share the results with y'all So Washington was the most picked answer good, and we had a little bit of all the other cities except Dallas Okay, so yeah, Washington is gonna be the right one But let me just let's just go over it and again like I said a table is gonna be very helpful for us so that we can analyze all the clues and then we can See like which one is really gonna be the right one All right, so let's just start from the beginning Paul and the boy from Washington met in Chicago. So let's just start from that Paul and the boy from Washington so that means since each of these four boys is from one city like It's a it's a one-to-one correspondence. That means that Paul isn't from Washington. So let's just draw like a little like X I'm just gonna do an X There we go Paul is not from Washington Okay, and they had never been to Pittsburgh or to New York So this mystery boy from Washington and Paul, they both had never been to Pittsburgh or to New York So what that means is that Paul isn't from Pittsburgh? He's not from New York. So Paul has to be from Dallas, right? So we know Paul is from Dallas. Okay, we got it Let me just outline this box Okay Now moving on to the second one Mika is not from Washington, okay No, and he came to Chicago later than the boy from Pittsburgh, so remember That means he isn't the boy from Pittsburgh. So that's what all these clues about coming later early in the morning are about They're just trying to like hint at us This isn't the same person and remember since this one-to-one if Paul is from Dallas Mika can't be from Dallas At the same time and that means Mika must be from New York Okay, good. So we have two figured out now. We just need to think about Darius and Jeff So Jeff liked the concert more than the boy from Pittsburgh again that tells us that he's not the boy from Pittsburgh Okay, he's not from Dallas since that's Paul He's not from New York since that's Mika and that means that the only place Jeff can be from is Washington so that means Jeff must be from Washington and from this same thing we can just you know Put X's here here and here Darius must be from Pittsburgh Which we can figure out. So if you look at this table, I can see that if you look at all the check marks Let me outline them in like red if you look at all the check mark boxes They're all in different rows and different columns because it's a one-to-one and so all of them None of them can be in the same row or column as each other. So this is another reason why Using a table really helps so instead of like Having to read and reread all the clues You can just like once you get a check mark put X's in the other row So like once we got Paul being in Dallas, we can just put X's over here because we know the others can't be in Dallas Yeah, so yeah Again this is just one of those one-to-one correspondence ones and like from the lesson with the tables Sometimes it helps you stay organized Especially if you have more clues because maybe in the real contest you'll have a problem with like five sentence clues or something So it's just helpful to stay organized Thank You Sharia hopefully students like that problem I like that problem Okay, Karen wants to place five bowls on a table in order of their weight She already placed the bowls Q, R, S and T in order Bowl T weighs the most so I'm going to write that this is the heavy end and this must be the light end Excuse my Tablet sometimes gives me a little Extra lines when I touch with the back of my hand. Sorry Okay, where must she place bowl Z? So we need to figure out where we can put bowls Z So it is in order correctly as we put them from light to heavy and you'll see the choices are to the left of Q Between Q and R between R and S between S and T and all the way to the right So we can choose any of the spaces to put it There is a poll I'll launch that in a moment Alright, I'll launch the poll. Try to think the different shapes have different weights. The triangles are all the same. Every triangle is the same as each other triangle. The squares are the same as each other. But the triangle and the circle and the square have different weights. Okay, I'm going to launch the poll. All right, most students have answered the poll. Thank you for doing that. I'll share the results right now. Looks like you did really well. This is another one where 3 4ths of you have the correct answer. It is between Q and R. We'll just double check for those who didn't get it or double check why that's the best answer. So if we compare the different shapes, we can see comparing Q and R, what changed, the square is in, there's at least one square in all of the bowls. So we can kind of ignore the last square since it appears everywhere. It doesn't affect the relative weights, right? It's like if everyone holds a 10 pound weight, we're still gonna put them in order and the 10 pounds you could subtract or add to everybody. Okay, so we can tell between the first and the second that when you switch from a triangle to a circle, the circle is heavier. So triangles weigh less than circles. Now we can look at either QS or the RT combination. You can see when we turn a circle into a square that the square is heavier than the circle. So the square is heavier than the circle. So now let's take a look at what we have in Z. In Z we have one triangle, one circle and one square. So again, we don't really have to worry about this last square because everything has a square on the right side. So if we look at the circle and the triangle and the circle, we can see that that is going to fit in between having two triangles and two circles, we have one of each. So the correct answer is going to be between Q and RB. Irina, Anne, Kate, Olga and Elena live in the same two-story house. Two of the girls live on the first floor. Three of them live on the second floor. Olga lives on a different floor than Kate and Elena. Anne lives on a different floor than Irina and Kate. Who lives on the first floor? Other than the tongue twisters of names, I think you can work out the problem. I'm always, I personally like drawing. So I would do, make two lines for the first floor and I would make three lines for the second floor and I would try to fill those in, but it's not necessary. It's just one approach. All right, last few seconds if you want to put an answer into the poll. Otherwise, I will close it in about three seconds. Thank you. You did really well. We have about the same ratio on all the polls that I've been talking about. We have about three-fourths of you getting it right, which is amazing. It's a really good job. If you're not getting the questions correct today, that is fine as well. Because the idea is to listen and to try and to explore how to do these problems. So the next time you have a problem like this, you'll have a good tool, a good way to read and analyze and solve. So if you don't get them correct on your first attempt, that isn't the problem, right? The problem is not trying to learn and improve and do better every time we meet, every time we have our webinars, every time you do practice contests. You're just trying to get better and better and trying to grow. All right. So remember, for these questions, sometimes you need to read them a few times to get all of the information. So I read that I have two girls who live on the first floor and three on the second floor. So that's why I drew this little diagram for myself, two lines on the first floor and three on the second floor. Then I have to split five girls into two groups, one with two, one with three. It says Olga lives on a different floor than Kate and Elena. Right. And then it says Anne lives on a different floor than Irina and Kate. So from that, I can see that Kate lives with both Elena and Irina. And the other two girls who are in this situation are going to be Anne and Olga. Now, nowhere does it say Anne lives with Olga, but by being able to figure out that Kate lives in a group of three, we can put those three on the second floor and then we can say that Anne and Olga live together on the first floor. We have enough time to go right to our bonus question, so let's do that. There are five children, Alec, Bartek, Tzarek, Derek and Edik. I don't want to say those again. So it's going to be A, B, C, D and E. One of them ate a cookie. OK, so we do know that one of these children ate a cookie. A says, I did not eat the cookie. B says, I ate the cookie. C says, E did not eat the cookie. D says, I did not eat the cookie. E says, A ate the cookie. This is a really important part of this question. Only one child is lying. Who ate the cookie? One child is lying. That means four are telling the truth. So you figure out who ate the cookie so that there are four truths and one lie in the five statements given. There is a poll, but I'll give you some time to think through this one. All right, I will launch the poll. If you are a person who does not see your screen and wants to jot these down before I launch the poll, I'll give you just a few moments to do that, and then I'll launch. Remember the question asks who ate the cookie not who is the liar. That would be a different question and would be an appropriate question as well is which of these statements is a lie but we're asking who ate the cookie. Okay, most of you have answered the poll. Thank you for participating. I think you found this question just a little bit trickier because about half of you have said that it was Bartek who ate the cookie. And we have 28% saying that Edith ate the cookie. So there's been a little bit of a difference here. Let's take a look at the problem. So remember I said there could be two questions to ask from this problem. The first is who ate the cookie. The other is who is the liar. So you might see either one of those questions on a future contest, right? Because we know only one is lying. So they might ask you to identify the lie. So remember I said that sometimes you just have to try things and then see if there's a conflict or if it works. So I'm going to say Alec ate the cookie. So in green, we're gonna assume that Alec ate it, okay? So Alec ate it is green. If Alec ate the cookie, he says, I did not eat the cookie. That is a lie. Bartek says, I ate the cookie. That is a lie. So we know that that's already two lies and only one child is lying. So it is not Alec. If I switch to blue, if Bartek ate the cookie, Alec says, I did not eat the cookie. That is true because we decided to try Bartek ate it. Bartek says, I ate the cookie. That is also true. That's what we assigned. C says, Edek did not eat the cookie. That's also true. He didn't. We decided Bartek did. Darek says, I did not eat the cookie. That is also true. But E says that Alec ate the cookie. That is the lie. So if we do that, we have the one lie that we are looking for. So that is correct. We can go through and we can test other responses as well. We can test what happens if C eats the cookie. Well, if C eats the cookie, Alec says, I did not. That is true. Bartek says, I ate it. That would be a lie. C says, Edek did not. That is true. Darek says, I did not. That is true. And Edek says, Alec ate the cookie. Well, that is a lie because we had decided that C ate the cookie. So we can eliminate this answer. The other answer we had a lot of was students saying was Edek ate it. So let's make Edek ate it in pink. If Edek ate the cookie, Alec says, I did not eat the cookie. That is true. Alec did not eat it. Bartek says, I ate it. That would be a lie. C says, Edek did not eat the cookie, but we said he did. So that would be a lie. Sorry, there we go, a lie. Darek says, I did not eat the cookie. That's true. And Edek says, Alec ate the cookie. That's a lie. So that would be three lies. So we know that that does not fit because we only had one lying child. So you can test for the D for Darek as well. Same ideas. We wanted to find only one liar. Now, if they had said which statement is not true, then the correct answer here would be Edek's statement that Alec ate the cookie is the lie. So let's go back to the wrap-up slide for this lesson. Come on. So I hoped you enjoyed the logical reasoning problem. Everyone might have their own type of favorite. You might like the picture one where we had a Sudoku. You might like the one-to-one correspondence where we made the table and where the boys are from. You might like the liars and truth-tellers types of problems. You will find lots of places to practice these types of logic problems online, but also in our Math Kangaroo past contests. We select the topics to teach in the webinars based on which types of problems occur frequently on the contests. So if you look through the past contests, you'll almost always find some sort of logical reasoning problem. These don't require calculations in general, but they do require you to use your critical thinking, finding contradictions, finding things that can't be true at the same time. Do you have any particular advice, Shoria, other than reading the problem statements? These can be reading comprehension problems. Look for small words that say not true instead of is true or exactly or at least. Yeah, I feel like the biggest thing for these types of problems specifically is just make sure that you're very organized when you're doing it. Like in this question with the five children, this was actually, I just want to share this because it's kind of funny. When I was in fourth grade, this was the question that I got when I took the test in 2019. And if I had known that I should have done case work on every single, by thinking which person ate the cookie, I probably would have done it really quickly. So just make sure that you're organized and that you try every case because it's not going to take you very long and you'll begin to see a pattern like as we did with Alec and Bartek. You'll begin to see which one is the one who ate the cookie based on the other conditions. So just make sure it's like, it's more about just making sure that you're reading very carefully, like Dr. Sagi said, and also that you just look at all the different things. These problems are kind of confusing. So if you're able to like make a table or write them out in some kind of list, that would really help. And of course, the best way is to just, do all the practice problems you can and you'll just get better and faster as you go. Thanks, Shurya. And admitting that we make mistakes when we try things when they're new to us, it's very natural. So if you were having a little bit of a struggle today, just keep at it. It will get better as you practice more. And as you look at hints and how other people solve things, that's the best way. That's one of the great ways that you can do learning and improving of your own skills. All right, last webinar next Sunday, I will see everybody there. And remember, there is another series of webinars starting in January. All right, have fun practicing math and logical problems. Bye everybody. Bye.

logical reasoning

truth-tellers and liars

contradiction

elimination

tables

problem statements

pattern recognition

critical thinking

logic puzzles

analytical thinking