Welcome to the Level 3-4 Math Kangaroo webinar. This is webinar number seven. And the title and subject today is Logical Reasoning. So welcome to our webinar. Good afternoon. And here's a warmup problem. You can begin with this one. It says, mother bought 16 oranges. Carl ate half of them. Ava ate two. And Sophie ate the rest. How many oranges did Sophie eat? And I do have a poll for this. I'll go ahead and launch it because I think it's pretty easy to read in the top of the poll. As usual, if you want to send me the answers in the chat, I try to give you a little thumbs up or smiley. I can't get to everybody, but I do my best. Ishan can also help you. I'm going to make him a co-host so that you can communicate with him. He's our teaching assistant. So if you have questions, go ahead and you can select to talk to Ishan in the chat and ask him. He'll help you out too. Yeah, the students have done really, really well. Almost everybody answered the poll and check this out. I don't think I've ever had this happen, Ishan. I do not think I have ever had a 100 percent unanimous answer in a math kangaroo poll before. Good work everybody this afternoon. That's amazing. So you are absolutely correct. So if we start with 16 oranges and we divide that by two, because we know that Carl ate half of them, that is eight, right? So the 16 minus eight gives us eight. Then we know that Ava ate two, and then Sophie eats all the rest, which is six. Perfect. Everyone did that. All right, our next problem. All right, let's go. Remember, logical reasoning is our topic today. One of the important things in these logical reasoning problems is going to be to read and understand the problem very carefully. I know that sounds obvious, but there'll be times when there's going to be some really important words in the problem statement, and that's usually where students get a little bit confused, is making sure they understand the problems exactly, and they're solving it just to get the proper answer. One of the things I like to do is, as I'm reading the problems with these logical thinking sentences, one by one sentence, I might take notes and see what that exact sentence means. A lot of times after I've written out the information in that sentence and gotten it all together, then the picture comes together and I know exactly what I need to do. So if you're getting stuck, you might try to go sentence by sentence and see if that helps you. Let's see what happens today. As you can see, this is our webinar number seven. There's going to be three more in this series. If you've missed any of the others, you can go into your Math Kangaroo account where you've registered for this webinar series, and there are recordings to the prior six webinars if you've missed any of them. Or if you want to see something again, if you want to double check on something, you can go ahead and go back. So in logical reasoning, we need to make conclusions based on the given facts and on some reasoning. So by reasoning, we mean, we're going to ask you to know some mathematical facts, like adding and subtracting. We're going to expect you to know what half is. We might make you need to know what older or younger. There's a question here. Maybe we might make you think that you know something about time or other things like that. So you're going to use those reasonings. Sometimes it's even and odds we might need you to know. So you might require guessing. You might say, if I do this, then I would expect this. Did I get it? If I do get it, then maybe my guess was correct. If I'm not able to get the expected results, I might have to change my guess. So we can do sometimes some guess, check, and revise in logical reasoning. And then sometimes we might need to make a list of all the different results. We've had a listing and tables webinar. So you might want to go back to using some of those. Sometimes I draw. And sometimes there's more than one way to solve the problems that we have here today. So keep in mind that the way I solve it or the way Ishan solves it, you might have a slightly different method or a completely different method. And that's OK, because we're all going to be thinking about them with our own brain experiences and our own set of what we like to do more than other things, right? Adam, Luke, Tom, and Alex went out for ice cream. Adam ate more ice cream than Alex. And Tom ate more ice cream than Luke, but less than Alex. List the names of the boys in order from the ones who ate the most ice cream to the one who ate the least. Most to the least. And I will launch a poll. I know a lot of you like to use the polls for your answers. Okay, I think we might have a genius group here today, Eshaan. So if I share the results of this poll, we have practically unanimous correct answer again. So this must be the super holiday version of a Math Kangaroo webinar where everybody gets the answers correct. That would be amazing. All right, so the way I read it, it says Adam ate more ice cream than Alex. So if we're going to list the most first, I'm going to put Adam to the left. And then somewhere, I'll leave some space just in case, I'll put Alex. All right, so if this is the most and this is the least. It says Adam ate more ice cream than Alex. And Tom ate more ice cream than Luke but less than Alex. So Tom has to be less than Alex but more than Luke. So I have Adam, Alex, Tom, Luke. That is B. Very nice. Some of you have done this one before, so I'll go through it pretty quickly. I'm not going to do a poll. I'm going to just solve it as I read it. How does that sound? Albert fills the grid with these figures. Each figure appears exactly once in every column and every row. So exactly once in the column and the row. This is like a Sudoku puzzle. Which figure must Albert put in the cell with the question mark? So I'll solve it for you right now with you together. What can go in the question mark? So in this column, we are missing the ghost and we are missing the rhino. So we need a ghost or a rhino here. And what's missing here? We have a ghost. We are missing a shark and a rhino. Okay, so I cannot put a shark in this row because there already is a shark. So there must be the shark in here. And then the rhino would have to go here. If the rhino goes in the fourth row, then the ghost must go where the question mark is. And the rhino must go up above. So the correct answer is that we are missing the ghost. So some of you have seen this before in our webinars earlier in the year. So I don't want to spend too much time. But if you haven't seen it before, it still is a really fun question. There are ten rocks in a pond. Some frogs and turtles are sitting on the rocks. Pardon that, I should say rocks. There's no more than one animal on each rock. More than half the rocks have an animal on them. The number of frogs on the rocks is three times the number of turtles on the rocks. How many frogs are sitting on the rocks? Be careful to answer the question is how many frogs are sitting on rocks. Very good. I'm seeing a lot of correct answers in the chat and in the poll, and some really good explanations for what you're doing in the chat. I appreciate you sharing your thoughts with me. So I'll end the poll here and share the results. So 70% of you have said the answer is 6. The next most popular answer is 9. So that's an interesting answer. 9 is a multiple of 3. That is less than 10. But you have to account for also that there are turtles on some of the rocks. So let's take a look. I can do it by two different methods. I've had students like both of those methods. So if I have frogs and turtles, and then of course the total is going to be that there are 10 rocks that we can be on. So if I select that there are going to be three frogs, because I know there's three times as many, I have to use multiples of three. Then there'd be one turtle, and that's four animals all together. So that is not enough, because it says that more than half the rocks have an animal. So this has to be at least half of 10. So I used to have to have at least five. If I have six frogs, that would be two turtles, and my total is eight. That works. That's at least the five. If I try to go up one more multiple of three, that would be nine frogs, three turtles, and that gives me a total of 12. So that is too many. So the correct answer is that there are six frogs. I'll switch colors, and I'll show you another way that some of you might like to do this problem, because I know that we're all slightly different in our thinking ways. So if I draw little spaces for the rocks, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 rocks, and I can put frogs and turtles on them. So for every three frogs, I'll have to put in one turtle. But as you can tell, that is not halfway across my rock path. So I can do it again. I can put in three frogs and one turtle, and that works. But then I only have three spaces left, and I can't do three frogs. Oh, I need one more frog, don't I? Frog, frog, frog, frog, turtle. Yeah, so there I have eight animals and two empty rocks, and that works. And that also would be 1, 2, 3, 4, 5, 6 frogs. So by both methods, I get the answer six frogs. I like it when my answers agree by different methods. It proves that I can check my answers, right? A secret agent wants to break a six-digit code. He knows that the sum of the first, third, and fifth digits is equal to the sum of the second, fourth, and sixth digits. Which of the following could be the code? So the first, third, and fifth, that's like all of the odds, and then you have the evens. You can think of it as every other. Remember, you can only have a single digit in each spot. There is no poll because I didn't want to have to try to get these little picture options in there. So just put your answer in the chat if you'd like to share it with me or Ishan. Remember, if you have questions, you can try to ask Ishan as well as me. All right I will start by examining possibility A. So if I look at A and I want to think about it as these three have to equal the sum of these three. All right so I have 1 plus blank plus 1 in green and in blue I have 8 plus blank plus 6. So so far I already have 14 here and here I only have 2 and 14 minus 2 is a difference of 12. There's no way to make up a difference of 12 if the largest digit that I can use is only a 9 because even if I put a 9 here I'm gonna get that my sum is going to be 11 so this is not going to work. I can use the same thing for all of these so for this one I have a sum of 21 on the top. On the bottom I only have a 2 plus I have 2 blanks. If I subtract 2 from here I get 19 and there's no way to get 19 with just a 9. The largest I could get is 9 plus 9 which was be 18 so it won't work. Can kind of do the same thing here 4 plus 4 plus 4 is 12 and on the bottom I only have 2. I would need 2 plus 10 that is not a one-digit number. On this one I have 1 plus 2 blanks. The bottom I have 2 plus 9 plus 8 is 19 but if I subtract that one I need 18 left. 18 I could do by putting nines into these two blanks here so that is the only one that is possible. You can see that I end up with the same issue here. I have only a total of 4 on the top and I already have 8 if I'm using the evens on the bottom so there's no way to subtract with the instructions we have which says to add to find the sum. So the only possible answer is D. In the park there are 15 animals cows cats and kangaroos. We know that precisely 10 are not cows and precisely 8 are not cats. How many kangaroos are there in the park? There is a poll for this one. It's a pretty short question to read. I think we'll launch the poll right away. This one tests your logical thinking a little bit more. I'll give you a hint. We know that precisely 10 are not cows. We still have about a third of you who haven't put an answer into the into the poll. Does anyone else want to give it a try? I'll give you a hint For that third of you who are still thinking about it if my light The light switch in my room can be on or off if my light is not on then my light is It must be off there's no other position, right? so if precisely 10 are not cows and There are 15 animals Then the other five that are not cows The other the five that are not not cows right if you're not not a cow then you are a cow And the poll The majority of those who answered are correct. It is that there are three kangaroos So that's very well done Let's take a look at what I was saying right out of 15 We have 10 in the not cow group And then we must have five in the cow group Likewise out of 15 if we have eight in the not cat group And We have seven in the cat group So out of 15 if I have five cows and I have seven cats Then I must have some number of kangaroos This is 12 right so this must be three kangaroos You So easy when you think about it like that But it's very difficult to if you've never experienced a problem like that getting into that if it's not 12 are not cows Then five are cows. It's once you get that then it falls together very quickly But if you don't reason that little piece out of it Then it's a very difficult problem because then you might be stuck doing a lot of guess-and-check Trying to figure out okay ten are not cows how many are cats how many are kangaroos? And you might be doing a lot of guess-and-check Until you make it work with that eight are not cats so that you have the sums that all add up together And that it is possible to do it that way It's definitely possible to reason through and make some guess-and-checks knowing that ten are not cows that okay I have to have a sum of ten that are kangaroos and cats and then you should be able to figure out that If would have to work that with this part as well, okay All right, Ishan is here and he has requested to lead the next two problems for you So first I'll read out this problem and then we will open the poll and then I'll give you guys a little bit of time to solve Question six states Olivia is 10 years old and her mother is six times older than her Olivia's grandmother is 14 years older than the ages of Olivia and her mother added together Olivia's great-grandmother's age is equal to the sum of Olivia's grandmother's and Olivia's mother's age How old is Olivia's great-grandmother? Okay, give you guys a little bit of time Okay, so I will end the poll now. Around 88% of people participated and of those 70% got it right, so it's pretty good. So I'll restate the question first. The question states, Olivia is 10 years old and her grandmother is six times older than her. Olivia's grandmother is 14 years older than the ages of Olivia and her mother added together. Olivia's great-grandmother's age is equal to the sum of Olivia's great-grandmother's and Olivia's mother's age. How old is Olivia's great-grandmother? Okay, so Dr. Siggy, could you write for me? I'm not sure how to annotate. Okay, I opened it for you, but I can annotate if you'd like. Okay, so first, let's define what we're looking for. We want Olivia's great-grandmother's age, right? And the third sentence tells us that Olivia's great-grandmother's age, let's denote that as GG, is equal to the sum of Olivia's grandmother's, let's denote that as GM, and Olivia's mother's age. So GGM equals GM plus M. That should make sense to everyone. Okay, and now we can simplify this further because we know that Olivia's Olivia's grandmother is 14 years older than the ages of Olivia and her mother added together. So here we know that GM equals O for Olivia plus M. And looking at that, we can just start with our original expression, GGM plus equals GM plus M, and turn that into GGM equals O plus M plus M plus 14, or GGM equals O plus 2M plus 14. Okay, and from here, we know something else. In the first sentence, we're given that Olivia is 10 years old and her grandmother is six times older than her. This means that Olivia's grandmother is 14 plus O plus M equals 60. And with some slight algebraic manipulation, we subtract 10 and 14 from the right and from the left, we have the mother's age should equal 36, if I did my mental arithmetic correct. And then from here, we know that O plus 2M plus 14 equals GGM, and M is 36. So what we have is 10 plus 2 times 36 plus 14 equals the grandmother's age, or GGM. And then from here, we know that O plus 2M plus 14 equals GGM, and M is 36. So what we have is 10 plus 2 times 36 plus 14 equals the grandmother's age, or GGM. We know 2 times 36 is 72. So we have 72 plus 10 plus 14 equals GGM, and that simplifies down into 72 plus 24, or 96. So that is our answer. Olivia's great grandmother is 96 years old. I hope that makes sense to everyone. Students may also want to try taking it one sentence at a time. Since you know that Olivia at the start is 10 and her grandmother is 6 times that, they can go ahead and start out with those numbers instead of using the variables. So sometimes that helps. Okay. And then number seven. Okay, I'll state the problem for number seven as well. Question seven states, each of the 10 bags contains a different number of pieces of candy. The number of pieces of candy in each bag ranges from one to 10. Each of the five boys took two bags of candy. Alex got five pieces of candy, Bob got seven pieces of candy, Charles got nine pieces of candy, and Dennis got 15 pieces of candy. How many pieces of candy did each of the five boys take? And we'll open the poll and give you guys a little bit of time, maybe till 3.57. Sorry about that. Sometimes I hit this button and it changes the screen. Only around 58% of people have answered, so we'll give you guys like 30 seconds or another minute. Okay, then I'll start the problem now. First, I'll reset the question again. Each of the 10 bags contains a different number of pieces of candy. The number of pieces of candy in each bag ranges from 1 to 10. Each of the 5 boys took 2 bags of candy. Alex got 5 pieces of candy, Bob got 7 pieces, Charles got 9 pieces, and Dennis got 15 pieces. How many pieces of candy did Eric get? So this problem seems very unapproachable, and if you don't know how to solve it, it seems very complex. But if you know how to approach it, it's actually very easy. So the first thing we have to take a hold of is the first thing the question gives us. Each of the 10 bags contains a different number of pieces of candy, and the number of pieces of candy in each bag ranges from 1 to 10. What that tells us is that these 10 bags contain in some order 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. And we can add these up together to get that there are 55 total pieces of candy within these 10 bags. That should make sense to everyone. And from here, we know the sum of 8 of those bags, because each of the 5 boys took 2 bags and were given the number of pieces of candy 4 of those boys have. Alex has 5, Bob has 7, Charles has 9, and Dennis has 15. Which means that however many pieces of candy are remaining must be the remainder, I think, from when you subtract 36 from 55. Because those 4 numbers represent the sum of 8 total bags. So there's 2 more bags left over, and that's what we're trying to search for, right? How many pieces of candy are in the 2 bags that Eric got? So we do 55 minus 36, and we get 19 missing pieces of candy in those 2 bags. Which also tells us that those 2 bags are the ones that contain 9 pieces of candy in 10. But that's not necessary. Thank you, Ishan. And just to share the poll, since students did that, we did get that two-thirds of you had it correct in the poll. So that's pretty good. This is one of our 5-point questions, so good work on that. I'm going to ask any of the students who are experimenting with the annotation to please not experiment with the annotations. If it is open, it is because I'm allowing Ishan to annotate, not the students. All right. I have had a student very successfully solve this problem in another way. And that way was to actually just make a guess. Come on, what's going on here? There we go. So if we make a guess that the 5 pieces of candy might have been a 4 plus 1, then you can say, OK, so 7 may be equal to 5 plus 2. Because it couldn't be 3 plus 4 if I've already used the bag with 4 pieces, and it couldn't be 6 plus 1 if I've used the bag with 1 piece. Then we know that Charles got 9 pieces, and the only way that we can make 9 pieces with the remaining bags would be 6 plus 3. And then Dennis got 15. The only way to do that with the remaining bags is 7 plus 8. And if you do it this way, you'll see that the bags with 9 pieces and 10 pieces have not been distributed to the children, so that there would be 19 pieces left. So that is an additional way to do it. That works out reasonably fast, so students may have tried that. There are 5 children in a family. Kitty is 2 years older than Betty, but 2 years younger than Danny. Teddy is 3 years older than Annie. Betty and Annie are twins. Who of them is the oldest? Remember I said there might be some assumptions that we expect you to know. You should know what it means to be a twin in a family. I'll give a few minutes. We do have a poll for this problem as well. Hey, we have most of you have put in an answer. Thank you for doing that. And you can see that 77%, three fourths of you think the answer is Danny. That is correct. The next answer would be Teddy. Let's see how we can get to it. I'm going to actually show two methods because I think that we have two really good ways to do it. And some of you may prefer one over the other or want to see both. So one is just kind of the relative thinking, kind of more abstract for some people. So the first thing it says is that Kitty is two years older than Betty. So there's two years difference between Kitty and Betty, but two years younger than Danny. So there's also two years in between Danny and Kitty. Okay. It says Teddy is three years older than Annie, but we don't know where Annie is until we read the next sentence that Betty and Annie are twins. So they're the same age if they are twins. They're usually born on the same day or very close together, right? So they'd be the same age. With Teddy being three years older, that puts Teddy here in between Danny and Kitty. But the oldest is Danny. Now you might say, what's the other method? The other method is to just assign an age. We don't need to know their exact ages. We just need to know who's the oldest. So if we assign an age, and we can assign an age just to the first person it says. So we can say, okay, for example, Kitty is five years old and she's two years older than Betty. So then Betty would be three and two years younger than Danny. So that makes Danny seven. Teddy is three years older than Annie. Again, we don't know where Annie is until we read that Annie and Betty are the same. And if Teddy is three years older than Teddy is four. So again, we get that Danny is seven and seven is the biggest number. So that would work really well to say that Danny is the oldest. So two methods give you to the same solution. Remember we always say that step number four is to check your answer, right? So what better way to check your answer than to try it another way and get the same thing? So these are the logical reasoning problems. We have these on pretty much every Math Kangaroo contest. There'll be at least one or two logical thinking problems. And you might say, well, this isn't strictly mathematics. But when you get good at logical reasoning, it does help your mathematical problem solving because you do need to be able to find things that are true, to be able to find things that are false, to find things that might be true if this is true. All of these things happen in our math challenges and in everyday challenges. We do have some really fun bonus questions. I really love this one. Mary has nine small triangles. Three of them are red, three are yellow, and three are blue. She wants to form a big triangle by putting these nine small triangles so that any two triangles with a common edge are a different color. So that's important. A common edge make different colors. Mary places some triangles as shown in the picture. Which of the following statements is true after she has finished? I do have a poll, but I'll give you a minute to look at it on the big screen before I open the poll because for some of you it covers it up. I will launch the poll now that you've had a chance to read that question. If it is blocking your view of the picture and you don't like it, you can always close it. The hint for you is to start with the triangles that only have one possibility, so they have more constraints, would be the more scientific or engineering term. They have greater constraints, so they can only be one color. This is a five-point question. It can be a little tough or tricky for some students. About half of you so far have answered it correctly. So let's take a look at it together and see what we can figure out. What was I meaning by start with the ones that have more constraints, where we know that there's going to be a particular answer? So the blocks that are already next to two others. So this is a red and a yellow, so this must be blue. We can't have any other color except for blue that's next to red and yellow. Likewise, we can't have any other color except for red that is touching a yellow or a blue. So that tells us two of them already, OK? So any of these choices have number four or number five. This one says number five is red. That is true, so we haven't eliminated anything yet. But one of the things you might have noticed is that it says that we have three of each. And we have already used our three blues, so we have no more blues. We've also used two reds, so we only have one red left that we can use. And we've only used one yellow so far, so we have two yellows that we can use. So how can we fill in the last three spaces with one red and two yellows and not have them touch? We would have to put the yellows in number three and number one and then the last red in number two. That's the only way to not touch them given what we have left. And then if you look at your options, you'll see that one and three are yellow. And if you look at them, you can cross out three is not red, so that's false. Number one is not blue, so this is false. Number one and three are both not red, so this is false. Number five is red, but number two is also red, so that's false. So E is the only one that would be true. All right. That is our time for today. I hope you enjoyed doing our logical reasoning problems. There's a pretty wide variety of things that we can ask with logical reasoning problems. So you could be asked like this picture about different colors. You could be asked about different ratios. You could be asked about ages. You could be asked about truth tellers and liars. That's also a very common logical reasoning problem. So you might practice looking at some problems with truth tellers or liars. Just a couple of hints to give you to help you with math kangaroo practice. Remember, the best way to practice math kangaroos, of course, come to our webinars, but also take a look at past contests. Past contests are really good indicators of the type of problems you will find on future contests, and they'll give you an idea of how much time you can spend on each problem, because one common area is that students run out of time. So learning how much time you can give to the three-point, the five-point questions, that will help you out with the pacing of your contest. Ishan, any final thoughts today? Nope, but I will say that I think logical thinking is probably the most important thing you'll take from math competitions in general. Yeah, it really helps us in, because mathematics is everywhere, and logical thinking is also one of those topics where even when you're writing in English, you want to make sure that you're not breaking mathematical rules when you're trying to prove a point in your writing. So some really important things here. All right, everybody, have a wonderful rest of your weekend. Probably most of you have tomorrow off of school, so have a good holiday. Happy President's Day.

Math Kangaroo

Logical Reasoning

mathematical puzzles

problem-solving strategies

logical deduction

numeric puzzles

candy distribution

reading comprehension

constraint-based logic

real-world applications