Hello, welcome everybody to the fifth webinar for our series. I'm trying to make sure the chat is open. Let me open the warmup problem for everybody. Which four digit need to be removed from the number 4,921,508 to 4,921,508? To get the smallest three digit number. What do you think? Can you put your answer in the chat? Let me see if I have a poll for this one. Sometimes I do. There you go, there's your poll. So, I know some of you just entered and I don't know that you can see the polls if you come in after it's already been launched. So, I will share the results of this poll and go over the problem. Sorry, just making, just making a note here, okay. So, the most popular answer was E with almost two-thirds of you saying E. So, let's take a look and see if that's correct or not. This is an interesting problem. I think that, in my personal opinion, maybe there should be a little more explanation about this. So, it's saying, which four digits need to be removed from the number to get the smallest three-digit number? So, we're going to remove numbers and then we're going to read what's left. We're not going to rearrange it. We're just going to see what's left, and that has to be as small as possible, okay? So, let's see what we can do. The smallest digit here that can be in the hundreds place is the one. So, I would like to make the one the hundreds, which means I have to remove the four, the nine, and the two. So, this is not a good choice. This is not a good choice, and this is not a good choice. I'm left with two of them now because I want to leave the one in the hundreds place. Now, I can use a five or a zero in the tens place. I would prefer to have a zero in the tens place if I want to make a really small number. So, I'll get rid of the five, and then I'll have the number 108. 108 is the smallest number, so the answer is D. Is that a tricky one? What I think a lot of students do is they think you can rearrange the numbers, but the simplest interpretation of this problem is that you just have to remove some numbers and leave what's remaining intact in order from left to right as your three-digit number. The correct answer here was D. Hopefully, that makes sense. Okay. I know. Once I explain things, it's always so much clearer than when you first look at the problem, huh? All right. So, to give you an idea, what we're doing this week is guessing and checking, but Math Kangaroo does not mean random guessing. We mean use your best properties of math principles. So, in this problem that we just did, we were using place value, right? So, we needed a small number in the hundreds place. So, we have to understand what the problem is asking. What plans can we use? What knowledge do we have that's going to make it so it's not a random guess, but it's an educated guess? And while it's been guessing, we might say, I'm not 100% sure. I'm going to go by, I know this, and that leaves me these possibilities that I can test. So, that's what we're kind of doing today is we're eliminating some possibilities. We're using good reasoning and logic, and then we're going to get to maybe where we have to try two or three different answers to see what works. And then this is really important when we look back and reflect, because what you'll notice is we have a strategy called guess, check, and revise. Revise means you're going to fix yourself, right? How many of you know that revision is like editing your writing? So, a good example that I like to use for guess, check, and revise is a game that I play with my kids. So, if we're doing something like driving in the car and it's getting kind of boring, we play a game called guess my animal. Okay, and you guys can play along in the chat with me if you want. So, I might say to my family, I'm thinking of an animal that begins with the letter W. And they have the option of asking me yes and no questions. So, they might say, is it a wombat? And I'm going to say, no, it's not a wombat. It's not a wombat. But they might say, is your animal a mammal? And I would say, yes, my animal is a mammal. So, now they know not to ask me about anything that is not a mammal. And then they might say, does your animal live in the ocean? And I might say, no, my animal does not live in the ocean. So, some of the guesses I'm seeing in the chat are animals that live in the ocean. So, now eliminate those. Do you get the idea where we have those yes and no questions? Oh, there's one that lives on land. So, then they might say, does your animal have fur? And I'm going to say, yes, my animal has fur. So, some of you are now guessing animals with fur. That's true, but no one has it yet. You might say, is your animal bigger than a cat? And I'm going to say, no, my animal is not bigger than a cat. So, now you know that you need a furry land animal that starts with W that's smaller than a cat. Do you see how each question helps you revise? That's what revise means. So, even though you might be guessing wrong, you're getting some information. Remember, my animal starts with a W. Still no one's gotten it. I'll tell you. I'll write it on the screen. See if I can write it on my screen. I forgot my other screen's not on. Okay. There's a weasel. Yeah. You see that? So, that's what we mean by guess, check, and revise. And now you know a new game you can play in the car when you're bored. Does that make sense, Ishan? Yeah. All right. So, like I said, just an example of one way that I use this strategy for fun. I think these math problems are fun, but, you know, not everyone is. Hopefully, you all love math as much as I do. Or if not, you're learning to love math as much as Ishan and I do. Okay. A measuring tape is wrapped around the cylinder. What number should be placed at the question mark? I do not have a poll for this one, so go ahead and answer in the chat for me. Oh yeah, these answers are coming in correct and quick, so I can't get high fives to all of you and thumbs up and stuff because there's just too many answers coming. You guys are awesome. Yeah, so by far most of you are saying 48C and that is correct. There are a couple, the basic way to do this is to look at, we know that the cylinder has the same diameter, so when we wrap the tape around it, it doesn't matter, it's going to be the same difference between this number and this number as the question mark and this number. So what a lot of students do is they say, okay, 27 minus 6 is 21, so then I just need to do 27 plus 21 and I get 48 and that's correct. Another way I've seen students do it is they tell me they looked at what is above the 1 and there's a 22, so they know that each time you wrap around, you have to increase by 21, so that works as well. Same idea, you can take any of these numbers and subtract the one above and below and get a difference of 21. So since it's the same diameter wrapped around, we still add 21, so we're going to add 21 to any number below to get the number above. Great idea. All right, we'll move along. Ishan, if you had replied which number you wanted to lead, it like it scrolled way up, so I don't know what it was. I said problems four and seven. All right, four and seven, thank you. Okay, so number two, each of the seven boys paid the same amount of money for the trip. The total sum of what they paid is a three-digit number that can be written as three blank zero. What is the middle digit of the number? And I do have a poll. I'll go ahead and launch it right away because I think it'll questions easy enough to read in the poll. This should not be a random guess. There's good information in this problem. We know that seven boys paid the same amount of money, right? So you can figure out what the sum would be if it's seven boys paying the same. While we're waiting to get the answers in the poll, just if this is your first week with me in our webinars, welcome. I know there were some new students registering during the week, so welcome to our webinars and welcome to Math Kangaroo Contest. Hopefully, you're registered for the contest that happens next month. So the actual contest is multiple choice like the questions are in this webinar, and you should always try to answer all the questions because even if you guess, we don't subtract if you get the wrong answer. So a guess is better than a blank. Of course, do your best before you guess. All right. I'm going to end the poll and share the results. This is a pretty popular answer. The answer is five. That is correct. Let's take a look at that. So we need to make some sort of three-digit number, three blank zero that is divisible by seven. So we know it's divisible by 10 because it ends in a zero. So we need the answer to actually be divisible by 70 because it has to be divisible by seven times 10. We can kind of ignore. Some students tell me, oh, just ignore the zero and make the rest of it divisible by seven because you just multiply it by 10 later. It's not a big deal. And that works. So the only two-digit numbers, let's see what two-digit numbers are divisible by seven. So remember, we have 28. Then the next multiple of seven is 35. And then the next multiple of seven is 42. So the only number we could put here would be 35. That gives us answer choice C. All right. So you guys did almost perfectly on that poll there. That was great. All right. The numbers are placed in cells of the four by four square shown in the picture. Mary found a two by two square where the sum of the numbers in the four cells is the largest. What is the sum? So we need four cells in a two by two with the largest sum. Remember, the answer should be the sum. I'll leave it big on the screen for a few seconds, but I do have a poll that I'll launch in just a little bit. Remember, it is guess and check, but use your best guess first, right? There's a logical place to start here. The poll does have the numbers, it does have this 4x4 grid in the picture in the poll. So, hopefully, even with that poll on your screen, you can still get everything done. Yeah, the poll is going really well, everybody. Anybody else want to put in a poll answer? All right, I'll end it. Almost everybody answered, so that's awesome. And not only did you answer, but 3 4ths of you have said 14, and that is the largest sum. So thank you for answering, and thank you for working very accurately. So on this one, remember it's guess and check, right? By far, the largest number in this grid is the 7. So it makes the most sense to try a square that includes the 7. And that gives you four choices, right? You can do this choice here. You can do this choice here. Does that make sense? So there's four choices that you can try. So let's try the different ones. So the top left, I would get 4 plus 1 plus 1 plus 7. So 11, 12, 13. I'm just going to put the numbers kind of like in here. So there's 13 for that 2 by 2 grid. If I do the next 2 by 2 grid, I'll switch colors. Over here, I get 7. Hopefully, you can see my little cursor here. 7 plus 1 plus 1 plus 3. This is only 12, so that's less. That's not going to be the correct choice. If I do the 7 plus 3 plus 3 plus 1, that is 7 plus 7 or 14. And then if I do the last one, 7 plus 1 plus 2 plus 1, that's going to be 11, right? 7 plus 4 is 11. So the largest number that I've obtained is 14. That is correct. You can try some other combinations, see if you ever get anything larger than 14. But in this case, you won't. So that is correct. And most of you did that very accurately, so good job. All right, so number 4 was a problem that Ishan wants to lead. If you're new, Ishan is our teaching assistant. If you have any questions as we're going through the lecture, he's also a co-host, so you can chat with him. All right, Ishan, go ahead. So question 4 states, Zoe has started to write some numbers in the table. She decided that each row and column will contain the numbers 1, 2, and 3. What is the sum of the numbers that she will write in these two shaded squares? We'll give you some time. Sorry, there is a problem, I will try to open it up. I think I just opened it up. Yes, I see it on my screen. I see that around 95% of people have answered this question, and around 85% of people got it right. So I'll go over it now. But 85% of people is very good. Okay, just to restate the question, Zoe has started to write some numbers in the table. She decided that each row and column will contain the numbers one, two, and three. What is the sum of the numbers that she will write in the two shaded squares? So if any of you have played Sudoku, you'll see that this is very similar, although it is significantly smaller. The key here is that each row and column will contain the numbers one, two, and three. And if you look in the square right beside the one and right above the two, that should be three, because in that square, there can't be one or two. And then if you think about the next square, the square that's right next to that three, in that same row is the numbers one and three, which means that the only number that can go in that square is the number two. And now you have to pay attention to what the question is asking for, because it wants the sum of the numbers that you will write in the two shaded squares, which means that we don't necessarily have to find those two numbers individually. Because we know that the number right above them is two, the numbers in those two shaded squares must be one and three and it doesn't matter which square they go in. It could be one and three like that or one and three the other way, but no matter what, the sum of those numbers will be four. And that's our answer, C, four. Students want to follow it all the way through. This square underneath the one and to the left of the two must also be a three, because I wouldn't be able to put a two in the same row as this two and I wouldn't be able to, the three is the only number that can go here. So that does lead you to make this B on the right-hand side a one and in the bottom corner, a three. But Ishan is also correct logically that we know they must be a one and a three with the same sum, no matter which way we do it. Okay. The queen tries to find out the three names of Rumpelstiltskin's wife. She asks her, are you called Adele Lily Cleo? Are you called Adele Laura Cora? Are you called Abby Laura Cleo? Each time exactly one name and its position are right. What is the name of Rumpelstiltskin's wife? Okay, most of you have answered the problem in the poll or sent me messages in the chat. Thank you for doing that. And let's see what we can do. I'm going to share these results. The top choice of your answers is A, Abby, Lily, Cora. But quite a few of you have chosen choice C, which is Adele, Laura, Cleo. Let's see if we can figure out which of those two is correct. So this is kind of a logical thinking question. The important parts here are it says that exactly one name and its position are correct for each of these guesses. So if I look at the first one and I say Adele, Lily, Cleo, if Adele is correct here, that means that Lily and Cleo are both incorrect. And if I say that Adele was correct in the first one, then Adele must be correct in the second one, which means that Laura and Cora are both incorrect. Now, how can that be that there's no second or third name? You see that the problem with having Adele correct is that you then eliminate both of the other choices. So we're going to erase that and we're going to start again. And we're going to say, OK, if Abby is correct, that means that Laura and Cleo are incorrect. OK, I know that Adele is incorrect because Abby is correct. Well, that means that Lily must be correct and Cora must be correct. And that gives me one in each place. So the name is Abby, Lily, Cora. That's choice A. All right, hopefully that makes sense. If you had gotten it wrong, that's OK, as long as you're learning from going through the experience. All right, so Math Kangaroo is not about being perfect on the first try. Math Kangaroo is about getting better and better as we practice. OK, the whole idea is to be challenged so that you're growing and learning new math. All right, this one is a long one to read, so I'm going to read it very slowly and carefully. And then I will give you quite a bit of time to solve this one because this one seems to be the trickiest for most students. Now, hopefully you're the exception. You get super fast. But if you don't, don't worry about it. You're not alone. In the land of funny feet, the left foot of each man is two sizes bigger than his right foot. And the left foot of each woman is one size bigger than her right foot. However, shoes are always sold in pairs of the same size and only in whole sizes. A group of friends decided to buy green shoes and buy shoes together to save money. After they put on shoes that fit them, exactly two shoes were left over. One size 36 and the other size 45. What is the smallest possible number of people in the group? So think about it for a minute. I'll give you a hint. If you were to buy your shoes all by yourself and you had two different size feet, but you had to buy pairs, you'd end up having to buy two pairs of shoes in two different sizes, right? So by sharing with your neighbor, maybe, you could buy fewer pairs of shoes. Okay, I'll give some more explanation now. We've got quite a variety of answers in the poll and in the chat. If the largest foot for every person in Funny Feet is their left foot, then on the biggest size pair that they buy, the left foot, the left shoe is going to be worn, but the right shoe will not be worn, right? Because there's no one who has that big right foot. So you're going to have a leftover right size 45. And by the same logic, you'll have a leftover left size 36 because the right foot is always bigger. The left foot is always bigger, so the smaller shoe is going to be leftover in the left size. So, I'm also going to say that if it was only men in the group, all of the sizes, since their feet differ by two sizes, all of the shoes would be an odd size or all of the shoes would be an even size. Right? Because they would always be skipping a size to make it fit their feet perfectly. But in this case, we're going from even to odd. So, we have at least one woman in the group. So, let's make the woman have the size 36 on her left foot, on her right foot, and 37 on her left foot. So, this can be one woman's pair of shoes. Does that help? So, there's a size 37. There's a right foot in 37. And if we have to make as few people as possible, we need to get from 36 to 45. So, going every other size gives us fewer people. So, let's try to match it up with a size 39. So, we could have a man who wears the size 37-39 funny pair. We would have that same man would need a size 41. We're going to have a size 43 and a size 45 because we have to buy the shoes in pairs. So, let's fill those in. So, we can have some more men wearing mismatched shoes, but they would fit the feet. So, the least number of people I can use those shoes is five. So, let's see how you guys did on the poll. About one-third of you said five, which is correct. Now, think about that. This is a number 24. This is at the end of the contest. It is a five-point question. And it was so tricky that only one-third of you got it right on the first try. But that's why you're here, right? Because now you'll understand how to do a problem like this. So, I hope that makes sense and that you're learning by being here. Okay, number seven. That was a really good one. Okay, number seven. That was for you, Eshaan, right? Okay, question seven states, write each of the numbers in 0, 1, 2, 3, 4, 5, and 6 in the squares to make the addition on the right correct. Which digit will be the gray square? And I'll open up the poll now. Okay, so I'll end the poll now. Around 70% of people answered and of the people that answered, around 60% of them got it right. So I'll just share the results. Okay, so this question, it is question 22, but I think this is significantly simpler than the previous question, the one with the funny feet. So to restate the question, write each one of the numbers 0, 1, 2, 3, 4, 5, and 6 in the squares to make the addition on the right correct. Which digit will be the gray square? So unlike question 4, I don't know of any ways to solve this without reinforcing it, so we'll just do that. Looking at this, it's very easy to ascertain that the first digit or the 100th digit of the sum must be a 1. And that should make sense, because if you ever choose two digit numbers and add them, the maximum the sum could ever be is 99 plus 99, which is 198. So there's no way you can get a three digit number that has a 100th digit of 2 or a number in the 200s when you're adding up the sum of two two-digit numbers. So we have the 100th digit is 1. And now we have to pay attention to the second thing. If we know that this 100th digit is 1, what must the numbers in the two squares above, what must the numbers in the 10th digit of these two two-digit numbers be? So there's two options. Well, there's three. We can check the first one, which is if you have a 6 and 5. If you have 6 and 5, and then the numbers in these two right-hand digits, the ones digits of both numbers, whatever those are, let's say we have 4 and 3, the greatest possible numbers, that means that the greatest possible number or the greatest possible sum must be 117. But that doesn't work because we can only use each number once. And we're using 1 two times and we're not using 2 at all. And whatever we choose for those ones digits, 4 and 3 or 2 and 0, no matter what, we'll always have a 1 in the 10th digit. This means that we can't use 6 and 5 as our two 10th digits for our two two-digit numbers. So what's the next one we can use? Let's try 6 and 4. If we try 6 and 4, then we have, well, we have the 1 in the 100th place, but let's think about the two numbers that would go in the ones digit. The greatest possible numbers could be 5 and 3. And if we put 5 and 3, we have 65 plus 43 equals 108. Well, that obviously doesn't work because there's no 8. So let's remove one of those numbers. Instead of 5 and 3, let's do 2 and 3. Now we get 5. And this works out perfectly. We use every single digit once, 0, 1, 2, 3, 4, 5, 6, and we have 0 repeats. This is most likely the answer, but it is the answer. If you want to, you can also double check by going through 6 and 3. But if you go through 6 and 3, you'll notice that on the right-hand side would have to be at max 5 and 4, so your number would be a two-digit number, but this is the answer. So the number in the gray square or the digit in the gray square should be 5. Thank you, Ishan. So there was a little bit of trial and error there. This is what we were talking about with guess and check, but it started off with the educated knowledge that if you have two two-digit numbers, you can only have a 1 here. So that limited what we did, and then we had to figure out how we can make it have a carryover to the ones digit. So obviously, we had to use some of the larger 60s or 50s or 40s. Good. Okay, next problem. We have some bonus problems, which I really wanted to get to today. The numbers 1, 5, 8, 9, 10, 12, and 15 are divided into groups of one or more numbers. The sums of the numbers in each group is the same. What is the largest possible number of groups? If you're not sure where to start, one thing to think about was I need the sum. Since 15 is the largest number, 15 is the smallest sum that I can have for a group. If the number 15 was in a group all by itself, then the sum of the other groups also has to be 15. Can I do that? Can I get other groups with a sum of 15? 12 would have to be 12 plus 3, so 15 is not going to work. Can I do the next sum up is 16, 15 plus 1 is 16, but 12 would have to be 12 plus 4, and I don't have a 12 plus 4. The next sum up would be 17, but I don't have 15 plus 2. We'll keep working that way and see what happens. Anybody else want to put an answer into the poll before I close it? Remember, if you don't try, you're definitely wrong, right? That's the one thing we can be sure of in a math kangaroo contest is blanks are zero points. All right, I'll end the poll and show the results. So most of you have said that you think the answer is three. Let's see if that works. So we've already discovered that it has to be at least 15, and we've been going up one from each. So if I did 18 would be the next one. There's no 15 plus three, 15 plus five would be the next choice. So 15 plus five is 20, right? So I can do one group that's 15 plus five. So I can cross those two numbers out. Then I can do 12. How can I get 12? How about 12 plus eight? 12 plus eight gives me 20. So I can cross those out. And then I can do 10, 10 plus nine plus one equals 20. And now I've used up all my numbers. So the largest number of groups that I can get is three. So in this case, my method was a little bit of guess and check. You can also add up all the numbers and start dividing to see what factors they have, and then see if you can arrange them into the groups. In this case, I think it might be a little faster because you don't have every single number. You only have a selection of the numbers. So in this case, some educated guess and check might just be the fastest way. All right. That is all of the time that we have for today. So we did get through one of our bonus problems, which is awesome. I hope you like what we called guess and check. So as you'll recall, don't take wild, crazy guesses. Look at what the problem is asking and use your mathematical principles as best you can. Use logical reasoning to come up with a good place to start your guessing. And as you go, revise your work. This didn't work because the sum is too small. This doesn't work because that's not one of the options that I have, right? And that will help you. So remember the guess my animal game. Keep making your answers better and better. Look at your answer choices. And I think you'll do real well on this. So hopefully you enjoyed the lesson. Thank you, Ishan. And thank you, everyone, for being here this Sunday. I'll see everyone next week. Bye.

Math Kangaroo

problem-solving

guess check revise

logical reasoning

interactive discussions

place value

number divisibility

strategic thinking

exam readiness