All right. All right. So now, good thing that we start recording for the meaty stuff right so right now it's just housekeeping. Okay, so, and the finally the polls will be anonymous, which means that there's going to not going to be name showing up in the polls result, so just do your best. Okay, awesome. So meet your instructors, and so I have really had the pleasure to spend an hour with you today on a beautiful Saturday afternoon. So I want to make sure I do our, my best to make this count, and fun for you. Okay. So my name ding ching chai. I have been a math kangaroo teacher for a couple years, I created this organization called West Chester math kangaroo week, which is based in New York. And I teach online in person classes on creative problem solving with math kangaroo with a bunch of third graders, fourth, fifth and sixth graders. Okay. And I also coach these children for math kangaroo competitions every year. I have been able to contributing to the online course development for math kangaroo USA, and I routinely teach classes for them as well. Now, over to Noah. All right. Hello, everyone. My name is Noah, I'm currently in 11th grade. And I've been participating in math kangaroo since around third grade. And I enjoy math. Math is my favorite subject at school. So Noah, everybody is dying to know how you managed to be so successful in math kangaroo competition. I know firsthand, it's not easy to get perfect scores every time. Do you have any secret you can share with our participants today? Well, not specifically for a perfect score, but to get a better score, it is. There's like 30 questions, right? Yes. So like, those 30 questions are ordered approximately based on difficulty, the later problems are probably going to be harder than the earlier problems. But that is not always the case. Sometimes you might find a problem that seems easier in the later sections, or problems that seem harder in the earlier sections. If you were to, for example, spend a lot of time for your problems on the harder problems on the earlier sections, you would not have the time to do what could be easy problems in the later section. So it's really important that you at least like read through all 30 problems and know that know which ones that would take a lot of time. And basically, you got to manage your time wisely. If you get stuck on a question for too long, I would suggest you skip it. And if you have time left, you can come back to it later. Because it's better to get some points rather than none. Well said, well said. So I am pretty sure there isn't a guessing penalty on the math kangaroo. So if you're so rather than leave a question blank at the end, at least put in some random answer because at least now you have like a one fifth chance of getting it correct rather than zero. And finally, it's just to read the questions very carefully since I found that math kangaroo, the way the problems are presented is more tricky than other math competitions. So it's very important that you read the question carefully and completely understand what they're trying to say. Well, thank you, Noah. This segues very nicely into our next slide, which is the four step problem solving strategy. So we all know by now that solving problem requires a careful thinking, but also following a method of four step method. And why? So the first step, as Noah said, is really understand the problem. Making sure you understand what is asking you and what the problem actually tells you. Because certainly you don't want to spend time solving the wrong problem. And the second step is children at elementary school or middle school age, they have a tendency to kind of jump into the math. And problem solving effectively actually meant that you let's think about it. Let's pause. Let's think about what is the best way to solve the problem. Not be in a hurry in calculating stuff. So think about how you can tackle the problem and what do you need to solve it. And do you have what you need? If you don't, then try something else, right? And only then, only after you plan out your attack strategy, you carry out your plan i.e. you start doing some math, right? You're doing some addition, some multiplication, et cetera, as needed. And do making sure you check your calculation is correct. Every calculation, okay? And then the last step after you solve the problem, you circle to your answer and you go back to check and you say, wait a second, the problem is asking for the smallest possible number and my answer choice is one of the largest in the five answer choices. Does that make sense? Okay, so you may want to go back and double check and look and think about if your answer makes sense, okay? So this four-step methodology, the four-step problem solving methodology really helps you to not only solve the problem correctly, but solve the problem effectively and efficiently. And that has a lot to do with time management, as Noah said, because even if you know how to solve all the problems, you may run out of time, right? Or you make a silly mistake somewhere and when doing these four steps and you can really show through your true understanding and your true problem solving skills. Okay, so now what I'm going to model for you for some of the problem, the four-step method, okay, and you will see in practice how that works. So before we get into the nitty-gritty and sink our teeth into some juicy problems, let's just to be clear, everyone on the same page on what pattern is. What do we mean pattern in this webinar? Okay, so in math, a pattern is basically a repeated arrangement of numbers, shapes, colors, and so on, right? In kindergarten, right, you may have seen repeated pattern of colors, right? You know, yellow, yellow, blue, yellow, yellow, blue, and you may see that and that's called AAB pattern, right? But in here, in level five, level six, we're working with more complex patterns, right? So which lead me to the second point. So the pattern in the basic sense can be understood as repeated arrangement, but in math, it could also be seen in a more formal way, which is the set of numbers are related to each other in a specific way or rule. And that rule or that particular manner of how the numbers related to each other is called a pattern, okay? And sometimes patterns are also known as a sequence, okay? So for example, in the sequence two, four, six, eight, okay? What is the next number that fit in that pattern, right? So as you can see, each number is increasing by two in this sequence. So the next number in that sequence will be eight plus two equals 10, okay? So finally, the number is not just about, the pattern is not just about numbers. It can also be related to any type of event or object. Okay, so we will see in the problems that we're going to use today, how that look like, okay? So at this point, some of you may say, well, what is the big deal with patterns, right? Why are we doing a special webinar on this? Okay, so I'm going to show you something will blow your mind. It's truly fascinating, okay? First of all, patterns are everywhere in nature. And patterns exist because they serve a very valuable purpose. And nature somehow figured it out. How amazing is that? So most of you probably familiar with the Fibonacci pattern, right? You see the seashell in this picture? That's a Fibonacci pattern, okay? You're also probably aware of the sunflower seeds, right? The way the seeds are arranged, right? It's also a Fibonacci pattern. So that pattern is fairly well known. Lesser known probably are the other patterns such as fractal pattern, spiral pattern, and Voronoi pattern, okay? So let me show you some amazing animation I found from the Franklin Institute. So this is the animation didn't work. Huh, okay. So, oh, the animation doesn't work. Okay, so this is a fractal pattern. And a fractal pattern is essentially a pattern that looks similar at any scale and repeats itself over time. So here's an example of how these patterns increases, but they look the same, okay? This example of fractal shows similar shapes multiplying over time, yet maintaining the same pattern. Examples of the fractal patterns in nature are snowflakes, trees branching, lightning, and ferns. Ferns is the type of plant. The spiral pattern is not showing in the slide here. So essentially it looks like a spiral. And the examples of spiral pattern in nature usually having to do with plants. And you see in the earlier slide here, the green plant here on the left is a spiral pattern, okay? And finally, we're talking about the Voronoi pattern. This is a pattern that provides clues to nature's tendency to favor efficiency. Okay, so what happens is that every Voronoi pattern has a seed point, okay? Everything inside that cell is closer to the seed point than any other seeds outside that cell. Okay, other examples of Voronoi patterns are the skin of a giraffe, corn on the cob, honeycombs, foam bubbles, the cells in a leaf like what's being shown here, and a head of garlic. If you open up a garlic, you will see Voronoi pattern in there. And remember, these patterns exist because it promotes efficiency in nature, okay? So that is the really amazing thing about studying patterns. So- I would like to add some stuff to the uses of those patterns. For example, something similar to a Voronoi diagram was used to a British physician named John Snow used something like a Voronoi diagram to discover that a cholera outbreak seemed to have been centered around one specific street pump. And from that, he inferred that cholera came from water and so forth. Also, about fractals, there's this very famous one called the Mandelbrot set, which is also related to this function called the bifurcation diagram, which is basically a function about the growth of a population. And the Mandelbrot set is that diagram over complex numbers. So this stuff is pretty interesting. Thank you, Noah. In fact, I'm showing this as my video image on the screen. I think everybody can see the Jing Qing Chai picture. That's the pattern that you just referred to. Awesome. So just like in nature, patterns exist to help plants grow more efficiently. In math kangaroo competition, recognizing patterns and being able to work with pattern problem also help us solving problem efficiently. Isn't that amazing? So it helps us solve problem by simplifying a complex problem. And you will see shortly, if you can identify a pattern and apply the pattern to solve a problem, it makes a complicated problem much simpler to handle. And also it helps you to reduce the number of calculations needed. And finally, obviously it saves time and increases accuracy because the fewer calculations you do, the less likely you're going to make a mistake, right? So in this webinar, we're gonna tackle four types of pattern problems are very popular in math kangaroo competitions. And the first one obviously is the number pattern problems. And the second one is image, which has to do with spatial reasoning or visual reasoning type of problem. And occasionally color gets thrown in the problem, which makes it extra fun to work with. And they're also patterns related to events, the events happening in a repeated manner. And interestingly, they are pattern problems that involve logic, okay? So in the next webinar coming up next Saturday, I'm going to do a webinar with Noah on logic problems. But today we're already showing you one logic problem where you can use patterns to solve it, right? And finally, last but not least, Noah is going to lead you on a problem that involves hidden pattern, right? So by the look of it, you don't think there's any pattern, right? But when you try really hard and you're clever about it, you can uncover a pattern which then help you solve a problem, okay? So that's what we're going to do together. All right, so number pattern, okay? So this is the first problem we're gonna do together. Now I'm going to model with you, right? And the next problem you're gonna be asked to do on your own, right? And then you're going to vote on your answer in the poll, which Noah is going to launch shortly. So for this problem, I'm going to model with you the four-step method, right? So let's do step one, really read the problem well and understand the problem, okay? So you will notice when I read the problem, I underline what I consider the key information, okay? So Peter went hiking the mountain for five days, okay? So I'm going to annotate now. I'm going to underline five days. He started on Monday and his last hike was on Friday. Every day he walked two kilometers more than the previous day, okay? When the whole trip was over, his total distance traveled was 70 kilometers. What distance did Peter walk on Thursday? Okay, now, so step one reminds us that we need to be very careful about the question, right? What question does this problem ask us? And what question it didn't ask us. For example, it didn't ask us for Monday, Tuesday, okay? So it asked us for the distance traveled on Thursday. Okay, so that's, now we have the information, we're ready to plan our strategy. Okay, so step two, plan our strategy. So what is a good strategy for this problem, okay? Now, every child approaches this kind of problem in their own way, which is great, right? Because that's diverging thinking. Diverging thinking is very important for creativity in problem solving. Having said that, my favorite strategy often is draw a diagram. And they say the picture speaks a thousand words, right? I love drawing a diagram to help me solve the problem. So let me draw a diagram, right? So you're welcome to do that on your own piece of paper if you like, okay? Or you can just watch me because we're still doing the modeling right now. So this is Monday. And on Monday, Peter walked this much. We don't know how much. I'm just drawing a bar, I don't know how much. On Tuesday, he walked two kilometers more. So I draw a bar the same as Monday. And now, on Tuesday, he walked two kilometers more than before. On Wednesday, I do the same thing, okay? So on Wednesday, this is the same as Tuesday, as before. But on Wednesday, I walked two kilometers more, okay? And now, I do Thursday, okay? This is the same as Wednesday. And I, or Peter rather, walked two kilometers more than the day before. And finally, it's the Friday, okay? I'm going to move this screen bar out of the way. So I draw first the same distance he walked before, but now this time two kilometers more than the day before, right? So each, the small bar represents two kilometers, right? And the bar of the same height on the very bottom is the miles, the kilometers he walked on Monday. Okay, now the problem tells us the whole thing is 70 kilometer, right? The total distance he walked over the course of five days. Now, for those of you who have done this in school, you would notice this is called the bar method. I basically just drew a bar method, a diagram, okay? So what would you go from there? So let me just do the mouse here. So we draw a picture of the bar models. And so now you can do a calculation, right? So if you add up all the bars, you will see there are five same height bar for Monday, and then plus how many bars of two. You have one plus two plus three plus four, which represents the increases in Tuesday, Wednesday, Thursday, and Friday. And each bar represents two kilometers. So you multiply by two, okay? And then when you subtract that amount, okay, from 70, essentially you get five bars represented by Monday, okay? So now what you have is 70 minus 20, so 50. 50 represents five bars of Mondays, right? The kilometers are walked on Monday. So you divide it by five, and this gives you 10 kilometer walked on Monday. Okay, so essentially this becomes almost like a visual problem, right? You're adding up the bars of the same height, and you know the whole thing is 70 kilometers, and then you were able to calculate the bar that representing Monday. Okay, so now you know how many kilometers who walked on Monday, then Thursday looks like Monday plus two, three sets of twos, right? Which is the Thursday shown here in the picture. So what you do is you calculate, oh, hold on, mouse, yeah. Mouse, please work, yes. Distance walked on Friday, on Thursday, sorry, there's a typo here. It meant to say Thursday, okay? And the answer is simply 10 plus three times two, okay? It equals 60. Can I add another interesting way? Go ahead. For example, in this one, you can see that we've started on Monday as our base day. Something interesting we can do is instead of setting Monday as our base day, we can set Wednesday as our base day, since Wednesday's in the middle. And then if we just say we walked like a certain distance here on Wednesday, we can say that instead of doing plus twos, we can also add in minus twos here and then we'll have two, two, two. And then if you add them all together, you'll notice it's just five times the number of the length you walked on Wednesday. And then you can do division 70 divided by five to figure out you've walked 14 kilometers on Wednesday, and then you can add two to reach 16 on Thursday. Just an interesting thing you can do since this is an arithmetic progression. Very good, very good. So this is exactly what addition patterns is going to help us do, right? As Noah said, that it gives you, the pattern helps you to simplify the problem, okay? So you will see that if you pick Wednesday as the starting point, you have a symmetry of negative two, negative two on the left side and negative two, plus two, plus two on right side and they cancel each other out. It becomes a simple problem and 70 divided by, 70 divided by the Wednesday number, okay? So this shows that there are multiple ways you can get to the same answer and patterns certainly help us improve the efficiency in how we solving it, okay? Excellent. So I hope you guys got the same answer as we do, okay? So interestingly, what Noah contributed, it just modeled a step four for us, right? Because I solved the problem in one way and I got the answer of 16 and Noah solved the problem in another way. He also got 16. So that's our step four. When you look back to check, in this case, using a different strategy, you got the same answer. So now you're 100% confident you have the correct answer. Congratulations, you just earned four points because this is a four pointer. Okay, let me clear the drawing here. Okay, let's go to the next problem. So this is also an addition pattern problem, okay? So this problem that I invite all of you to try your hands at solving it, okay? So let me again, model the step one because understanding the problem is very crucial for any problem. So this problem says, what number should replace X if we know the number in this upper row is the sum of the numbers from the two circles right below it? So the key word I would like to circle is X because that's asking, this is the question, right? This is something that I need to solve for. And the other one I wanted to circle is the sum, okay, of the numbers from the two circles right below it and the number in the circle in the upper row, okay? So this gave me the sort of the positions of how different numbers relate to each other and the kind of the rule that decides on the pattern, okay? So I would love it if you guys could just give it a try, okay? And Noah shortly is going to launch the poll and then you get to vote on the answer based on your own problem solving. Wow, we have some students are quite quick in lending the answer. Noah, would you like to just maybe launch the poll since we already have a lot of people? Yeah. Wow, I didn't realize no at the results is shown automatically in real time. I thought it was a share result button. But this is cool as well. I think the participants can only see it after we click share results. Yeah, okay, you're right. So I am seeing it, but they didn't. Yes. Okay, that makes sense. Wow. Well, based on what I can see on the results on this group is doing extremely well with this problem. Now don't forget to use that for to look back to check and see if that number if the answer makes sense. Okay. All right. Um, Noah, do you think we show it or. Okay, let me, let me do that. I mean we can do this either way. So, so, um, so let me see what. Oh, okay. Yes, go ahead. All right. Wow. Wow. So for this problem most of you think at two is the right answer. Okay. And that's like 89% of you who responded, I think, at two is the answer. So let's see what we got. All right, let me just click on the mouse so I can show you. Okay, so, so this is what we did. Okay, just quickly since most of you got the correct answer. So, the trick with this problem is that you wanted to pick your starting point quite wisely. Okay. And you will see on this end, you cannot go very far. Right. But if you start with this end, everything just falls into place. Right. So in pattern problems, or any other kind of problem, you need to be very careful with where you pick your starting point. Okay, if you pick your starting point carefully, the whole thing will go much faster and much more smoothly. Okay, so I started, I realized the right side is is more promising. Okay, so I started with this side. And then you see that you have nine here you can immediately fill out and 12 here you can immediately fill out. Right. And this is a subtraction pattern. And right here, 12 minus nine is three. Okay, so so far I'm moving into the middle of this pyramid. And then this allows me to go up now with addition. Right, look how interesting this problem is right it parts of the problem requires you solve with subtraction and parts of it is you solve it with addition. So, I get eight here already and seven plus eight is 15. Awesome. And now I see 812 would allow me to calculate this circle on top of that. So that gave me 20. And from here on, we can just add our way up. Right, so you have 35 here from 15 plus 20, and you have 47 here from 20 and 27. And finally, for the X, our big prize, that's the prize we're asking, look us going for is simply 35 plus 47, which is 82. Now, if you were to do step four, you're gonna check your answer. One more time by going over your calculations and if if for some people, it might be more effective if you start alternating the addition with the with the subtraction for example instead of redo the calculation with 35 plus 47 equals 82, I might decided to do subtraction. In my step four, I will say how much is 82 subtract 35 and it's 47. So that's how you know you have the correct calculation. Okay, so this problem is kind of an interesting mix between the addition pattern and subtraction pattern. And it's really quite fun. And you guys did a wonderful job. So we're going to build it up and progressively more challenging and I hope you're ready for it. Okay, awesome. Let me clear the drawing first. And we're going to go to the next. The next problem. Okay, so I need to shrink the poll here. Okay, great. So for this problem, again, we start with step one, right, we wanted to understand the problem really well. We wanted to understand the problem really well. So this problem says starting with a list of three numbers, the change some procedure, create, creates a new list by replacing each number by the sum of the other two. For example, now you see math can grow problems make it easier for you sometimes by giving you examples. So you really would like to spend time on understanding the example, right, because it illustrates what the problem actually means, right. So for example, from three, four, six, that set of three numbers change some procedure gives 1097. Okay, and a new change some leads to 1617 and 19. Okay, so what does that mean? Um, what this statement actually means so you can already kind of play with, right, so you have four, three, four, six, right, the change some simply means in the place of the three, you're going to add up the other two number, which is four and six. Okay, and that's going to give you 10. In the place of four, you're going to add up the other two numbers in the set, right, which is three plus six equals nine. And in the place of six, you're going to add up the two other numbers in the set, which is 3.4 equals seven. Okay. And bingo, that's what actually the change some procedure will give you this new set of three numbers, right. So this is what the problem actually means right now you truly understand what change some procedure does. Okay, and then the problem continues. Okay, so if we begin with the list. 21 three, what is the maximum difference between two numbers of the list after 2013 consecutive change sums. Okay, so let's go. Okay, so as we can all see here that it. Give me a moment some things. All right. It asks, after 2013 consecutive change sums. You obviously can't be expected to do this thing 2013 times, which means that the problem wants you to look for a pattern. Right. So if you see something ridiculous like this in your, in your questions, you should immediately think to look for any sort of pattern rather than going through the problem 2013 times. So this is kind of a tip off. Right. So, you know, it gives you the impression is a very complex problem. Right. And that's when you know, no, we're not going to take your bait. We're not going to take the bait of complex we're going to go try to simplify by finding a pattern. Now, let me just finish the step one by slowing down on the question, right, you notice the question is asking for maximum difference. It's not minimum difference. It's maximum difference. And it's a difference. Difference can mean positive or negative. It did not say max maximum positive difference. Right. So, slowing down on the question really ensures that you're solving for the right question. Okay, and we'll see this has a bearing on how you solve the problem. So let me model on the how we can solve this problem. So I wanted to pick my strategy. Okay, so, interestingly, I have a different strategy from what's shown on the math kangaroo official solutions. Okay, because I decided I wanted to use the strategy of solving a simpler problem first. And from there, I'm going to try and find a pattern. Right, so I'm not even solving the 21 three set problem, I'm trying to solving the smaller problem which is the fourth, the three, four, six. Okay, so what pattern do I see. Okay, so I'm going to do that. So, first, I'm taking the simpler problem, I basically say okay, this set three, four, six. What is the maximum difference. The maximum difference in this simple set is simply six minus three. Agreed. Right, this is the maximum difference which is three. Okay, and now I use the change some set in giving in the problem 1097, and the biggest difference in this set is also three. Agreed. Right, I mean because it's a such a simple problem, everybody, everybody who knows how to do addition and subtraction can quickly see the biggest difference. Okay, now I'm going to check yet the third change some set from the simpler problem, right, which is 16, 17, 19. So all these is written in the problem, it was given in the problem. Right. So from that, I can see the maximum difference, again, is three. Okay, so what this tells me is that in the change some problem, in a change problem using change some procedure. Okay, the maximum difference actually does not change, depending on how many times the change some procedure is applied. Okay, and not only it doesn't change, it remains the same value as the very first set before you apply the change some procedure. So in this particular problem, the biggest difference I found from the three, four, six set is three. Okay, so what this tells me that if I apply this pattern, no matter how many times I apply the change some procedure, the maximum difference for this particular set starting with three, four, six is going to be three. Okay, so I just apply the same pattern to the question that being asked. Remember, the question is asking us to solve this set. Right, the list of 21, three. So when I look at the 21, three, I noticed that the biggest difference from that set is 20 minus 119. Okay, again, it's pretty simple math. I just see it. It's the biggest difference. So if I were to apply the pattern from the simpler problem, I would know that this is the same difference that's going to apply to all the set of numbers after 2013 consecutive change sums or other numbers of consecutive change some procedures. Okay, yeah. So at this point, I'm fairly confident the answer to this problem is 19. There is a way to rigorously prove that with algebra, but that would take a bit of time. Correct, correct. And I'd certainly encourage all of you to try to build an algebra solution for this problem that would apply to any initial set of three numbers. Okay, it's not hard to do, but it's a little tedious and it does require you have studied, you have pre algebra knowledge. Okay, but that's all the fun right with the fun with the math problems is that you can start with a simple solution and go all the way up to high school. For some problems at level five, level six, you can solve it with high school math. That's what gets me my creative juice going always. Okay, so I hope this problem was helpful to you. Okay, let's now moving on. If, oh, by the way, if you have any comments or you have a some burning questions or disagreement, you can chat it over to Noah, who's monitoring the chat box. Okay, he may not get back to every single one of you. But he will share some common resources if there are enough sort of common theme that coming out of your questions. Okay, so let me go down to the next problem. So now you know the drill. The next problem is your turn, my friends. Okay, so again, we start with step one. On Monday, Alexandra emails a picture to five friends for several days, everybody who received the picture emails that the next day to friends who haven't received the picture yet. So don't forget to underline your key information, five friends, and everybody emails the next day to two friends who haven't received the picture yet. On which day? Now let's slow it down on the question. On which day of the week does the number of people who have received the picture become greater than 100? Now, slow it down on the question tells us that the number of people who have received the picture is not just the people who received the picture at that very last day, but everybody who has received the picture from the first day, which is Monday, when Alexandra emailed the picture, right? So you're looking at some cumulative number. Okay, so keep that in mind. Okay, so now you have, I will give you a couple minutes to try this problem out. Okay, and then when Noah launch his poll, you can vote on the answer. Okay. So since you haven't launched the poll. Yeah, you should launch. Oh yeah. So, because the problem is too long for the poll you I only included the question. All right, but it should be sufficiently helpful for you. If for some reason the poll is blocking your view of the screen. You can click on the, the, the logo of which says polls, and then there's a move feature there you can just move the poll to the side. Okay. Well we have some really quick kids who react right away to, to, to this problem. So that's always good to see but I do would encourage you to do step four, and double check your calculations and check to see if the answer makes sense. Okay, so, um, in the interest of time, Noah, maybe you can launch the results. All right. Mm hmm. This one's more spread out than the other. Yeah. It definitely does. Hmm. So are you sharing the results yet or I cannot quite see from here. I have shared them. Okay, so yeah so for this problem that we have some sort of splitting on between Friday and Saturday, and was a slight more votes on Saturday than Friday. So let's look at the problem. Okay. So, So, so what is a good strategy for this problem, right, I like to make a table to help me organize the information to keep track of what happens on each day. Okay, so this is the table I made. So on Friday, the number of people receiving the picture is five, right, I copied this down from the problem. And the total number of the people receive the picture on Monday, obviously, is five. Right. So on Tuesday, these five people who received the picture from Alexandra. They emailed to two more friends right so on Tuesday, 10 more people received the picture right five times two. So, and there, the total number of people receive the picture as of Tuesday is five people from Monday, plus 10 people received the picture on Tuesday that's 15. So I continued on with this multiplication pattern. Right. So on Wednesday, that 10 people email the picture to two, two, two friends, each. So that's 10 times two gives you 20 and accumulatively by Wednesday, the number of people receive the picture is the number of people previously received the picture plus the 20 people received the picture that day that 35. So I continued on until the total number of people received the picture exceed 100. Right, because that's what the question is asking. So that would lead me to Friday. Okay, so the correct answer for this problem is Friday. Now you notice that the multiplication pattern in this problem is not hard, but it is important for you to keep track of information using using a table or some other schemes, you know, some people like to make a list, that's fine as well. Okay, so that is how this problem, the pattern is going to be tracked and it's going to be solved in a relatively effective way. Okay, so let's move on to the next problem. So this is an example of the multiplication pattern problem. Okay, so the next problem I will model for you. This problem is interesting because it's not only looking at the numbers, how they relate to each other, but also locations. So there's a special spatial dimension to this problem. Okay, Piotrek is writing the numbers from 0 to 109 into a five column table using a rule, which is easy to figure out. See the picture shown. Which of the pieces below cannot be filled in with numbers to fit Piotrek's table. Now remember, as Noah said earlier, that you kind of don't want to draw out the whole table, because that will take too long, right? So by recognizing pattern, you can solve this problem in a much more efficient and timely manner. Okay, so when you look at the picture to the right, you notice the rule, right? And the rule is simply that the first rows are essentially increasing consecutive numbers, and then every two rows, they actually add 10 more to the previous two rows. Okay, so for example, 10 over 0 is 10, 12 over 2 is an addition of 10, and 11 over 1 is an addition of 10. So that is the rule that the problem was referring to. Okay, so what's really interesting about this problem, if you're focusing on the question itself, it's asking you which of the pieces below cannot be filled in with numbers to fit Piotrek's table, right? So one of them does not fit in this pattern. Okay, so a common strategy for this problem is simply guessing and checking, right? So you wanted to guess each answer choice and see if it fits in the pattern. Okay, so for example, if you guess 8, you will see that it's 65 and 68, right? So it's almost like you can use a similar, you can find a similar pattern over here. I'm going to draw it for you. So you see here, 5 and 8, okay? You see how this 5 and 8, the two of them fit into the pattern, and here you have 65 and 68, okay, in the exact same position, right? But obviously they belong to other rows, right? Because it's not being shown here, and we don't actually need to show the numbers in the other rows, right? So from here, you can see that the 65 and 68 actually fits the pattern. Okay, now then move to B. Let's guess B. Could B be the possible answer? Now this gets really fascinating, right? If you look at the number, the number 67 and 78, and you were like, yeah, of course it looks like they fit the pattern, right? Because for example, if you pick, I'm going to use a different color, if you pick the 7 here and you pick the 18 here, right? They seem to relate to each other in the same room, right? Which is an increase of 11, and they're on this relative position, right? They're basically one row down and one cell to the right, right? So you're like, okay, so B is possible as well. Not so fast. If you look at the image on the right, you notice the numbers that end in eight is on the very last column of that table. And there's no, there's nothing beyond that column, okay? So this guy has no place to go. It does not belong in the five column table shown in the problem. Okay, so even though 67 and 78 relates to each other using the correct rule, but its positioning is wrong. This guy does not belong in this problem. So the correct answer is B. Now, of course, if you have time, you want to double check the other answer choices are possible, okay? But after you do that, you would have seen that the other answer choices fit the rule as well. And B is the only answer that cannot be filled in this table, fit in this table. So the correct answer for this problem is B. Okay, so this problem is almost like asking to thinking out of box, which is why I say there's a quite a bit of spatial reasoning component to this problem. So don't forget to link the number pattern rule with the spatial positioning and the both come together to help you solve this problem. Okay, so let's move on to the next problem. So this problem, again, is your turn to show your problem solving skills, okay? So let's read the problem carefully. The positive integers have been colored red, blue, or green. One is red, two is blue, three is green. Okay, and now four is red, five is blue, six is green. Now, do you see some color patterns here? The color patterns that mix up with numbers, right? And so on. So Renee calculates the sum of a red number and a blue number. And the question asks, what color can the resulting number be? Okay, so this problem, you're working with both the numbers and the numbers, right? And already, I think you can see some color pattern involved. Okay, so now it's your turn. Two minutes, or maybe one minute, maybe one minute. Okay, and then Noah will launch the poll shortly. All right, should I launch the poll now. Yes, please. Wow. Okay, since we have more than 50% reply, can you launch the results? All right. Okay, great. Wow. Okay, so this time we have the majority answer being only green. Okay, and let's see if that is the correct answer. Okay. So, again, this problem gave you the simple example of one, two, three, and four, five, six. Right. Even though the question does not specify what number is being what numbers are being added together. But you can certainly use the simpler problem to find a pattern, and to be able to answer the question for the sum of any numbers that with the right colors. Right. So, let's see what the smaller problem looks like. So, in the smaller problem of one, four, seven, 10, you will see that any number, right, that fits the pattern of one plus multiples of three is a red color, right, because the problem tells told you so. Right, because one is red, and four is red, and you know seven is red, if you were to compute the pattern, compute the sequence. Right. And similarly, similarly, any numbers that fits in the sequence of two, five, eight, 11, means that two plus multiples of three is going to be in the blue color. Right. And you know that because you can see that two, five, eight, 11, they're all going to be in blue colors. Okay. And three, six, nine, 12, that sequence, and you can see from the example given that three and six is going to be in green color, and this means that any multiple of three is going to be in green color. Now, this gets now, what comes next is getting really interesting, right, because if you add the red number with a blue number, essentially you are adding one plus multiple threes with two plus a multiple three, and that simply gives you a three plus two sets of multiple threes. Okay, and since three is a multiple of three, and two sets of multiple threes are definitely multiple of threes, that means you always going to have a green number. Okay, so the correct answer for this problem is green. Okay, good job class. Okay, now let me clear this. Let me turn the next problem to Noah, who is going to lead the problem on the hipton pattern. Go ahead. All right. Are we going to do a poll for this one or am I just going to talk through it. Um, maybe you can talk us through. Yeah. All right. So, pretty sure when faced with this problem. Everyone would want to just start attempting to plug in numbers into the circles and see what works. Okay, so there's a hidden pattern here. If you've noticed, we can label all these circles. We, we can see that this, we know that there are three sums we're looking at here right and there. Oh, that lines not straight. There are three sums we're looking at. And these are all the same number, right, which means if we add up these three, some, these three different sums, we're going to get some multiple of three. Oh, that is, give me a moment. My handwriting is a bit bad. All right, we're going to get a multiple of three here, right? And then you can also notice that this center number is added three times. So if we were to, for example, take out the center number, the sum of the remaining six numbers would still be a multiple of three, since if you take a multiple of three and subtract a multiple of three, you're going to end up with a multiple of three. So basically, you can use this pattern here to see which number out of these seven we can take out to keep it being a multiple of three. And if you add up all these numbers together, you'll notice that it's a multiple of three, which means we must take out a multiple of three to keep it being a multiple of three. Otherwise, it's not going to be a multiple of three. For example, if I were to take out four, it would have a remainder of two. So that means we can only put three, six, and nine in the center so that the remaining six numbers add up to a multiple of three, which we need for, since given by what we find earlier, these six numbers must add up to a multiple of three. Great. Thank you, Noah. So this is an example of a hidden pattern, right? Most of us would like to jump in the problem and start experimenting and try to find three number pair that have the same value, right? And it turns out when you try to discover the pattern used here, you don't actually have to do that. And that saves you a lot of time. It's a difference between spending three minutes on this problem and spend 30 seconds on this problem, okay? So that I hope to encourage you to continue to try your best to find a pattern and use pattern to solve a complex problem, okay? Let me clear the screen here. So we're now at the end of the webinar. We were not able to get to that logic pattern problem, but if you can join in next Saturday's logic problem webinar, Noah and I are going to offer you, we will definitely cover that logic pattern problem in there as well, okay? So today we solved these number patterns and image pattern problems and by using strategies such as drawing a diagram, making a table, solving a simpler related problem and some logical reasoning involved and finally guessing and checking, right? So these are the common strategy you can use to help you solve these problems. And just to remind you that pattern recognizing ability is among the most important abilities in solving problems in both math and everyday life. And remember, the mother of nature certainly agrees with that, okay? So hope you have fun. And if for whatever reason you have to leave early or you just would like to review some of the problems we've covered here, there's going to be a webinar recording link provided in Math Kangaroo USA website under the webinar page, okay? And Noah and I wish you have a wonderful weekend and continue doing math kangaroo problem solving. Bye-bye.

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