First of all, we want to congratulate all of you Math Kangaroo 2024 winners. Congratulations. Give yourselves a round of applause. So as a sort of reward for you guys, we wanted to introduce you to our MethaPlus production called Cutting Cakes, Laser Eyes, and Racing Shopping Carts, Mathematical Adventures in the Grocery Store. Hopefully you'll be introduced maybe to some new, interesting, exciting math topics, have the chance to answer some questions, and win prizes at the end. So I'll introduce myself. I'm Bhagirath Metha. I completed my bachelor's and master's in electrical engineering and computer science in four years at Stanford. I'm the co-founder of MethaPlus along with my sister. I've interned at Amazon and Microsoft in the past, and currently I'm working as a software engineer at a Silicon Valley startup. And I am a Math Kangaroo alum, like all of you. I placed first place internationally twice with the perfect score, and was top 10 nationally several times as well. Hi, I'm Ms. Haripriya. I did my bachelor's and master's in four years in electrical engineering and computer science at MIT. I'm also co-founder of MethaPlus, where we teach a lot of fun camps, and we like to host productions like this. I am a software engineer at Microsoft, and I placed top 10 nationally at Math Kangaroo. Okay, so, I, hold on, am I getting a call? I, what the? Oh no. Ms. Haripriya, do you know who's calling me? Uh, I don't know. I think you should turn your phone off, um, you know, it's, it's during the presentation. Okay, I think I'm getting a call from Dr. Rue. You must be Dr. Rue, the world's greatest mathematician. Do you want to take the, um, change the, uh, resolution? Uh, yes. Oh. Ah, yes, and you are? Ah, you must be Dr. Rue, the world's greatest mathematician. Ah, yes, and you are? I'm Ms. Paws, one of the store managers of Prime Produce. Are you in need of my services? Yes, we have an emergency. Please follow me to Composite Cafe. We need your help with... cutting a cake. Ugh. Let go. That's my slice of cake, no fair. You keep on taking bigger and bigger slices of cake. No, you keep hogging all the frosting. As you can see, it's a non-stop cat fight between these two. They look like horses to me, but no matter. What exactly is the issue? Jack and Jenny are twins having a birthday party. However, they can't decide how to cut the cake in half. Every time we let Jack decide where to cut the cake and he picks a slice of cake, Jenny argues his slice is bigger. But when we let Jenny choose where to make the slice, Jack says the same thing. And their brother, Nettie, is not even here yet. I'm almost out of cakes. And their brother, Nettie, is not even here yet. I'm almost out of cakes. Usually, when Prime Produce caters a party, our cakes come pre-cut from Prime Produce's bakery. But my supermarket has been a mess today and I ordered it from another bakery. For three people, or I mean horses in this case, I would have made three cuts with my knife in the cake. Hold your horses. Three cuts for three people? That doesn't sound right. Okay. Okay. So, hopefully you all are following along. It seems like Miss Paws needs some help with her grocery store because two kids are arguing about how to split some cake. And Miss Paws, or three kids, Miss Paws said that she's tried to make three cuts in the cake for these three people, but Dr. Roo said that's not quite right. Hopefully, you guys know why. I need to present from this slide. Sorry. I'm not sure why I keep starting at the beginning. My apologies for that. Yeah, so that's not quite right. Is the cake a rectangle or a circle? For the purposes of all the problems today, we're just going to keep it easy and simple, and the cake is going to be a rectangle. Hey, kangaroos have enough voices. Yeah, kangaroos have high voices. They are annoying. So, what I'm going to do is I'm going to launch a poll. This is quite a broad age group. We have grades one through five, so not every question might be super easy for you, and that's completely okay. Hopefully, we will take grade one into account. Let me launch the poll, and then maybe, Mr. Bagheerat, you can talk them through the question here. Yeah, so is Dr. Roo right? Does it make sense? Is he right when he says that it's confusing when Ms. Paz claims that three cuts allowed her to form three slices? How many slices will we have if we make three cuts that are parallel to each other? That's question one, and then question two is what are the most number of slices we can have if we make three cuts in different directions? So, hint, think, making cuts maybe in different planes or axes. So, is Dr. Roo right? Do you have more or less cuts when you, or more or less greater or fewer slices when you have three cuts? If you have a question for us, if the question doesn't make sense, then you can type it in the chat. Otherwise, please use the poll function. We will determine final scores using the poll function only. I don't think it's possible, unfortunately, to change the answer, so yeah, just make sure that you're careful when you press it next time. If you already made a mistake, that's totally okay. There are enough questions for you to get on top. I am going to close the poll in maybe 30 seconds, so. So there's, yeah, three children or horses. I probably should have just said three horses, but yeah, there are three horses. We want to have a slice for each of them. And Ms. Paz said that she made three cuts in the cake for these three horses. So is that correct or not is the question. And my screen sharing stuff seems to have stopped working. So let me share again, apologies. The poll is on Zoom. So it should have popped up. And why there's two questions. There's two questions, two answers because there's two questions, right? How many slices will we have if we make three cuts parallel? That's question number one. And what are the most number of slices we can have in three cuts in different directions? That's question number two. Never saw the poll. Did anyone open up the poll in some sort of special way? Because I can't see that on my end. Like I can see my version of the poll, but no. It just popped up. Yeah, I don't know if you're on the phone or the Zoom. Here's what you can try doing, but this will be difficult because there's a lot of you. You can try answering in the chat or here, how about this? You can, there's a button, there's a poll button. Can you try the poll button? If that doesn't work, you can try maybe emailing me, but that might be a lot of emails. Just try maybe putting in the chat. We'll try to get it. There's just a lot of, we're getting a lot of chat messages, so it might be difficult, but just try to write in the chat and we can look for it. And then maybe at the end, we can ask for whoever the poll wasn't working, just let us know. Okay, I am going to end the poll. So Mr. Bagheer, if you are ready to present again, I'm ending the poll now and I will share the results. Okay. Okay. Okay, so yes, first let me go through the answer. So if we have three cuts parallel to each other, we have- Excuse me one second, the music keeps on playing. Do you want me to share screen and start? Yes, can you share your screen? It keeps fighting with me, keeps crashing. Thank you. Yeah, sure. And it's still playing, gosh. Yeah, if you want to close the tab and- I have closed the tab. Oh no. Okay, let me share my screen instead. Okay, hopefully we got D as the answer. It looks like most of you said, B, and honestly, that was my first answer as well. Some of you did say D though, and fewer of you said A or C, so let's discuss. Mr. Bagheer, take it away. Oops. Can see my screen, Mr. Bagheer? Yes, the music is still playing in my background, but hopefully you guys can hear me. Can you still hear me? Yeah, I can hear you. Yeah, I can't hear the music. Okay, excellent. So there are, this is like one rectangular cake. Imagine like, I don't know, a sheet cake or something. So this is from the side view. You can see Dr. Roo trying to slice the cake. So if we cut it once and then we cut it, yeah, so now we have two slices, right? And we cut it a second time, then we have three slices, and then we cut it one more time, and now we have four slices. So that's four slices, and there's something called like an invariant, or sorry, a variant. It's basically something, a quantity that always changes by the same amount every single time. And so every time you make one slice, as long as you're doing it in this way, you'll always increase the number of total slices by one. Since we start out with technically one slice, right? We have one entire cake. So every time we make one cut, we're adding one more slice. We do three cuts, so we add three more slices. So there's four slices total at the end. Okay, yeah, and if you could go to the next slide, please. And this is the second question. So if you look at it from the top view and side view, imagine you're looking at it from the top for that first image and the side from that second image. Okay, so if we make the cuts, let's start by making a side cut. So we split the cake that way. Okay, and then that gives us two slices. So now let's say we now slice though on another axis or another plane. Okay, so we cut from the top. We cut like on the X-axis and then that gives us four slices. And then if we make another cut, that's, yes, I think, how should, exactly, yes. So that gives us eight slices. Does that make sense how it'll work? So you have like one slice in the X-axis, Y-axis and Z-axis. Okay, excellent. And so Ms. Paz, you mean two cuts for three people. Dr. Rook corrects Ms. Paz. And let's her know that she only has to make two cuts. Okay, so cake cutting is a branch of mathematics. There's a number of- Oh, just one second. Yeah, people still didn't get the- Oh, people still didn't get it, okay. Let's see, let's see. What questions do people have? Here's the way. So let me just go back here to this example. Essentially, think of it this way. Yes, for those of you who said six, if you're thinking of just making a cut on, like three cuts, kind of like the way you'd make cuts on a pizza, then that would be good. But instead what we're saying is, hey, let's make two cuts on the cake. So you have four pieces and then we're gonna cut the cake in the middle. So we're gonna, if you can see my hand gestures, we're gonna cut the cake in the middle. And so now those four pieces become eight pieces. Okay, let's move on. Okay. So yeah, cake cutting is a branch of mathematics and it refers to like a number of different ways to fairly divide resources between people that may not see eye to eye, like these siblings. So if you have like different opinions as to like which slice is bigger or like what do you value more? Maybe you value the frosting more. Maybe you value like the chocolate more, the decorations, right? There's all of these. It's difficult to divide things fairly, but this branch of mathematics kind of gives some ways in order to make that happen. Okay. Excellent. So six is not correct. Eight is the correct answer because we're looking for the most number of slices you can get with three slices. So I cut, you choose. I think Rowan said that in the chat exactly. That's the kind of solution to this cake cutting problem. So let's say Jack is the one who cuts. So Jack makes that first cut and let's say there's two slices. Then Jenny gets to pick which slice she wants. So let's just imagine we have just these two horses for now. So Jenny might say that she wants the one on the left. Well, Jack knows that Jenny's going to be the one who gets to pick the slices. So Jack has every incentive to make the slices as fair as possible and as evenly as possible because he doesn't know which one she'll pick. And if he makes one much bigger than the other one, he knows that she'll take that one. So he'll try to make both slices fair so that no matter what she picks, he'll be happy with whatever he ends up getting. So Jenny takes the one on the left and then Jack will be left with the one on the right. And yeah, that's how you fairly split it between two people. So now Mrs. Paz says, hey, I get it. What if there's three people though? What happens when Ned arrives? Can we do the following? And now this is Mrs. Paz proposal. It's a little complicated, so listen up. Okay, so first, Ned makes a cut. So Ned is the third horse. Okay, so now Ned, after Ned cuts, there's two slices, A and B. Okay, great. So next, Jack gets to make a cut. So Jack decides which slice he wants to cut into two pieces and Ned gets to keep the other one. So Jack, in this case, is deciding to cut B based off of Mrs. Paz proposal. And that means that Ned will get to take A. So, okay, Jack cuts B. Meanwhile, Ned is given A. And so now Jack cuts B and there's two slices that get split into, let's call these C and D. So now this is similar to I cut, you choose between two people. Jenny will get to choose which slice she takes. So in this case, Jenny will take slice C. And then Jack takes the last slice. And so the distribution of slices will be fairly even. You know, everyone should get one third, one third, one third. Now, so Ned cuts first, okay. Then Jack decides which slice to cut and the slice that he hasn't cut, that goes to Ned. And then, oh, sorry, yeah, let me go. Let's go from the beginning, because since there are questions. So Ned cuts first, okay. So Ned cuts first, okay. Then Jack decides which slice to cut into two pieces and Ned gets to keep the other one, okay. And then once Jack cuts it into two pieces, Jenny gets to choose which slice she takes and Jack gets the remaining one. So this is just Ms. Paz's proposal for what she thinks might be a fair suggestion. There are some interesting questions in the chat. I would advise you to, before you write any more in the chat, just look at this quiz question so that you might not be, you don't inadvertently give out any answers. So is Ms. Paz's suggestion fair? Answer A, yes, because Jack and Jenny still do the I cut, you choose method for whatever piece of cake is left after Ned cuts. B, yes, because it extends the solution for two people to three people. C, no, because Jenny can force Jack into taking a small slice. And D, no, because Ned can always take at least half the cake for himself. So let us know what you all think. yeah, Miss Paws' suggestion was that Ned cuts first. Actually, Mr. Bruguier, do you mind putting that in the chat for everyone to see? Sure. Just like the summary of Miss Paws' suggestion. But essentially, yeah, Ned cuts first, then Jack gets to pick one of the two slices, which he's gonna cut into two pieces. Ned gets to have the other one, Jack gets to cut the two slices, then Jenny gets to pick, then Jack gets to pick. And we'll give you 30 more seconds. Okay, some people are complaining that horses don't have brains. I will say horses do have brains. Let's, let's be respectful to horses in the chat, please. Okay, I'm going to end the poll in 321. Okay, so a lot of you said this, and it's great. D. So, no, because Ned can always take at least half the cake for himself. Excellent for those of you figured it out. A lot of you figured out almost instantly. But yeah, let's go through why that is. So let's say okay, Ned cuts first. Now Ned can cut whatever he wants, wherever he wants. So what if he just cuts in the halfway point? Okay, so Ned does this, then Jack decides what slice to cut into two pieces, and Ned gets to keep the other one. So if Jack says he's going to cut B, well, Ned gets left with A. So Jack is always going to decide to cut the larger of the two slices, right? So if both sizes, slices are exactly the same, then both will be exactly 50%. So the maximum that can get is 50% of the cake. Now let's say Jack tries to be as fair as possible when cutting the remaining slice of cake. Well, now he's decided divided into quarters of the original cake, right? So C and D are both a quarter of the original cake. So now when Jenny comes along and chooses a slice of cake, she'll take slice C. They're both roughly the same, right? But she's still getting only a quarter compared to Ned, who's getting half the cake, right? Okay, so Jenny takes slice C, and then we're left with Jack taking the last slice, which is also a quarter of the cake. So altogether, we see that Ned gets half the cake, Jack and Jenny each get a quarter of the cake. So this is not fair. Okay. Okay, so this proposal doesn't work. And Ms. Paz says, okay, I see that's not fair, but how about the following? So now here's a new proposal. Okay, so Ned cuts first. Okay, so we still let Ned cut first, but, and now Jack decides what slice to cut into two pieces, but Ned does not just get to take the remaining slice. So Jack decides which slice to cut into two pieces. And so Jack will cut slice B into two pieces. Yep. And so Jack cuts slice B, and we get slice C and D. Okay. And then Jenny gets to choose which slice she takes, so she's going to choose to take slice C, or slice A, sorry, slice A. They're all roughly the same, basically, at least in Ms. Paz's idea right now. She thinks they're all one third. So Ms. Paz, so Jenny takes slice A, then Jack gets to choose which slice he'll take. So he'll take slice C, and then finally, Ned will take, or will have to take slice D, because that's all that's left. So I'm going to write this proposal in the chat just because there's a lot of maybe confusion. So Ned cuts first, then Jack cuts, then Jenny chooses, then Jack chooses. Hopefully that proposal makes sense. So now the question is, is Ms. Paz's new suggestion fair? So A is yes, because Jack and Jenny still do the I cut, you choose method for whatever piece of cake is left after Ned cuts. B, yes, because it extends the solution for two people to three people. C, no, because Jack and Jenny can conspire to cheat Ned. And D, no, because Ned can force Jack into taking a small slice. Okay. Oh, okay. Thank you, Toby. Let me try sending it to everyone. Thank you, Toby, for letting me know that not everyone could see this. Okay. And as for the question of why the bunny is the smartest mathematician, well, actually, it's a kangaroo that somehow sounds like a bunny or a chipmunk. And that's because, well, does anyone have any guesses in the chat why the kangaroo is our animal of choice for the smartest mathematician? And the kangaroo has all your help. That's a hint. Yeah, mad kangaroo. That's why, Dr. Roo. And Dr. Roo isn't smartest on his own. He has all of your help so he can, you know, do the fair cake cutting. Okay, I will close the poll in three, two, one, zero. Okay, did pretty well. We did have another question in the chat about like, why can't Miss Paws make the cuts? Well, Miss Paws could make the cuts, but what if the participants don't feel like they're evenly divided? Then how do you choose who gets to choose the piece of cake first, who gets it second, right? Whoever gets to pick the last piece of cake may feel cheated. So you need to figure out like, well, even if you involve Miss Paws, right? It's not necessarily that simple. So, okay, so is Miss Paws' new suggestion fair? The answer is no, because Jack and Jenny can conspire to cheat Ned, and then let's see how. So Ned will cut first. Now Ned can cut as fairly as he thinks he needs to, which is cutting the slices of cake into 1 3rd and 2 3rds, okay? So he expects Jack to then choose the 2 3rds slice of cake and then cut that into two pieces. But what if Jack says, hey, I'll cut A? So Jack will cut A, and remember A is only 1 3rd of the cake. So, and let's say Jack decides to cut it fairly, or that 1 3rd of slice. So we will see that when Jack cuts slice A, we get two slices, C and D, that each of these represent 1 6th of the cake. Jenny gets to choose which slice she takes. So she'll take a slice C, or no, will she though? She'll take slice B because it's the biggest, right? It's 2 3rds of the cake. So she takes slice B, 2 3rds of the cake. Then Jack gets to choose which slice he takes. And so he'll take slice C, which is 1 6th of the cake. And then finally, Ned has to take slice D, which is 1 6th of the cake. So the distribution of slices isn't fair. Now you could ask like, okay, why would Jack do that? What incentive does he have? Well, it could be that Jack conspires with Jenny, right? So if you go to the next slide, please. So Jack and Jenny may have been colluding with each other, to share their slices of cake. So together they have 5 6th of the cake, and then they can split it amongst themselves, this 5 6th of the cake, and each of them will have 5 12th of the cake. So nearly half the cake. The older siblings cheating the younger sibling. Yes. So- Also conspire is actually the technical mathematical term that's used if you look at proofs and stuff in high school. Conspiring is essentially when two people or multiple entities are working together and kind of cheating someone else. Okay. Very cool. And so the principles of cake cutting does not have to be limited to dividing actual cake. So you can extend it to many, many people, many entities, you know, like countries, companies, different organizations and multiple resources that each person values differently. So maybe it could be about like two countries that have some land to divide up, and they might be figuring out, okay, you get, you know, these oil fields and I'll get, you know, these mountains, things like that. Or imagine companies trying to divide up an asset, you know, who gets, you know, what part and what's fair. Some of my friends, they have roommates. And so each of the rooms in their house may be different sizes. So how do roommates split the rent in a way that's fair? So these are all questions that can be answered by cake cutting. Yeah, there's a way to do all of this fairly. But for the, this case, if you're asking Bernard, how do you cut three slices of cake for three horses or three people? What you can do is basically take a knife and you start from the left side and you keep moving it towards the right. And when the first person says stop, you stop, you cut a slice of cake there, and then you give them that slice of cake. And then you repeat that for every single slice you make. And then whatever is the last slice of cake leftover, that's given to the last person. Does that make sense? So slice, each person is incentivized to kind of like yell, hey, cut the cake now, once it reaches one third of the cake, because if it extends past one third of the cake, if it goes to like, I don't know, like the halfway point, then they may get that half slice of cake. But if someone else yells out, hey, cut the cake now, that other person will get the slice of cake. And now there's half the cake left to divide between two people. So each person gets less cake. Does that make sense? Why each person is incentivized to yell out like at exactly the right time that, hey, I want one third of the cake? Okay, cool. So this is called the moving knife method. Well, Rianchi is asking, what if someone says stop at one half of the cake? Well, the thing is, let's say Ned yelled stop at one half of the cake, okay? Jack and Jenny missed out on saying stop. But now what's left? Only half the cake. So whenever Jack and Jenny say stop, whoever yells stop after that, they'll get like whatever's remaining of the cake, but they probably won't get much more than a quarter of the cake, right? So the thing is, no one wants to be the person who's last, who ends up getting the smallest slice of cake. And so people want to yell like, as soon as they think it's a fair slice of cake, they'll yell, hey, yeah, cut the cake. Okay. Now, some of you are also like finding like, what if this and what if that? And the what if scenario that we sort of haven't mentioned here is envy, right? Let's say these cake cuts have been made. I'm a very small eater. So maybe just a tiny bit of cake has been, the knife is over a tiny bit of cake and I say, okay, cut it. And someone before me has cut equally thinking, oh, this is one third of the cake. Let's say it's me, Mr. Bagheerath and you. And so let's say you pick first, right? And so you stop at the one third, you say stop. And then I'm not a big eater. So, you know, I have a very small piece, like maybe a quarter and I say stop. So now Mr. Bagheerath has more than one third of the cake. Now you could have envy because now that you realize that I'm a small eater, you could have definitely asked for more cake. And then you can be jealous of Mr. Bagheerath and that's a problem. But here the assumption is it's an envy free problem. Cause then that gets into a lot of complicated stuff that you cover in high school and stuff. So for example, I was in my high school's math team. And because I was the top student in my math team, I was in something called oralist. And basically what you had to do is the cake cutting was one of our topics. And so they gave me three problems of cake cutting and I had to solve it within 15 minutes. And then I had to give a lecture on it and solve the problems live in front of the judges. So yeah, this is something you'll definitely encounter in the future. Should we go to the next page or? Yeah. Let me, Mr. Bagheerath, do you want to share the video or does the sound still come on yours? Yeah, I can share the video, sure. Okay, cool. Let me stop sharing the screen. Just give me a second. Zoom is kind of acting up. share? Okay there. Okay um give me a moment to share please. There's not any issue if you get all the questions wrong. There still will be prizes for everyone. There will be special prizes for the people who do well. Thank you doctor, you saved the party. Can you yeah share screen yeah and uh just keep your eye on the time it's uh 10 40. Check in yet my brother who's one of the other store managers also needs your help in the supermarket. Can you please go next door? Thank you doctor, you saved the party but you can't leave yet. My brother who's one of the other store managers also needs your help in the supermarket. Can you please go next door? I guess I can. I hate being so popular. Dr. Vu, someone is sabotaging our business. Someone has ripped off the numbers underneath the barcode and we can't use our barcode scanner because it isn't working either. Well we should check the circuit breakout back but first you should check these prices for the customers. How can we do that with my laser eyes? No no we can do the calculations manually. It is quite elementary my dear whiskers. Okay let me stop sharing. Okay let me start just share my screen again. Okay. Okay. Oh, sorry. Okay. So how do you read barcodes without lasers? So a barcode is made up of several groups of alternating bars and spaces. So there are 12 groups of bars in the picture above, and each individual group of bars and spaces represents a digit. And that's why you can see that there are 12 digits. So next slide, please. Okay, so we'll just look at one of those individual groups of bars and spaces, and each group, by the way, has four total bars and spaces, and they're alternating. So it could be bar, space, bar, space, or space, bar, space, bar. And the white things are the spaces, the black things are the bars. So look at the relative widths of the bars and spaces. Now, let's say the smallest width, let's call that one unit wide. Then you can measure basically all the other bars and spaces with greater widths relative to the smallest width. So, for example, the large bar on the right of this group of bars and spaces, that's four units wide because it's four times as wide. Can everyone see how it's like four times as wide? And I've also written out below. So like the first space is one unit wide. The second bar, that's one unit wide. The third thing, that's a space again, that's one unit wide. And then finally, we have a bar that's four units wide. Okay, excellent. Thank you. So next slide, please. And so all bars and spaces are either one, two, three, or four units wide. And there's basically a lookup table that you can look at and figure out what each group of widths corresponds to, what digit it means. Next slide, please. Okay, good. So these are the different kinds of widths. So like if we have a bar of three, followed by a space of two, followed by a bar of width one, followed by a space of width one, that represents the digit zero. So hopefully that makes sense with this chart is showing. So next slide, please. Yes, okay. So now try to sum up the digits shown here. Next slide, actually, because I think I put below what the widths are here. Yeah, this might be easier to see for people. So here you can see there's two groups of bars. So the first group of bars, the widths for the bars and spaces are two, two, two, one. The second group, it's one, three, one, two. So figure out what digit that corresponds to and then sum up those digits. And that's your answer. Okay. Let me launch the poll. Okay, there. This is question four. Oh, this one, everyone's doing really, really well. So, yeah, basically, we just want to sum up the digits here shown on the barcode. We have the lookup table. So let's just sum up the digits using the lookup table. So see what 2, 2, 2, 1 represents in the lookup table, what 1, 3, 1, 2, and then whatever those things represent, I want you to add them. How to read the barcodes? Essentially, you just count the number of white spaces and black bars. So you have two white spaces, two black bars, two white spaces, one. So that's how you get 2, 2, 2, 1. But even if you can't read it, the numbers here are below for your convenience. Okay. 3, 2, 1. I'm ending poll. Mr. McGeers, are you muted? Yes, I'm muted. Thank you. So it's B, 8. Thank you guys for those of you who answered correctly. So the first digit shown by that first group of bars, it's 1. That second group of bars represents the digit 7. So 1 plus 7 is 8. Excellent. Next slide, please. So now Mr. Whistler has a question. Without the numbers, how can you tell which side of the barcode is on the left and which is on the right? What if he's reading it upside down? Well, this is actually pretty clever. So next slide, please. So the total width of the bars is always odd on the left and even on the right. So if you look at each group of bars, right, let's say, for example, for the digit 0, we look at the group of bars and spaces should be of widths 3, 2, 1 and 1. Well, on the left, it'll look like space, bar, space, bar. And then on the right, it'll look like bar, space, bar, space. Why is that? Because the total width of the bars always has to be odd for any group on the left. So if we do space, bar, space, bar, we see that the total width of the bars is 2 plus 1. Right. So that's 3. That's an odd number. So that works. And that works on the left hand side. But if we want to represent the digit 0 on the right hand side, it has to start with the bar because and then, you know, alternate bars and spaces because 3 plus 1 is 4. Right. And so the total width of the bars is 4. Does that make sense to everyone? No. Do you have an example? Yes. Let me see if I can add an example. OK. Can you... Should I refresh? Yeah. Or do you want to write on the slide? Let me just paste perhaps two pictures that might make it clearer what's going on. OK. So... Let me add a new slide here. Do you need me to copy the table down again? No. I think that should be fine. OK. So here... OK. What's the total width of the bars? If you go to slide 70, please. What's... If you calculate the total width of all the bars combined. So we have two black bars, right? No, no. Just the bars. Not the spaces. Not the white spaces. Just the black bars. Just the black bars. OK. It's just 5. Excellent. Yes. So because the second thing that you see, that's a black bar. The fourth thing that you see, that's a black bar. So you just want to see the width of those. So that's 1 plus 4, right? That's 5. So since it's 5, right, this would be an example of what this group of bars and spaces would look like on the left-hand side. Does that make sense? We wouldn't see this on the right-hand side because the total width of the bars is odd. Not even. Does anyone have any questions about that? Is everyone with me so far? So how would you see this represented? Do the students need the table? I keep on getting that question. Oh, sure. Yeah. Can you show the table again, please, from the previous slide? This one, right? Yes. So the table doesn't really matter in the sense that this is true for any digit. It's true for any digit, but the point is that if you're trying to figure out how to represent a digit to be on the left-hand side, well, you have to make sure that you do it in a way that the total width of the bars is odd. If you do it on the right-hand side, the total width of the bars should be even. So here we start with a space, then we go to a bar, then we have a space, then we go to a bar. And so this would be on the left-hand side, and this is this digit six. But if we wanted to represent the digit six on the right-hand side, we would have to start with the bar, then have a space, then have a bar, then have spaces. Does that make sense? So okay. Can you give an example? Yes. Let me paste in a picture. If you could. Okay. Okay. So these also have a width of one and one and one and four. But here, if you add up the total widths of just the black bars, what is that? It's two. Yes. Excellent. So this, if you want to represent a digit six on the left, you would do it the way shown on the left. If you do it the way that's on the right, then that's how you show it on the right side of the barcode. And ignore the table. This really is not relevant for this question. I just put it there because you guys were asking me, but this is just a mapping table. So it just says this number represents this number. It has nothing to do with this part of the problem. Yeah. So okay, so which of these contains two numbers that are both from the right side of the barcode? And so basically all you have to do is add up the black bars in each image, find the total widths of the black bars in each image, one, two, three, and four. And if it's even, that image is from the right side of the barcode. Which right side? Yours or mine? Just write one right side. I get a lot of questions, when does class end? It was on the invite. Class ends at 11.15. CST. 11.15 EST. Yeah. So right side is even. That's right, Krish. This is kind of cool that barcodes use basic, you know, math. Yeah. So next time you go to a grocery store now, you'll know exactly how that scanner works. I can read out loud the question for those of you who can't see the question, but I guess there's an image in the question. You might have to move your poll aside or something. Okay, I'm going to end in three, two, one. Okay, excellent. So which of these contains two numbers that are both from the right side of the barcode? Well, if you look at image three, the, oh no, give me a second. If you look at image one, I mean, one and three, that's the first and third bar. The width of those is one and three. So if you summed that up, that's four. And so then the total sum of the black bars is even. And the same thing is true for image two. What is the total sum of the widths of the black bars? It's one and three. So that's four as well. So those are both even. And so then the answer is A. Does that make sense? Some people are saying, okay, I see. Yeah, I think we can move on. Yeah, let's move on. Because of the time. So how do you tell if the barcode is fake? Well, this is a concern that Mr. Whiskers has. So Dr. Wu tells him, well, you can sum up the first five digits and then sum up those digits, et cetera, until you get a single digit. So this single digit number should match the six digit of the barcode. So the next slide, please, for an example. So let's say our code is 1, 2, 3, 4, 5, 6. So you take the first five digits, 1, 2, 3, 4, and 5. You sum them up. That's 15. Then you sum up the digits in 15. That's 1 plus 5. And that gives you 6. And 6 is equivalent to the last digit. I don't know what that W-I-K-P is. Sorry about that. So but yeah, thank you. So that means this number is correct. However, one of these barcode numbers is fake. So try to figure out which one it is. And the sixth match is this last one. So just make sure to keep that in mind when you have the poll, if I can ever get the poll ready here. Oops. Sorry. Yeah. So people are asking about the spaces. Like how do you tell there's a space? Well, it's a white space right between black bars. So that's how you tell it's a white space. Which of these numbers is fixed? Again, you need to keep on adding the first five digits. Keep on adding them until you get to a one-digit number and make sure it matches with the last digit. And this is the final digits you get from reading the barcode, right? So the original like groups of bars and spaces, they don't matter here. That chart of widths to digits, that doesn't matter here. So there's no need for the chart. Yeah. Oops, sorry. Not possible to change answer. I'll give you 20 more seconds. Okay. For those of you who are saying don't understand it. Your code is 1, 2, 3, 4, 5, 6. You are summing up just the first five digits, 1, 2, 3, 4, 5, which you get 15. 15 is not a one-digit number. So you have to add the 1 and 5 and 6 is a one-digit number. So that's where you stop. This 6 matches the last digit. So you're all good to go. Now let's go back to this question. Hopefully that made sense. So just add these first five-digit numbers. If what you get is not a one-digit number, then whatever your output is, again, keep on adding the individual digits until you get a one-digit number. And that should match with this last digit. If it doesn't match, then it's a fake. Okay. One more time. And this is the last time. So your code is 1, 2, 3, 4, 5, 6, right? You only take the first five digits. So we're just going to take 1, 2, 3, 4, 5, and we're going to consider 6 separately. So for 1, 2, 3, 4, 5, we're going to sum up each of these digits. 1 plus 2 plus 3 plus 4 plus 5 is 15. But that's still not a one-digit number. So now you take the 15, and 15 is composed of 1 and 5. So you add 1 and 5, right, 1 plus 5, and you get 6. Now is 6 matching with the last digit? It is. So this is a real barcode. If it didn't, then it's a fake barcode. Okay. Now I'll give 20 more seconds. Okay, three, two, one, I'm ending poll. Excellent. Thank you. Um, Oh, uh, I think the poll has the incorrect answer, but, um, we'll have an incorrect answer. Yes. Uh, yeah, it should be C, um, C should end with one if it were legitimate. So how do we know that? So zero plus two plus seven plus four plus six. We can do the math on that. That's equal to 19. That's not a one digit number. So let's add a sum of the digits one plus nine. And that gives us 10. That's not a one digit number. So let's sum of the digits of 10. So one plus zero, that gives us one. Uh, the last digit is not one, so it's invalid. Okay. Next. Thank you. So yeah, we'll fix the polls. Thank you for that. So, Oh, sorry. Next slide. Yeah. Okay. Uh, do you want me to play the video for that or? Yeah. Yeah. If you want to play the video. Excellent. Thank you. Okay. So we're almost done here. Nice. okay so we know that the laser was broken so how do they fix the laser well we're locked out oh my gosh it's mr. Taurus the manager of composite cafe but why would he do this to us he's like a bull in a China shop running around and destroying everything but how the doors are locked we'll need to figure out how fast we need to go to get them to open okay so they're locked out they need to ram the doors with the grocery cart now how do they do that well they can drive a grocery cart into it so what we need to know is how much energy does it take to open up the doors well kinetic energy that's the energy of motion you can calculate that with the formula half times mass times velocity squared now the cart is 10 kilograms that's the mass it takes 125 joules of energy to open the door so if a jewel is uh can be expressed in kilograms times meters per second square what speed does the cart need to be traveling at to break open the door and uh yeah we'll give you i guess 15 seconds for this one since i think this is a quick one One second. Let me give the poll. For the video the idea is just they want to ram the shopping cart into the door since the door got locked by Mr. Taurus. And for those of you who think that this is a little bit cringy I would say that you know the grade 6 to 12 are better sports than you. Somehow they always love it. We do this every year for them. So maybe you guys are just too cool for school or I don't know because even adults love cartoons. Okay. Some people are still saying... This one is difficult. This one is a little bit difficult so I can walk you all through this as well. The formula for kinetic energy is half mass times velocity squared. So you need to multiply half times 10 kilograms which is the mass times velocity squared. And that should equal to 125. So you need to solve for velocity. This one is a little bit difficult. This is one of those questions that maybe the older students will be better at. Well you don't need to know the formula. We gave it to you. Yeah so this is half times mass times velocity squared. We gave you mass. You need to solve for velocity. And this should equal to 125. Yeah volume. That's the that's the problem with math right? The same letter represents a million things. Yeah M means mass and meters. So it's a little confusing. I'll give you three more seconds. Three, two, one. It's the weekend. Yeah all of our brains are officially broken. It's B and we can quickly go through on the next slide or skip one slide please. And on the next slide. Yeah so it takes 125 joules. So that's 125 kilograms times meter per second square of energy to break open the door. We need to generate that much energy with the cart. So half times mass times velocity squared needs to equal 125. So if we substitute 10 kilograms in for mass that gives us half times 10 kilograms times velocity squared equals 125. In other words half times 10 that's just 5 right? So 5 times velocity squared needs to be 125. And so V squared that's velocity squared that's 25 and therefore velocity is 5. So we have to travel at 5 meters per second. So we do know the mass range. It's 10 kilograms. Yeah we gave the mass and velocity. Mass is 10 kilograms and velocity. We gave the joules the total energy and so you can calculate. Yeah the total energy exactly. Okay excellent. We're gonna move on. If you didn't get it that's totally okay. This is a little more complex question because it involves physics but it's at the end of the day it's just math. Okay and now we have a smart cart that we can program a specific speed into but only in revolutions per minute. So let's look at how that works. Next slide please. So what's the exact RPM Dr. Roon and Mr. Whiskers need to be traveling at in order to go at 5 meters per second? If they go too slow they won't be able to break through the doors. If they go too fast they'll not be able to stop and they'll cause too much damage. Next slide please. Okay so the wheels are 1 over pi meters in radius and the circumference of a circle is 2 times pi times radius. During one revolution a wheel travels the equivalent of its circumference on the ground. How many RPM does the cart need to travel at to reach 5 meters per second? Hint, consider how many meters the car traveled at 1 RPM. What speed does the cart need to reach in meters per minute? There are 60 seconds in a minute. And again this is one of the more complicated questions. So yeah if you have any questions about it just put this in the chat. Okay. Do you think you can help with the formula, Mr. Brigert? We'll give them a... Sure. So what's if we are going five meters per second, what is that in meters per minute? Consider that. So how many meters per minute is that? And then we know the circumference of a circle is two times pi times the radius, which is one over pi. So what's two times pi times one over pi? That's the circumference. And we travel one circumference in one rotation or one revolution, right? So how many meters do we travel if we're going one rotation per minute? So one rotation per minute equals to two times pi times one over pi meters per minute that we're traveling. So if that's one rotation per minute is this much, how many rotations per minute do you need to go to reach five meters per second? And a lot of you are on the right track, whoever is messaging us. So think about the circumference. And for those of you who don't understand, maybe just message me what the circumference, sorry, what the, yeah, what the circumference is, because you need to multiply two times pi times one divided by pi. What would that be? Yeah. So I get the, that's a, yeah. And now five meters per second, how many meters would a person who's going five meters per second, how many meters in a minute? Yeah. So now see if you can do the rest of it. So we know that it's however many meters per minute, and we know that we see how many rotations you need to do per minute. If during, you already calculated the circumference. So the circumference is how much it goes in one rotation. RPM is rotations per minute. So in one rotation is whatever the circumference is. So maybe I should write this. One rotation equals whatever the circumference is. How many rotations per minute? We have already given you the sort of the speed here. Okay. I will end the poll. Again, these last two questions we know are very difficult. They were more for, geared toward our older students. So I'm going to, where's my poll? Oops. Okay. So I can explain the answer. So first of all, five meters per second. Okay. So, okay. The circumference is, we said two times pi times radius. That's two times pi times one over pi. That's equal to two meters. So at one RPM, the cart would be moving two meters in a minute. Now the cart needs to move at five meters per second. There are 60 seconds in a minute. So this is equivalent to saying five meters per second times 60 seconds per minute. That means we move 300 meters in a minute. Now, if one RPM equals two meters per minute, then how many RPM do we need to travel at to go to 300 meters per minute? Well, 300 divided by two is 150. So we have to travel at 150 RPM. Some people are suggesting using X. We didn't, I was trying to avoid using algebraic notation because that might confuse people even further. So, yeah. Yeah. It's a huge age group, right? First to fifth graders. So think about when you're in first grade if you knew about X. So, okay. Mr. Bagheer, it's all you. Excellent. Okay. We got in, Dr. Rue. Thanks for your help. I thought we'd be stuck out here until the cows go home. Mr. Torres, we caught you red-handed or red-hooved, I guess. And I would have gotten away with it had it not been for you meddling kids. And Kangaroo too. Whoa, what a mess. But why would you do this to us, Mr. Torres? Your customers always take up all the parking spaces in our shared parking lot. So none of my customers can ever park. I thought maybe if I drove you out of business, I could have the parking lot all to myself. That's why I distracted you by asking you to cater for those misbehaved horses while I caused mayhem here. Well, the cat's out of the bag, Torres. It's time to leave. Wait a minute. Why can't we solve this problem civilly and share the parking lot? Cake cutting, anyone? Okay. That's it. So congratulations, folks. You definitely helped here solve all the issues that Miss Pauls and Mr. Whiskers were having. I'll give you 20 more seconds. But if not else, that's all we had for you today. Congratulations again. And thank you so, so very much for being a part of today's Math Mystery Theater. Have a great weekend. Thanks for coming, everyone. Bye. Thanks for participating, all of you.

Mathematical Adventure

Math Kangaroo 2024

MethaPlus

fair division

cake cutting problem

barcode structure

kinetic energy

revolutions per minute

interactive session

mathematics in everyday life