Today we're going to be drawing or interpreting pictures and diagrams. So sometimes Math Kangaroo Contests will include a picture or a diagram, and sometimes you might just want to draw one to help you personally understand the problem. We'll practice both of those today. So remember as we go through today, you want to make sure that you understand what the problem is asking. One way to do that, like I said, might be to draw. If you happen to be like me and you are a good visual learner, then sometimes drawing a picture can really help you out. So our plan today is going to be to look at pictures or to make pictures. I'm telling you the plan because that's how we do our webinars. We do a different type of tool each week. And then I have to be very careful. I might want to check my answer. One way to check your answers is to put your answers into the chat so that I can see what you have. And sometimes I'm able to give you a little thumbs up or smiley face to let you know you got it right. We do have a lot of polls today, so that will be another way you can check your answers is to put your answers, your responses into the polls. Remember polls are anonymous, so I cannot tell what you have done in the poll. I can't tell it's you. I can't tell if it's somebody else. You don't have to worry about that. Let's go. This is just a timeline of what we do in each lesson, so you can see that we're here on webinar number three with our pictures and diagrams. Here is our first one. I will read it and then I'll launch. There is no poll for this first one because obviously the answers are these pictures. So just put your answer into the chat. We start drawing segments connecting every dot on the circle until we are back at number one. The first two segments are already drawn as shown in the picture to the right. What figure do we get? Okay, I have a lot of correct answers and then I also have quite a few questions and that's natural. Don't worry about it. So what does it mean when it says the first two segments are already drawn? Well, we have to look at how they drew them. They drew a connecting from 1 to 2, 1 to 3, but skipped number 2. And then they drew from 3 to 5, but skipped number 4. So we are going to connect every other point with line segments. So I will connect from 3 to 7. Oops, that's a very quirky line. Maybe I can undo that one and do straighter lines. Okay, so if we connect from 5 to 7 with a nice straight line, and we connect from 7 to 9, now what happens? I don't have a number 11. But my pattern was skipping one. So if I continue that pattern and I skip, that will get me to number 2. And if I skip from number 2 to number 4, number 4 to number 6, 6 to 8, then 8 would come back to 1 and you can see that will start it again. So the best answer here is what picture will I get matches E. So that will be the answer. I have a student who cannot see, you need to change your view. If you cannot see the problems, the screen is shared, you'll need to change your view. And it's, I can't really debug your view. You can ask someone in your house to try to help you with your particular version of Zoom. Okay. So today we are going to be focusing on the drawing. So if a problem is not illustrated, it could be helpful. It depends on if it's an easy problem to draw. It depends on if you like to do it visually. I do. So you'll notice that even when it's not a drawing week, I frequently will draw. When you do draw, make sure you do something as simple as you can. Okay, so you'll notice today that I might use some letters instead of fancy pictures, or I might just use little dots instead of a fancy shape. Often you can also use these in a multi-step problem. So you might, sometimes we'll have like people changing places, and so you might have to do the different steps. All right, let's see what else. Mike sets the table for eight people. He must set the table correctly for each person sitting at the table. Setting the table correctly means that the fork is on the left of the plate, and the knife is on the right of the plate. For how many people did Mike set the table correctly? I will launch a poll here in a minute. Remember, the poll is anonymous, so I cannot tell who is answering the poll, but it's a great way to see how everyone is kind of as a group is doing. Is this the first question? We had a warm-up problem, but this is the first problem of the lesson. So if you can see that poll, you might have to scroll up and down to see the picture or to see the answer choices. That's fine. Everyone has answered, so we can end the poll and share those results. What you can see is that over 70 percent of you think the answer is five, and that's a good choice. That is correct. Let's take a look at the problem because not everyone got the same answer. We want the fork to be on the left and the knife to be on the right. This one is correct. This one is not correct. I didn't get to answer. Is correct. We cannot participate that way. We can only participate in the chats and in the polls. This one is correct. Now, how do we figure out if this one is correct or incorrect? Well, I've had some students who told me I printed the handout, and so I can just turn the handout around. And then you have to imagine that you're actually sitting at this table, right? So imagine that you're sitting at the table over here, and you're looking this way. So this would be on your left, and this would be on your right. So that is correct. You can see this one is backwards. This one is a mirror image of this one. So this one is incorrect. This one is just like this first one. It is correct. This one, being a mirror image on the opposite side, imagine if I was sitting over here and looking at it. I'm looking at it from here. And this is on the right-hand side when it's supposed to be on the left. So this is incorrect. So if I count up my checkmarks, I have 1, 2, 3, 4, 5 checkmarks. Five are set correctly. Number two. Zach was dividing a chocolate bar. He broke one row of five pieces for his brother, and then one row of seven pieces for his sister. That's shown in the picture. So here's the row for the brother on the top, and here's the column for the sister. How many pieces were there in the whole chocolate bar? This is the whole chocolate bar before he divided it. So this is before, not now, but before the division. I'm sorry, I can't annotate real well with this mouse. All right, I have almost all of you have participated in the poll, and I'm going to share those results right now. So about half of you, a little more than half of you think the answer is 40, and the next most popular answer is 35. So let's take a look and see if it's 35 or if it's 40. So we know that for his brother, he broke off one row with five pieces. So this is five pieces, and we can even draw this out. We can draw little dividing lines, one, two, three, four, five pieces. And we know that for the sister, there were seven pieces. So this color is a little hard. We could try to draw this off with seven pieces, one, two, three, four, five, that's very uneven, but six, seven. Now if I look at this all together, I can continue these lines. Very crooked today, wow. So if I continued those lines and I counted all of those, I would find out that there were 40. But another way to think about this, without having to draw all the lines, is if there were five across the top, and then, remember I've removed this piece. This piece is gone now, and I still have seven. So originally there was one more piece here. So I'd have seven plus one equals eight. So originally in the bar, I started out with eight by five. Eight times five equals 40 pieces in the original candy bar, D. So if you said 35, what you probably did was seven times five, but you forgot that this one piece was broken off at the beginning. So we really had eight pieces going up and down to start with. All right, clear that and go to the next problem. Peter and Paul went to a boy scout camp. During a meeting, the scouts stood in a single row. On one side of Paul, there were 27 scouts, and on the other side were 13 scouts. Peter was standing exactly in the middle of the row. How many scouts were there between Peter and Paul? It's a multi-step problem. Drawing might help you. I do have a poll. I'll give a few seconds to see the problem big before I launch the poll. Anybody else want to answer in the poll? I'm not seeing everybody yet, which is fine. But just keep in mind that on a Real Math Kangaroo contest, you will not want to leave any of the answers blank, because blank is worth zero points. And if you guess correctly, you could get points. If you guess incorrectly, we do not subtract anything. So it's better to guess than to leave blanks. I'm getting a lot of different answers for this one. This is a really cool problem. Let's share the results because I think that's very interesting. Look at that. The most popular answer, I guess, is seven. So let's take a look and see if seven is correct. But every answer choice got some numbers, got some votes. So let's take a look. Okay, stop that and come back here. All right, hopefully you see the problem now. Let's see what I can do to draw it. Let's look at one sentence at a time. On one side of Paul. All right, so I'm going to put Paul here. I'm going to write Paul. And on one side of Paul, I'm just going to start on the left. I'm going to make a long line because that's the biggest one, are 27 scouts. And on the other side, it says there were 13. That's a shorter line for me. Peter was standing exactly in the middle of the row. So if I do another one, Peter is exactly in the middle. Now remember, they're actually in the same line. I'm just drawing it this way so it's easier for me to see. You might have drawn yours slightly differently. That's all right. If he's exactly in the middle. Now I have to figure this out. In the first picture with Paul, I know that there were 27 on one side of him, 13 on the other, and Paul. So 27 plus 13 plus Paul equals 41 scouts. You see? Don't forget to add in Paul. So in this one, if Peter is exactly in the middle, I'm going to take 41 minus 1. That equals 40. So I know that there were 20 scouts on Peter's left and 20 scouts on Peter's right. So hopefully everyone follows along to that point. That point's pretty good, right? Now we want to know how many scouts are there in between Peter and Paul in the line. Well, there's a few ways to do this. You could do 27 minus 20 equals 7, but that includes Peter. Because from Paul, when I count, Peter's actually in that line. So then I have to subtract Peter and I get 6. The other way to do it is to think about it in this way. Paul, if this is the first person in line, Paul is number 28 in line. Peter is number 21 in line. So who is in between them in line would be numbers 22, 23, 24, 25, 26, and 27. So that is 6 people, right? So I would need numbers 22 through 27. That is 6 scouts. This is through, not subtract. This is through. Okay, so the answer is A, 6 scouts. Kind of an interesting problem. So remember, if you just take the 27 minus 20, you're actually subtracting Peter, right? So we want to make sure we get rid of that one. All right, very interesting problem, and I think the pictures kind of help you out. If you can figure out what their numbers are in line, that's one way to do it. A square is divided, sorry, a square piece of paper is white on one side and green on the other side, and divided it into nine little squares. She labeled some edges with natural numbers 1 to 8. See picture 1. This is picture 1. What is the sum of the numbers along, she labeled some edges with natural numbers 1 to 8. What is the sum of numbers along the edges which she cut? See picture 2. I'm not going to give any more explanation right away. I want you to try it just the way Math Kangaroo wrote it, but in a few minutes I'll give another clue and then another clue, and I think that will help you. But first let's try it just the way it would appear on a contest. Okay, I'm going to offer some hints now. You can see in picture two, we see some white parts. We know that one side is white and one side is green. If we call the green the top, then the white is the bottom. If we see the bottom, then it must be folded, right? So we can see that some parts are actually cut. This green piece here is cut. And then some pieces are folded either up or folded down. This is a down fold, right? And this is folded up. So that's the first hint is that some of these lines get cut and some get folded. I'll leave you with that and I'll provide another hint in a few more seconds. OK, final hint before I launch the poll. For me, it helps if I imagine that this corner is the same corner as this corner here. So that helps me. And I'll go ahead and I'll launch the poll, the little pictures in the poll as well. Okay, most of you put your answer in the poll, that's awesome, and most of you have it correct. Not everybody, and I know there were some students who still have questions. I hope the hints helped. You can let me know in the chat if those hints were good, were helpful for you, but it looks like we have some really smart answers here. 20 is correct. Let's take a look together. So if I match up the corners the way I was talking about, then I can see that this first cut here is along the line with the number two on it. If I go clockwise, I'll see that I have a fold, a fold at number three, not a cut, but I have a cut at number four, a fold at number five, a cut at number six, a fold at seven, and finally a cut at number eight. So now it says that I need the sum of the numbers that are cut. That's two plus four plus six plus eight. And I'm a little lazy, so two plus eight is 10, four plus six is 10, that gives me a total of 20. I'll ask all the students to stay muted during the sessions, okay? A squirrel went through the maze gathering nuts. You can see the picture. This is the picture of the maze and there's one nut in each box. It could only go through each door. It could only go through each door once between the rooms of the maze. What's the greatest number of nuts it could have gathered? I think an important thing here is it can only go through each door once. So step one is always understand the problem. So we can go through the door, we want to get the greatest number of nuts. So we don't want to know the fastest way through the maze, we want to know what's the greatest number of nuts that you can get. I do have a poll. See if I have a poll for this one. Anybody else want to put the answer into the poll? Now we have just about everybody participating. Thank you. And this is one where you did really well. Most of you have said that it is 11, if that is correct. I have a few students saying 10 and 12. That's kind of common that sometimes we miscount or we miss a step here. So the idea is to go from the in arrow to the out arrow, going through as many squares as possible without backing up, right? Cause you can't go backwards. That's what it would mean when you can't make circles, you can't back up. So we want to avoid the dead ends, right? So if I was to go across or I could go down, in both of those ways, I would end up right here. So I have to choose one of them. So I'm going to choose to go across. Doesn't matter. Cause I get one, two, three nuts, or if I went down, I get one, two, three nuts. Now I can go down and out of the maze, but that would leave behind a whole lot of nuts. So I want to try to collect as many nuts as possible. So I think I'll go all the way across. If I go up, I hit a dead end. So if I go up and across, it's a dead end. So I'm going to go down. Now I could come out, but if I come out of the maze, again, I won't collect as many nuts as possible. So I can go across all the way. If I keep going across, I'll end up in the dead end. So I could come down and then go out. And if I've done that, I've collected one, two, three, four, five, six, seven, eight, nine, 10, 11 nuts. Now, some of you may have done this slightly differently, and that is okay. Remember, we can go either way at the beginning. There is also this path that some students use. I'll go down at the beginning just to show you. It doesn't matter. And some students may have come down instead of across and then come back up. You notice I go through exactly the same squares and I still have one, two, three, four, five, six, seven, eight, nine, 10, 11 nuts. So you could zigzag side to side or you could zigzag down and up. You'll always have 11 nuts and you'll always want to avoid these. Let's see if I can change my color. And you'll want to avoid these dead ends, right? These are all dead ends. So you can't go into those dead ends. All right. We are making a sequence of triangles out of diamonds. The first three steps are shown. On each step, a line is added to the bottom. In the bottom line, there are two outside diamonds, one on each side that are white. All the other diamonds in the triangle are black. How many black diamonds will the figure have in step six? So remember, these are the, oh, sorry. These are the step numbers, step one, step two, step three. And that's basically, they're showing you how to interpret these words into a picture. They're asking us about step six, not step four. So make sure you go all the way to step six. And they want to know how many black diamonds there will be. And remember, the bottom corners are white. I'll go ahead and launch that poll now. Thanks to the students who are putting their answers in the chat. Hopefully, you find my little smiley faces or my little comments helpful. Just a few more seconds, because I'd like to explain this problem by more than one method, so it'll take a little time to explain it. All right, we're going to stop the poll here. I see that not everyone has answered, and that's OK. This could be a difficult problem. We've had about 85% of you actually given an answer, and of those, 42% have 26. That's the most popular answer. And I guess we have a tie between 21 and 34. Let's see what we can do. As I said, there is more than one way to go about drawing this solution. So I'm going to show a few methods, because I think that's helpful. Because the way I solve it, my favorite method might not be yours, and I think it's good to see some different choices and opinions. So what some students might have said is that there are 1, 2, 3 diamonds in the first figure, and there's 1, 2, 3, 4, 5, 6 diamonds in the second figure. There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 diamonds in the next figure. So if you take a look at this, we have added 3. We have added 4. So for the fourth figure, we would probably want to add 5 diamonds, which would take us to 15. For the fifth figure, we would probably want to add 6, so that takes us up to 21. And for the sixth figure, we'd want to add 7, because it says we add one more in each, right? So that takes us up to 28. So this is the total diamonds. So I know that the two bottom corners are going to be white. So 28 total diamonds minus the two white diamonds gives me 26 black diamonds. That is correct. And this is a great way to solve the problem. But what if you didn't do it that way? What if instead you counted just the black diamonds? That's OK. You can count the black diamonds. What you'll find is the same pattern, that you have 1 here, you have 4 here. So you can see you added 3. Here you have 1, 2, 3, 4, 5, 6, 7, 8. So you can see that you've also added 4. So you would follow this same pattern, and you would get 26 instead of 28. Because each step, you're going to be 2 less, because you're not counting the white diamonds. This is, sorry, a little boo-boo there. That should be 19. I can add. I just can't write. There we go. So that's a good way to do it. Nothing wrong with this method. There is another method, which is perhaps, I'm going to erase some of this out of here. This is why recordings are great, because you can still see, if you like that method, you'll still have it. So you might have tried to draw it. You might have tried to draw that sixth figure, and that's fine. If I draw that sixth figure, I have 4 rows in the third figure, 3 rows in the, so I might draw 1. That's 2 rows. Here's 3 rows. Here's 4 rows. Here's 5 rows. Here's 6 rows. And in the sixth figure, I have 7 rows. But I'm going to make these open circles, because those are white. So you could draw it this way, and then you could count up the blacks. That's a very good way to do it. Draw and count. You can also take a look at this figure that you've drawn, and you can discover that there's 1 black here, 2 blacks here, 3 blacks, 4, 5 blacks, 6 blacks. I would say that there are 7 blacks, but the 2 outside ones are not black. So this is 7 minus 2, which is 5. And then if I add up all those numbers, I also get 26. So you can do some counting. You can do some adding. You can do some patterns, several ways to solve this problem. I hope that helps. Sometimes it's confusing when I give more than one way. Sometimes I'll be like, that's the way I did it. I got it that way, Coach Sagi. The hexagonal stained glass tile is flipped. So it's glass. Oops. My mistake. I'm in the wrong. Click wrong. It's stained glass. So remember, glass is see-through, right? So it's flipped. One of the flips is shown. What does the stained glass look like at the far right? By the far right, we mean this piece here. We want to know what that one looks like. I do not have a poll for this one because I wouldn't have been able to put those little picture choices in it. So just put your answers right in the chat. Well, you sure did put those answers in the chat. Good job. Have a hard time keeping up with all those little high fives and smileys that I give to students sometimes. So let's think about what it means to flip. So you can see that the ladybug is right next to the ladybug. So that likely means that when I flip it, the dolphin will be right next to the dolphin. I'm not going to try to draw all these little animal shapes. They would be not very recognizable. I'm going to use letters. If you're better at drawing, that's fine. If you have colored pencils and you want to try to match the colors, that's fine too. You can do red for the ladybug, gray for the dolphin, however you like it. All right. Opposite the dolphin, as far away as possible, I see what I'm calling the wiener dog. So you can see it's a dog with short legs. I think it's a wiener dog. So I think that the wiener dog, WD, is going to go right there. And I'm going to just give you a little decoder key here. So, so far I have that D, capital D I should say, equals dolphin and WD equals wiener dog. All right. Next to my dolphin, over here, I would have an ant. Between the ant and the wiener dog is the butterfly bee. Okay. Let's see. Next to my dolphin on this side, when I flip it, this I'm going to call a cat. And then the only other one is the ladybug. So if I'm going to finish my little decoder key over here, I have B equals butterfly, LB equals ladybug, C equals cat, and A equals ant. So now my letters are easy for everyone to understand. Now I'm going to flip it one more time. Remember, on the place where I flip it, the dark line, I get the exact same animal, just flipped over. So there's the, the butterfly. Opposite it, see if I make a straight line across, that will be the cat. So if I look for a picture that has a butterfly and a cat, I see that this one does, and none of the others do. None of the others is butterfly and cat in the same orientation. So the correct answer has to be B. Okay. And I could draw in the rest of the animals, but it's not going to change my answer. So we won't waste the time to do it, but you could. All right. We have a lot of bonus problems today. So we're going to keep using this drawing strategy. We're going to have a lot of fun on these bonus problems. Now remember, bonus problems are usually tricky, hard five-point problems. This one happens to be a four-point problem, so maybe it's not too hard. Let's take a look at it together. The room in Kanga's house are numbered. Baby Roo enters the main door, passes through some rooms, and leaves the house. The numbers on the rooms that he visits are always increasing. Through which door does he leave the house? And I do have a poll. I'll launch it in just a minute. Remember, you have to have increasing numbers. If the number's not increasing, you can't go through that room. Anybody else want to put their answer into the poll? Most of you have it correct. All right here are the results of the poll and like I shared most of you do have it correct. The correct answer is D. So good job. Let's take one quick look at this one and then we'll go on to some other problems. So if I enter the house at 1 I could go to 2. So let's see what happens if I go to 2. If I get to 10 that's pretty much a dead end because they're like there's no numbers greater than 10 in that area. I go to 4 I can go to 6 that's okay. I can come to 10 again remember I didn't want to enter 10 it's too high. If I go to 9 that's all right. I can't go to 8 from 9 but I could go to 12 but now I'm at a dead end because there's no group number greater than 12 nearby. So I've got to back up a few steps here. So perhaps I don't want to go to number 2. I probably don't want to go to number 9 because it's so large. So let's try number 3 first. From number 3 I can go to number 4 and number 6. Then I'm stuck I can't get any further because number 5 is lower than number 6. So again let's back up a few clicks. I go to 3 now I've decided I have to come to 5. I don't want to go to the 6 and the 5 I discovered that was a dead end last time. I could go 5 6 can't come to 5 but I can go 6 7 that works. I can go to 7 8 either this way or 7 8 that way and then either way I want to come back to 9 and then the only way out would be through D number 10. So a little bit of trial and error going through the maze kind of one of these kind of almost like a game problems right a little bit of logical thinking just increasing your numbers kind of a fun one. All right let's move along. Bonus question number 2. How many more triangles than squares? I want more so remember more means a difference. There's a subtraction. How many more triangles than squares are shown in the picture? So I have to figure out how many triangles there are and how many squares there are. Don't forget all the hidden ones. Okay. I do not have any more polls for these bonus problems so you'll have to use the chat which is fine. It's a great way to communicate with me. I'm seeing several different answers. Some of you are correct, good job. I just wanna go through and let's see if I can help you find the squares and maybe that will help you find the triangles. So of course, if I do just the smallest squares, I have one, two, three, four, five, six of them. But I also have a two by two square. So I have seven squares. I'm gonna just draw a figure for squares, seven squares. See if you can use that same kind of idea to figure out how many triangles there are and then you'll need to subtract to find the difference. Okay, let's see if I can show you the correct solution. We have some of you getting it right, some of you not quite, but that's all right. So let me clear what I had put up here before. All right, so here are the squares. This is what I had shown just a moment ago, that there are six of the 1 by 1 squares, and then there's one of the 2 by 2 squares. So seven squares total. Now let's look at the triangles. There's one that's the entire thing. Some students miss that. Don't forget to take the entire thing as a triangle. Then I have the smallest little triangles, which are the ones that you see up along the edge. And now I can make triangles that entail three pieces. You see those three-piece triangles? So there's one, two, three more of those. And then I have triangles that take almost the whole thing. They're missing one row. So all together, I have 10 triangles. So 10 triangles minus the seven squares is that the difference is three. I think when you see it color-coded like this, it really helps. Does that help everybody? Sure helps me. Yes. Good. All right. What is the largest number of small squares that can be shaded in the figure on the left so that no square, like the one shown on the right made of four small shaded squares, appears in the figure? This is going to be—this is from 10 years ago, but it was the second-to-last question, and I think it's a pretty tricky one. You want to shade as many squares as you can, but you cannot have four together that get shaded. So you will need to leave some white. We didn't make this easy with our answer choices, did we? Because the answer choices are all very similar, right? So it's not like you could say, oh, some of these are just not reasonable at all. That's always a good strategy on a multiple choice test is to say some of those don't make any sense. All right, I think I'm gonna start this one by shading all of the outside squares. So excuse my little scribblies, but that's how I'm gonna do it. I could probably use a stamp or something, huh? So if I scribble in all of the outside ones, that works because I don't have any little four square, right? Okay, now let's see. I could do this one as well, that works. Can I do this one here? If I do this one here, that works, but which other ones can I shade? I can shade this one. I can shade this one. I can shade this one, this one, this one. Can shade this one. Now, if I tried to shade, for example, this one here, that will give me squares that I don't want, right? Because that would give me, I'll change color, I think that will help. That will give me like this one here, right? That doesn't help me, so I can't shade this one here. Let's see what I could shade. I wanna keep trying to shade things. What if I shade this one here in the middle? That works, but if I try to shade any of these that I make blue here, if I try to shade any of those, that is going to interfere, right? So let's see how many I have shaded. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. That works. Now, there is another slightly different pattern that you can make that also works. So some of you might have done this. Let me try to get this. Some of you might have said that I can shade here, That gives you four unshaded, but it's the same total shaded. So there are two patterns that you could do. In either case, you have four white squares. So with the four white squares, there are 21 shaded squares. Just like to show you when there's more than one option, because some students might say, but I did it a different way. Different way does not mean incorrect way in Math Kangaroo, okay? A large cube, we're out of time if you need to leave, but I do have this one more problem and I really like it. This is actually an even more advanced problem if you wanna stay. A large cube has an edge with a length of seven centimeters. On each of its six faces, the two diagonals are drawn in red. The large cube is then cut into small cubes with edges one centimeter long. How many small cubes will have at least one red line drawn on it? So you can think of, I don't want a diamond. I want just a square. I think of just the square. And then I think of the red lines drawn on the diagonal. See why drawing helps on this one? If I didn't draw this it would be pretty hard to visualize it, but there's a drawing for you. Okay, I'm going to keep helping. I'm going to go to the next slide where somebody has drawn this a little bit cleaner. So you can see on one face of the cube, so this is one face of the cube, and on one face of the cube I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13 squares that have some red marking in them. I didn't count the center square twice, I only counted it once, right? So I have 7 plus another 6, 13. And I have how many faces on a cube? I have 6 faces on a cube. So if I did 13 times 6, that gives me 78. So I might have some students who choose to circle 78. But take a look, I'm trying to change the color, take a look at what happens with the cubes on the corners. Do you see how the cubes on the corners get counted in each face? So one of the ways I like to think about this is that the corner cubes are counted three times in that 78. So I need to subtract the extra counts. Okay, so I have how many corners on a cube? I have eight corners. They were each counted three times, but I really want to count them one time. So if I take eight, okay, so I have eight corners. I don't want to count them so many times. I want to count them one time, not three times, so that's 8 times 2. 8 times 2 equals 16. So now if I take my 78, which is the total count, I subtract the 16, which was the corners that got counted too many times, I would have 62 of the small squares. And that's the end there. So I hope you liked it. I'm proud of you guys who stuck around to do all the bonus problems. I like to get to do the bonus tricky ones with you, get your brains thinking really hard. So don't forget, if you want to do really well in the Math Kangaroo contest, try to practice past contests and keep coming to our webinars. So I'll see you again next Sunday. Have a great week. Bye.

Math Kangaroo

mathematical problem-solving

diagrams

visual learners

interactive exercises

critical thinking

geometric transformations

pattern recognition

problem-solving techniques

creative interpretation