Hello, and welcome to the Math Kangaroo 2017 Levels 3 and 4 Solutions video series. First of all, I'd like to congratulate you on participating in the Math Kangaroo. When I was in elementary school and in high school, I used to participate in the exams myself every single year. So I'm very familiar with how big of a mathematical challenge this can present. So you should definitely be proud of yourself for taking on this huge mathematical task. It's going to be very beneficial for you in the long run, and it's going to be great for your brain. Now the purpose of this video series is to show you a path to get to the correct answer of every single question on the exam. Keep in mind, sometimes the method that you see in the video will not be the only method to get to the correct answer. Sometimes you might get the correct answer using a completely different method, in which case your method might be equally good, or it might just be that you were lucky and your method doesn't normally work. It just happened to work for that particular problem. If you have any questions about whether your method is equally good or not, please feel free to email me at thomas.mathkangaroo.org. In fact, if you have any other questions about the exam, please don't hesitate to get in touch with me. Now before we move on to the actual videos, I'd like to offer a couple of pieces of advice. The first piece of advice that I always give is to make sure to actually try doing the problems on your own before looking at the videos. Because if you just look at the videos right away, then you're going to have me telling you this is how to do this, this is how to do that, and you'll hear it and then you'll go, oh, okay, that's easy. I can do that. But you won't have actually learned anything. You'll have just seen how I solve a particular problem. It's much better to actually struggle with a problem on your own for a little while, so that way you understand why a problem is difficult, maybe where your weaknesses can be, and anything of that nature. So definitely try solving the problems on your own first. Now the second piece of advice I have to offer is to make sure that you write down as much of your work as possible. Because a lot of times people who are good at math try doing too much work in their heads because they think they can, and that's just a way to make silly mistakes. Plus, if you actually write out all of your work and then review your answers later, it's easier to see where you might have gone wrong because you have all your thoughts written down on paper. So there are only advantages to writing down all of your work. It makes it easier than keeping things in your head, and it makes it harder for you to make a mistake. So it's a win-win. Now with these two pieces of advice in mind, I hope you find the video lectures super helpful, and I hope you have a wonderful time working your way through this exam. Good luck and have fun. Problem number one. Which of the pieces A through E will fit between the two pieces shown above so that the two equalities are true? So first we have this 8 minus 3 expression on the left for these two pieces, and we know that this is supposed to equal 5. So for all of these middle pieces, we can go ahead and eliminate any of the pieces which do not have an equals 5 on the left side. So this equals 3 and this equals 4 can both be eliminated. And now we know that we need the other expression to equal 2 based on the piece on the right side that we have. So for answer choice A, we have 5 minus 1, which equals 4, which does not equal 2. We have for answer choice C, 1 plus 2, which equals 3, which is not 2. Oh, and then for answer choice E, we have 1 plus 1, which definitely equals 2. So that's going to be our final answer. Problem number two. John looks out the window. He sees half of the kangaroos in the park. See the picture. How many kangaroos are there in the park? So what this problem is telling us is that the 1, 2, 3, 4, 5, 6 kangaroos we see make up half of the kangaroos in the park. If we want the total number of kangaroos, then logically all we have to do is double the number of kangaroos. So in other words, we do 6 times 2 or 6 plus 6, whichever you prefer, and that's going to be equal to 12. So there are 12 kangaroos total in the park. So our final answer is A. Problem number three. Two transparent grids have some dark squares as shown. They both slide into place on top of the board shown in the middle. Now the pictures behind the dark squares cannot be seen. Only one of the pictures can still be seen. Which one is it? So what happens when we actually connect these two grids together? We end up getting rid of these five squares. These are all out of view. The star, the dog, this circle, this flower, and this car all become out of view because of this. And then also this square, this square, and this square get eliminated. So this ball, this bird, and this cup all get eliminated. And when we overlap these two figures, that's all that gets erased. The only square that is white, or in other words transparent on both of the grids, is the top middle one. So this is the only image that gets preserved. It's the butterfly. And so our final answer is E. Problem number four. A picture of footprints was turned upside down. Which set of footprints is missing? So when we look at this picture, we see for instance these two sets of feet get rotated around over here. And so that's why they're over here in the top right corner of this image. Now looking at all of these, we see that there's something missing in the top row here, which means it was missing from the bottom row here. We see that there's one of this bear-looking footprint, one of these kind of funny-looking footprints, something that's missing, and these kind of human-looking footprints. And here when we look at the row, there's a bear-looking footprint, this kind of weird-looking footprint, this kind of weird-looking footprint again, and a human-looking footprint. So it looks like the prints right before the human ones are the ones that are missing, and that would be these ones right here. So that's where we have the match. And so our final answer is going to be C. Problem number five. What number is hidden behind the panda? So I'm not sure how different this question looks when it's printed out on paper, but on the screen it's a little bit difficult to see that there are actually numbers here, here, and there's some operations here. Plus is minus, equals. If we were to actually write it all out, here we have 10 plus 6 is equal to 16, so that's this line of work, plus 8 is going to give us 24, and then minus 6 gives us 18, and then plus 8 gives us 26, and then minus 10 gives us 16. So once we work our way through all of the operations, the last number here should be 16, which means that's the number that the panda is covering. So our final answer is A. Problem number six. The table shows the correct sums. What number is in the box with the question mark? So first things first, we need to figure out what number belongs here, so that we can add it up with 13 to figure out what this number is. We know that 2 plus something equals 11, and that something is going to be 9, since 11 minus 2 is equal to 9. What that means is now all we have to do is take the sum of 9 and 7, and that's going to be equal to 16. So our final answer is E. Problem number seven. Dolly accidentally broke the mirror into pieces. How many pieces have exactly four sides? So this is just a matter of checking which pieces have three, which pieces have four, and which pieces have five sides. I've marked a circle in each of the pieces that actually has four sides. The others are all either triangles or pentagons. So these four pieces each have four sides. So our final answer is going to be C. Problem number eight. Here is a necklace with six beads. Which of the pictures below shows the same necklace? So basically the question is, if we were to unfold the necklace, which of these would it look like? Let's start at one of the beads and follow the path to see what we get. So we have black, then white, then white, then black, then black, then white. So again, it's black, white, white, black, black, white. And if we take a look at all of these, we see that A is the one that is consistent with this. Black, white, white, black, black, and white. So that's going to be our final answer. Problem number nine. The picture on the left shows the front of Anne's house. The back of her house has three windows and no door. Which view does Anne see when she looks at the back of her house? So again, all we're looking for is a house with no door and three windows. And we also want it to have the reverse image of this chimney. This chimney is on the right side. If we're looking from the back, it should be on the left side. So instantly we can eliminate the answer choices with doors or with too many windows. So there are two houses with three windows on them. But again, we care about where this chimney is. Since we're looking at it from the back, it should be the reverse of wherever this chimney is. So here it's on the right side, we want it to be on the left side, and that means it's consistent with answer choice E. So that is our final answer. Problem number ten. Circle plus circle plus circle plus circle plus square equals square plus square plus square. Which of the following is true? So before we even look at any of the answer choices, we want to try to find a relationship between circles and squares. How many circles make one square, or how many squares make one circle? Well, we can sort of eliminate a square from both sides since there's a square on each side. So if we just ignore this square and this square, we have four circles equals two squares. And if four circles equals two squares, then that should mean that two circles is equal to one square. And in that case, and that's just because every circle is worth the same amount and every square is worth the same amount as any other square. So, you know, if we have, if we break the number of squares in half, two divided by two is one, here we have four divided by two is two. So two circles correspond to one square. And when we look at the answer choices, we see that E is consistent with this. Circle plus circle equals square. So that's going to be our final answer. Problem number eleven. Balloons are sold in packets of five, ten, and twenty-five. Marius buys exactly 70 balloons. What is the smallest number of packets he can buy? So we want to buy as many of the packets filled with lots of balloons as possible. This is the way to get as few packets as possible, because if we just buy packets of five over and over again, we're gonna have to get a bunch of those. But for instance, we can buy two packets of 25. If we were to buy three packets of 25, that would be three times 25, which is 75, which is too many. So we can fit two times 25 in there. So two packets of 25 will give us 50, and then that'll leave 20 balloons left. And 20 balloons can easily be answered with two packs of 10. So 10 times 2 is equal to 20. So we're gonna have two packs of 25 and two packs of 10 for a total of four packets. And that means that our final answer is going to be B. Problem number 12. Bob folded a piece of paper. He cut exactly one hole in the paper. Then he unfolded the piece of paper and saw the result as shown in the picture. How did Bob fold his piece of paper? So first things first, when reading this problem, please try not to mess it up just as I did. But more importantly, let's focus on the actual solution. If we were to go ahead and fold up this sheet of paper and then punch a single hole in it and then unfold it, then the way to think about it is it would be as if the dot were reflected and mirrored over each of the folding lines. So for instance, if we add a dot here, it got mirrored by unfolding and then both of these dots got mirrored by unfolding this way. Now which of these sets of lines and folds actually seems to be consistent with that? Well, the one that seems to lend itself the most is C. Because it's like you have four quadrants here, four spots, and here it looks like you have these four points here. So if we were to have a hole cut out here and then unfold it this way, you would have two holes here and then unfold it again, and you can see that there would be one hole in each in each of the quadrants. And it should look something very similar to this. The other shapes don't really give any such visual impression. So our final answer is going to be C. Problem number 13. There's a tournament at the pool. First 13 children signed up and then another 19 children signed up. Six teams with an equal number of members each are needed for the tournament. At least how many more children need to sign up so that the six teams can be formed? So the question basically amounts to how many children do we need until we get a number that is divisible by six? And we know that first there were 13 children and then another 19 children, and if we add that up we know that there are 32 children. Well, the next number up from 32 that's divisible by six is 36. And we know this from our multiplication tables. Five times six is 30. Six times six is 36. So 36 is going to be the next number divisible by six. And to get to 32 from 36 we need to subtract four, which means that we need four more students to join. So that means our final answer is going to be D. Problem number 14. Numbers are placed in the cells of the four by four square shown in the picture. Mary finds the two by two square where the sum of the numbers in the four cells is the largest. What is that sum? So looking at the square we see that there are a bunch of ones, twos, and threes, and there's also a four and a seven. So the natural guess would be to see where these two clearly largest numbers would share a two by two square. And in particular see what other squares have the seven, since that should clearly be the area where we will have a square with a large sum, unless it's surrounded with a bunch of ones, which sort of offset how big the sum will be. But if we have numbers like threes and fours and stuff then then we should expect the sum with the seven to be the largest. Just because seven is so much larger than all of the other numbers. So we have this two by two square here, 4, 1, 1, 7, and we see that that's actually bigger than 1, 1, 7, 3, because the only difference is that there's a three instead of a four. And similarly 1, 1, 2, 7, there's a two instead of a four. So this square is bigger than those two other squares. But then we also have the square 7, 1, 3, 3, and what that's going to be equal to is 7 plus 7, because 3 plus 3 plus 1 is 7. And here we have 1 plus 1 plus 4, which is 6. So 1, 3, and 3 actually add up to a bigger sum than 1, 1, and 4. So we see that this is going to be the largest of the two by two squares. Any of the other ones would be much smaller because they don't contain the seven. So if we add up the four numbers in the square, we get 7 plus 1 plus 3 plus 3 is 7 plus 7, which equals 14. So our final answer is D. Problem number 15. David wants to prepare a meal with five dishes using a stove with only two burners. The times needed to cook the five dishes are 40 minutes, 15 minutes, 35 minutes, 10 minutes, and 45 minutes. What is the shortest time in which he can do it? He may only remove a dish from the stove when it is done cooking. So the key here is to find a way to distribute the time, the total time that we're going to cook, as evenly as possible between the two burners. For example, suppose we took up all of the time needed to cook all of the dishes and added it together, and we got 100. Like it takes 100 minutes to cook everything. That's not the case here, but it's easier to think about it this way. If you have 100 minutes total to cook all the dishes, then if you were to spend 80 minutes cooking on one of the burners and 20 minutes cooking on the other burner, well then collectively, you're going to be spending 80 minutes using that stove, because you're going to be burning both of them at the same time. The 20 minutes are going to finish, and then the 80 is going to finish, so you'll have spent 80 minutes total. Now suppose you break it up into 70 and 30 minutes. In that case, the total time you're going to take is going to be 70 minutes, and so on and so forth. Basically, the closer you get to the halfway point, so for 100, that would be 50, the lower the time is going to be. And that's what we would call the best optimal time, making the best use of your resources. Actually, if you're kind of curious, this is not just in the case of this problem, but if you want to find a rectangle with the biggest area possible based on how much how much of a perimeter you have to work with. Basically the idea is it works best when all of the sides are of equal length and so that would make a square as opposed to having one side of the rectangle being longer than the other. So that's that's a different problem altogether but that's a little interesting aside. But basically our strategy here is to find out what the total time is go ahead and cut it in half and try to get as close to that halfway mark as possible. So if we add up all the times together we get 145 minutes and divided by 2 that's 72 and a half minutes. Of course there's no way to break it down into exactly 72 and a half minutes per burner but there is a really close way we could actually break it up into 70 and 75 like so. The circles with red add up to to 35 minutes the 40 and 35 minute dish that's going to be 75 minutes and then the blue circles added up give 70 minutes of cooking. So 70 and 75 on the burners will mean that we're going to be burning for a total of 75 minutes. The 70 minute dish is going to finish and then we're going to spend five more minutes finishing up the 75 minute dish. So this is as close to the halfway point as we can get so this is going to be our optimal answer and that means our final answer is C. Problem number 16 which number should be written in the circle with the question mark? So at first this problem might seem a little bit difficult because they don't give us any numbers at all to start with but if you look at these operations one that should really stick out to you is this times zero. Remember if you multiply anything by zero you're always going to get zero as the result. So whatever this question mark is if we multiply it by zero we're going to get zero here. Now if we add six that gives us six. If we multiply by four that's six times four equals 24. If we subtract 15 that gives us nine and if we add four that gives us 13. So if we go ahead and fill out that circle as follows we end up going from zero to 13 in the question mark point. So our final answer is D. Problem number 17. The picture shows a group of building blocks and a plan of the same group. Some ink spilled on the plan. What is the sum of the numbers under the ink spills? So comparing the plan to the actual figure that we have here we see that there's one block here corresponding to the number one here. There's three blocks in this area corresponding to this three here. There's a one and then another one and one and another one. So we see that whatever number we have here corresponds to how tall that particular set of blocks is. So if we go ahead and look over for this ink blot that's going to be right here we see that there's one block being covered right and then there's the second block here. It's one block shorter than this set of three right here this block. So this is two. This ink blot here is covering a two and then we have this four right here one two three four and we can see that the height of these blocks is one less than the four and four minus one is three. So this is three blocks tall this thing in the back here. So this ink blot is covering a two this one is covering a three and if we add those two up we get five. Two plus three is equal to five. So our final answer is C. Problem number 18. How long is the train? So from these two pictures we have a little bit of information that we can gather. First of all the length of the bridge is a train plus 110 and then that means that 110 plus a train plus another train is equal to 340 meters. That's the way to think about this. So we have train plus train equals 340 plus 110 is equal to 340. So we want to figure out how long the train is. That means we want to basically get everything else onto the right side of the equation and only have a train on the left side. So that way we have train equals some number. So if we subtract 110 from both sides we get train plus train or 2 times train is equal to 230 and if we want to find out what train is equal to we can just divide both sides by 2 because if we divide both sides by 2 this equality remains true and 230 divided by 2 is equal to 115. So the train is 115 meters long. So our final answer is B. Problem number 19. The ancient Romans used Roman numerals. We still use them today. John was born in February of the year MMVII. How old is John today? Today's date is on this booklet. So for this problem the date is sometime in March of 2017 I believe. In other words this person this John already had his birthday this year. It's not happening sometime later so we're counting this year as one of the years where his age goes up by one. Now if he was born in the February of year MMVII and we want to convert that, well M is equal to a thousand so it's 2000 and then V is 5 and then II is 2. So it's gonna be 2007 because 2005 and 2 so 2007. So that means that you know since his birthday already got counted for 2017 that his age should be 2017 minus 2007 which equals 10 which then can be converted back to X. So in this case our final answer is going to be E. Problem number 20. A small zoo has a giraffe, an elephant, a lion, and a turtle. Susan wants to plan a tour where she sees two different animals. She does not want to start with the lion. How many different tours can she plan? Okay so there are how many different kinds of animals? There's a giraffe, an elephant, a lion, and a turtle. So there's four different animals and there's going to be two different animals visited on the tour. So for our first animal that we see it's not going to be a lion meaning we have three choices either a giraffe, an elephant, or a turtle. Now let's suppose that she starts with with the giraffe. After that she won't be able to go see the giraffe again so she's gonna need to pick between the elephant, the lion, and the turtle. There's going to be three more choices. In other words if the first animal she picks to see is the giraffe there are three different tours that start with a giraffe. Similarly if she picks the elephant first there will be three different tours she can do that way. In fact for any animal she chooses there's always going to be three animals left. So in other words the total amount of tours that can be selected are three choices for the first animal times three choices for the second or nine. And again that's just because for whatever animal you pick first you're going to get three choices for your second animal. So the total is nine and our final answer is going to be D. Problem number 21. Four brothers ate 11 cookies in total. Each of them ate at least one cookie and no two of them ate the same number of cookies. Three of them ate nine cookies in total and one of them ate exactly three cookies. How many cookies did the boy who ate the largest number of cookies eat? So the first thing that we can try figuring out is okay three of them ate nine cookies and one of them ate exactly three cookies. Is this person who ate exactly three cookies a part of the three who ate nine cookies or not? Well think about it this way. If this person who ate exactly three cookies was not part of the group of three that ate nine cookies then we already know that the group of three ate nine and this fourth person ate three and nine plus three is twelve. But the four brothers ate 11 cookies in total and 11 clearly isn't equal to 12. So what this means is this person who ate exactly three cookies he can't be the separate fourth person. He has to be a part of the nine people who ate cookies. So we need to find different ways to add up to nine where three is one of the three numbers we're adding and there's really only two possibilities here. You can add one three and five or two three and four because all of the numbers have to be different. Now which of these cases is actually true? If we go ahead and look at the two plus three plus four case well again there's going to be nine cookies total and there's a fourth person who ate who ate enough to make it 11. Well nine plus one equals 11 it's going to be nine plus two. So this fourth person has to have eaten two cookies but remember every single person eats a different number of cookies. So if we use this situation here with the two plus three plus four then the problem is there's a person who ate two in the people who ate nine cookies and there's a person who eats two as the fourth person. So we can't have two people eating the same number of cookies so we eliminate this and this is maybe one of the trickier parts of the logic to keep track of so I would recommend re-watching this part of the video until you understand why there can't be a person who ate two here and also a person who ate two as the fourth person. But in any case what we've determined is that there's three people who ate one three and five cookies and then a fourth person who ate two cookies so the biggest number out of one two three and five is five so our final answer is C. Problem number 22. Zosia hid a smiley in some of the cells of the table. In some of the other cells she wrote the number of smileys in the neighboring cells as shown in the picture. Two cells are neighboring if they share a common side or a common corner. How many smileys did she hide? So I really love this problem because it's basically Minesweeper and if you've ever played Minesweeper it's it's a game that comes on any Windows computer I think they still get them you can get it as an app for free on your phone as well it's a fun game where basically you want to click all of the tiles which don't have mines on them and this is basically Minesweeper where you can figure out which of the tiles have smileys and which of the smiles don't so let's go ahead and get started. We want to find a point where maybe like here's basically the point that we focus on first and the reason why is because there's only three neighboring cells that are that are unknown like this cell is clearly just a number but these three cells are unknown and this cell says three so we know all three of these cells have a smiley in them. Now this is this has to be our starting point because otherwise we're doing guesswork this is the only tile we can start with where we know everything that's surrounding it so once we fill that in since this tile has two surrounding it and we already see that these two cover this one we can eliminate these two squares right so there's a smiley here a smiley here and a smiley here and because there are two smileys already around this guy then these two can't have smileys. Similarly we can go ahead and eliminate all of the other squares around this two because there's already two smileys here neighboring this guy so all of the other ones get eliminated. Now this three right here we have one smiley two smileys and one more unknown so this should also be a smiley and this one right here we eliminated everything around it already so that gives us our final smiley and if we go ahead and add up the smileys we see there's one two three four five of them so our final answer is B. Problem number 23 each of ten bags contains a different number of pieces of candy the number of pieces of candy in each bag ranges from 1 to 10 so we know right away that there's going to be 10 bags each with the different number from 1 to 10 so every number is going to be covered before we continue reading this problem. Now each of five boys took two bags of candy Alex got five pieces of candy Bob got seven Charles got nine and Dennis got 15 pieces. How many pieces of candy did Eric get? So there are four boys who each took two two bags of candy that means Eric got his own two bags of candy. Really all this amounts to is finding out how many pieces were taken total so far and the rest of them all go to Eric so all we have to do is add all the numbers from 1 to 10. Now there's a cute little shortcut here I forget who which which person came up with this but he was like in middle school or something and he came up with a super fast way to add a whole group of consecutive numbers so for instance 1 through 10 basically what you do is you you take the maximum number if you're starting at 1 and this works for even numbers in particular for odd numbers it's a little bit trickier but for even numbers you divide this by 2 to get to the midpoint which is 5 and now you add the outermost two numbers 1 and 10 and that gives you 11 and similarly the two that are second outermost like like going 1 in from the left it's 2 going 1 in from the right is 9 2 plus 9 is also 11 3 plus 8 is also 11 and 4 plus 7 is also 11 and 5 plus 6 is also 11 so all of these all of these couples are adding up to 11 and because there's 10 numbers total and we're breaking them up into couples well 10 divided by 2 means that there are 5 couples which add up to 11 and that's why this adds up to 55. Now this might sound a little bit complicated at first and of course if you'd rather do it you can just add up all of these numbers by hand but this 1 plus 10 2 plus 9 3 plus 8 shortcut can be really really useful if for example you wanted to add all the numbers from 1 to 100 in that case the shortcut will help you solve it way way faster than adding up all of the numbers by hand so try to keep that shortcut in mind if you want to learn more go ahead look it up but that's something that can be pretty helpful as a time saver for these kinds of problems but anyway the total number of pieces of candy is 55 and we want to subtract the 5 the 7 the 9 and the 15 from the other 4 boys and these numbers all add up to 36 so we get 55 minus 36 equals 19 remaining pieces of candy so the final answer is E. Problem number 24. Kate has four flowers one with six petals one with seven one with eight and one with eleven petals. Kate tears off one petal from three flowers. She does this several times choosing any three flowers each time. She stops when she can no longer tear one petal from three flowers. What is the smallest number of petals which can remain? So for this problem there are really multiple different paths you can take for taking off all the petals from the flowers. I would say if there's any advice that I would give it would be to try first taking away as many petals as possible from the from the flowers with more petals on them until eventually they all get close to having a pretty close to equal number of petals. That's sort of the general strategy I would give. So listed here is the is the numbers on each petal or on each flower the numbers of petals on each flower on each turn that Kate is doing this three petal removal thing. So for instance she starts with one with six one with seven one with eight and one with eleven and she'll take off one from the six one from the seven and one from the eleven and that will give five six eight ten. And we basically want to get these two smaller ones close to each other and these two bigger ones close to each other. So four five eight and nine reducing this one this one and this one to four five eight and nine. Then we see that we're actually able to bring it down to three four eight and eight. We could have also just as easily done for instance three four seven and nine but we would like these all to be equal to each other ideally just so that they all get close to each other. So that way we don't have like two flowers which have like almost all of their petals and then two flowers which have like zero petals because then clearly there would be too many petals there. Anyway if you continue along this path the idea is if you try removing the petals in a fairly even fashion not giving too much attention to any of the flowers giving more attention to the flowers with more petals you can eventually work your way down to a situation where there are two flowers with zero petals and two flowers with one petal each and that total is going to be two petals so that means that the final answer is going to be B.

Math Kangaroo 2017

exam solutions

problem-solving

note-taking

logical elimination

arithmetic operations

mathematical proficiency