Hello, and welcome to the Math Kangaroo Media Library. You are about to view interactive solutions to levels 1 and 2 of the 2014 competition. You will likely notice that some of the solutions given here are slightly different from the suggested solutions you may have already reviewed. So as you follow along, please compare your own solutions with the suggested solutions and the current presentation, and make sure you understand any differences. If you have any questions or comments, please feel free to contact me at the address provided below. My name is Luke, and I'm a past Math Kangaroo participant, and I hope you will find this presentation useful in your next Math Kangaroo competition. Question number 1. The ladybug, pictured over here, will sit on a flower that has 5 petals and 3 leaves, and which of the flowers below will the ladybug sit? We have to make sure that the ladybug finds the flower she likes, and that flower has to have 3 leaves. So let's go ahead and count the leaves here in each of the flowers. In flower pictured in A, we have 1, 2 leaves. In the next flower, we have 1, 2, 3 leaves. The next flower has 1, 2, 3, 4, and 5 leaves. Flower D has, again, 1 and 2 leaves, and in E we have 1, 2, and 3 leaves. And we can right away go ahead and cross out the possibilities that have less than 3 or more than 3 leaves, and that leaves us with 2 flowers. Of those, the ladybug also has to have 5 petals, otherwise she will not sit on that flower. So now let's go ahead and count the petals. So here we have, in B, 1, 2, 3, 4, and 5 petals, and in E, we have 1, 2, 3, and 4 petals. Now the ladybug would like to have 5 petals, so we can right away cross out answer E over here, and now we find out that the ladybug will sit on flower B that does have 5 petals and 3 leaves. So the answer to number 1 is B. Question number 2. If you start at the arrow, which is over here, and move along the line following all the loops and intersections, in what order do you meet the shapes? So let's do exactly that. Starting at the arrow here, we move through the first loop, through the second loop, one more loop, and then we come to an intersection, and we follow the line here through the intersection like that until we meet a triangle. So the triangle here is first, and we can go ahead and check our answers. So in A, the triangle is first, in B, the triangle is first, in C, the circle is first, so C can be crossed out, in D, the square is first, we can cross out D, and in E, the square is first again, so we can cross out E again. And then we keep on moving. Next, let's move past the triangle here, along the line, and we make one intersection and stop here at the square. The square is next, and of the two remaining answers, the square is second in A, but not in B. In B, the circle is second, so we can cross out B. And so, now we have our answer. One choice remains, that is choice A, but let's just make sure that it is the correct choice anyway. The third shape we meet, the last shape, is here, the circle, so let's make sure that's exactly what's happening. So we follow here the curve from the square, make two intersections, make a third intersection, and then we end up here at the circle. So A is indeed the correct answer, first we meet the triangle, then the square, and finally the circle. Question number three. How many more small gray squares are there than small white squares? So in the picture, which I have enlarged here, we have a square shape, which is made out of smaller gray on the outside and in the center, and then smaller white squares. So let's first count the gray squares and see how many we have. Starting from the upper left-hand corner, I will count them counterclockwise, so we have here 1, 2, 3, 4, 5, 6, 7, 8, and 9, and then 10, 11, 12, 13, 14, 15, 16, and in the center one more for a total of 17. So we have 17 gray squares. And we can count now the white squares, again from the upper left-hand corner, now going clockwise, we have 1, 2, 3, 4, 5, 6, 7, and 8, so 8 white squares. And then we can subtract the difference, so 17 minus 8, that gives us 9 squares. So the answer is 9 more gray than white. And that comes out to answer D. Question number 4. Put the animals in order from the smallest to the largest. So here we have five animals, and we see that they're arranged in a grid so that by counting the number of cells, here we can see how high each animal is. Then we have to give the number of the animal in the middle when they are arranged according to height. So let's put down the number here by each animal that corresponds to the number of cells or their height. So animal number 1 over here is like a small cat, and he fits in just one cell. So number 1 here has height 1. Number 2 fits in 1, 2, 3 cells. So that would be the height of animal 2, which is a larger cat. Next below is a monkey, and he fits in 2 cells. So that would be his height. On the right here we have a bird that looks like a stork, possibly an ostrich. And that's 1, 2, 3, 4, 5, and 6 cells. So that's the height here of this bird, 6. And finally we have a giraffe, and the giraffe takes up all of the cells here except 1. And we have exactly 1, 2, 3, 4, 5, 6, 7, 8 cells. So the height of the giraffe is 1 less, and that would be 7. So now let's arrange the numbers here according to height. The smallest height we have is 1, and that is animal number 1. So 1 is the smallest, and then here we'll have largest. And the next animal after that is the monkey. He has height 2, and he is numbered 5, so 5 goes next. After the monkey we have the larger cat right here with a height of 3, and the number of that animal is 2, so 2 goes next. He is followed by the bird with a height of 6, and the giraffe. But the bird is labeled as animal number 4, so 4 goes next. And the giraffe is labeled as number 3, and that is the largest animal. So in the middle here we see that we find number 2 in the center. That is 2 units, 2 positions from the smallest and 2 positions from the largest. And that looks like this larger cat, which possibly is a leopard or possibly a cheetah. And that is our answer here. Animal number 2 is in the middle according to size, and that corresponds to letter B. Question number 5. Anne has 12 of these tiles shown here, where in the lower left-hand corner we have a semicircle. She makes a design that is one continuous line. Anne starts at the left side of the grid as shown in the picture here, shown in my enlarged version of that picture. The question is how does the line end on the right side of the grid? So as this curve here, this line is continued through all the 12 pieces, what will the right side look like? And so with this piece being the starting piece over here on the left-hand side, on the right-hand side, just on the opposite end, we will have a piece, and our job is to match the piece with the given choices. So let's simply see what would happen if we were to fill in the next missing piece and continue the curve. We would have to copy the piece given in B and stick it into that spot. So let me do exactly that. There is a copy of that piece, and if I put it here, I can continue the picture, and I have an unbroken line over here. And I continue thus, it looks like C should be next, so let me make a copy of C and move it into the right position, like so. And now I have an unbroken here picture like that. And we see that this is symmetric, so I can simply copy the whole thing now over itself. So that whole piece can go in the next position, and I have an unbroken curve, and I can do that once again, like so. And now we're done. Now we see that the piece we were looking for, opposite of the starting position, so the starting position here is marked by the dot, in red, this ending position would be on the very right-hand side, opposite the red dot marked here in blue, and that looks like piece C. So we choose that as our answer, and that's the missing piece here on the very right of the grid that Anne has to fill in with the 12 shapes. Question number 6. Which of the pictures below is the shadow of the girl? So here we have a girl riding a tricycle, and if we were to imagine that all the colors are removed, we would just have her silhouette or a shadow, and we have to match the picture with the correct shadow. So it should be very clear that choice E is not the right answer, because the front wheel here is up. In the picture, all three wheels are on the ground, and here, in E, the wheel is up. So that's the easiest one to identify as the wrong choice, and then let's go through each of the remaining possibilities here, and see if we can spot differences. Now, between A and the picture, we see that the arm is visible, the two ponytails are visible, the front wheel is visible, but the back wheel here is missing. So it cannot be A. We then continue, and in B, it's very clear that the ponytail here, her left ponytail, is missing. So it cannot be choice B. In C, we have the wheels in the right position, we have the ponytails present, but here, the girl's leg, her left leg, is up, and that's not what is pictured, so we cannot have choice C. What remains is choice D, so let's just make sure that it is the correct picture. We have the two ponytails, we have the arm not raised, we have the leg not raised, and all three wheels are on the ground. So that is the best match, and we choose that as our answer. In picture D, the shadow matches the color picture of the girl on the tricycle. Question number 7. A square here, in my enlarged picture, was made out of 25 small squares, some of which are pictured over here, but some of these small squares are now missing, and we have gaps here in the picture. How many small squares are missing? Since we know that originally we had 25 of them, and now some are missing, we can count how many there are remaining, and subtract that number from 25. So let's just count all the squares that we have. I will do that starting from the lower left corner here, this will be square 1, and then I will count them by row. So 1, 2, 3, 4, 5 on the bottom, next 6, 7, and 8, next 9, and 10, 11, and 12 are next, and in the top row 13, 14, and 15. So we have 15 of the 25 squares remain, so by subtraction, we subtract 25 minus 15, we obtain 10, so 10 squares are missing. And that answers our question. If 10 squares are missing, we can choose 10 as the answer, and that is letter D. Question number 8. How many ducks balance the crocodile? What we have in the first picture is a crocodile on a scale, on the left hand side of the scale, and on the right hand side are two lions, and the scale is balanced, so that means it takes one crocodile to balance two lions. In the next picture we have one lion and two ducks, so one lion balances two ducks. And the final picture here, we have to balance the crocodile with ducks, so how many do we need? Let's take the two that we have here in the picture, let's make a copy of them, that will be a little easier to work with, and what we have is that these two ducks over here balance one lion. So if I were to double the number of ducks, for example put two more here, next to the two that we already have, I would have to have another lion in that picture. So if I make a copy of that lion, I can put him on the scale here, and now everything is balanced. If two ducks balance one lion, then four ducks will balance two lions. And now we see that these here quantities of lions are exactly the same. So these are equal, and therefore the number of ducks over here should match the crocodile, because on the left hand side we still have just one crocodile. And so that gives us four ducks. And in the picture below, we have four ducks exactly in the second picture B. So we can move these right away into the correct spot, and so we have here four ducks balancing one crocodile.

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