Question number 9. When the ant, pictured over here, goes from home, which is positioned here on the grid, following these arrows, three arrows to the right, three arrows up, again three arrows to the right, and then one arrow up on the board, the ant comes to the ladybug, so he ends up over here. And let's check again, if we go one, two, three arrows to the right, one, two, three arrows up, again one, two, three arrows to the right, and finally one arrow up, we do indeed find the ladybug. The question is, which animal will it come to if it goes from home, so again starting from this position, but now following a different set of arrows. So let's draw them in, the first arrow is to the right, and the number two here means that we have to move two arrows to the right. So I will draw that in red. One arrow to the right, second arrow to the right, then two arrows down, so one arrow down, second arrow down, then three arrows to the right, one, two, three, three arrows up, one, two, three, two arrows to the right, one, two, and finally two arrows up, one, and two. And we see that at the end of that path, we have found a butterfly. So that is the animal in picture A, and that is the animal that the end meets, following this set of directions and starting from home. Question number 10. We have in the picture here a kangaroo, and the kangaroo is sitting inside of some circles. The question is, in how many circles does the kangaroo sit? The kangaroo is inside how many circles? So let's, in different colors, mark the boundaries here of all the circles in which the kangaroo sits. I'll begin with blue here, and we have this large circle, and I can sketch a circle on top, like that, in blue, and so here the kangaroo is sitting in the blue circle for sure. Then, in the same color as the kangaroo, here is a circle in orange, that's the second one. In red, we'll have another circle over here that contains the kangaroo, like so, roughly. And then we see that there is one more circle here, let's just color it in black, and that circle here does not contain the kangaroo, so the kangaroo does not sit inside of this black circle, but it does sit inside of the red one, the orange one, and the blue one, and that makes a total of three circles, or answer C. Question number 11. We have a square here that was cut into four parts, as shown, so first a rectangle was created, and then three triangles. The question is, which of the following shapes cannot be made using only these four parts? So let's begin with the large triangle, which I will outline here in blue, and then match that shape in all of the pictures. So in A, that large triangle here appears on the top, it does again appear in B, on the top, in C, it makes the top piece here, and in D, I can't really find that shape, so I'll just put a question mark here, and then in E, it's again found on the top. Next, let's look at the rectangle, so I'll trace that out in red, and we see that in E, I can find a copy of that red rectangle, in D, I can find the copy, in C, I can fit in that rectangle, in B, I can do it also, and in A, we have the red rectangle as well. And what remains in the original square are these two small triangles, which I will shade in orange, so two of these, and we have exactly two of them in E. So now, we have created E, in D, I can shade two of them again, but we're not done. In C, there are two of these triangles, and so C is complete, B is complete, because I have two of these triangles again, and A, I can complete as well, if I draw the triangles here, right next to one another, here is one, and here is a second one, so now A is also complete. Let's go back to D, in D, we see that two triangles remain, I will shade them here in blue, like so, and now, looking at this picture, we would have one red rectangle, two orange triangles, so that makes three shapes, and then we would need two more shapes in blue, which could be obtained by cutting the blue triangle here in half, but that would create five parts, and we're only allowed to use four. So overall, it is not possible to make shape D out of this original square, so that is the impossible arrangement, and we choose D here as our answer. Question number 12. Which of the shapes shown below will fit this shape to the right exactly to make a rectangle? So what we have to do is imagine that the shape here in the right corner is being rotated or flipped upside down, and then all the teeth here, all the jagged pieces, line up exactly correctly with the jagged pieces in one of these choices, so that there are no gaps or no overlaps. So what I can do here to show you the answer is to cut out the shape exactly, and then move it about, and see which is the best matching choice. OK, so I have traced it out, now I will make a copy, I will rotate that about, and then move it to each of the pieces. So we see that A here, if we try to line these up, there will be a gap with A, if I move that to B, and try to line this up, there will be three gaps, if I move on to C, there is an overlap here, now the shapes are overlapping here in the middle, so that cannot be correct, it doesn't fit exactly. In D, we have a larger overlap than in C, they start overlapping right now, and now if I move it down lower, there is a gap, two gaps in fact, and so the only valid or possibility remaining is E, so let's check, and if I move these closer and closer together, we see that the gap is filled up very nicely, and so this is a good choice, if I could cut out the shape nicer, I would see that in fact, it does make a perfect rectangle with choice E, so let me just leave this over here, and mark the answer, and that would be E. Question number 13. In the diagram shown here, we have to walk from K all the way to O, along the lines of the grid, and pick up letters in order that spell out kangaroo. What is the length of the shortest walk in meters? In order that spell out kangaroo. What is the length of the shortest walk in meters if between each corner here in the grid, the distance is 1 meter, or in other words, the grid is made up of 1 by 1 by 1 by 1 meter cells. And so, the first letter that we see, or that we have to pick in order, is a K, so starting at K, we have to walk to A, and I will draw a rectangle here in red. The reason for that is, now when I walk from K to A, I have two choices, I can walk along here the top edge and then one unit down, or straight down and then three units to the right. But either way, that would be the shortest path, and I will indicate that simply by just writing a 4 on top here, so that's four moves I have to make to go from K to A. Now from A to N is another rectangle like that, now in blue, and moving from A to N, again I have a couple of choices, I can move down first and then two units to the left, or first two units to the left and then one unit down. But either way, the distance here will be plus 2. Now from N to G, I can move pretty easily, and either one to the right and one down, or one down and one to the right, so that is two moves here that we make. Next is an A, not the A I already used, but the other remaining A, so from G we move to A, and that is going to be three moves, two to the right and one up, or possibly one up and two to the right. From A, I have to move to R, so that will be indicated here like that, with a purple rectangle, and again that's three moves, one to the left and two down, or backwards, two down and one to the left, for a total of three. And then finally from R to O, and then O again, the moves are simple, there aren't many choices, we just have to move two units to the left and one unit down. So from R to O, that's two moves and then one move down. And then we sum up all of these distances. So we have 4 from K to A plus 2 from A to N plus 2. Question number 13. Walking from K to O in this grid along the lines, we pick up letters that spell out kangaroo in the correct order. The length between each vertex here is one meter because the grid is made up of one by one meter cells. The question is what is the length of the shortest walk in meters that takes us from K to O and spells out kangaroo in the correct order. So let's move from K to A, the first move we have to make, and I will indicate that by drawing in a rectangle here and I'll explain why. From K to A here, we can move in the shortest possible way by either moving three units to the right and then one unit down or one unit down and three to the right for a total of four moves here. So I have four. And I'll sum up all of these moves as we find our letters in the correct order. From A, I have to find my way to N. So let me do that here. From A to N in blue, I move two units to the left and one down or backwards, one down and two to the left for a total of three moves. From N, I have to find G, which is two units away. I have to move one to the right and one down for a total of two. Then from G to A, I already used up one A, so let me move to the next one and that will be in green. From G to A here, two units to the right and one up, again for a total of three. From A, I have to find R, so let me indicate that in purple. From A moving to R, I can move one unit to the left and two down or two units down and one to the left for a total of three moves again. And then from R to O, we don't have much choice. We just go along the straight line here from R to O and then O again is one unit away. So we have two and one. And that's all the distances, all the moves that we have to make. If we sum those up, we end up looking at the total of four plus three, that's seven, two more is nine, three more is 12, two, three more is 15, 17 with two more and plus one here gives us a total of 18. And as we have done this, we're picking the shortest distances each time, so we're sure that our answer here is indeed the shortest walk. And that comes out to 18 meters or answer C. Question number 14. How many numbers that are greater than 10 and less than or equal to 31 can be written with only the digits one, two, and three. And as we do this, we can repeat digits. So let's just write them down. The first number that appears in our list is the smallest number greater than 10 and that would be 11. And then we continue until we find 31. So 11, then 12, 13, 14, 15, 16, 17, 18, 19, and 20. And then 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and 31. So we are searching among these numbers here for the ones that can be written only using the digits one, two, and three. So we can right away cross out some possibilities. Anything with a 9 is not going to work. Anything with an 8 is not going to work. Anything with a 7 or a 6 or a 5 or a 4 is not going to work. Now 0 is also not allowed, so we cross out 20 and 30 and then we check that the remaining numbers are constructed in the way that we like. So here we have only ones in 11, ones and twos in 12 and 21, twos only in 22, in 13 we have a 1 and a 3 and so in 31, and then in 23 we have a 2 and a 3. And so digits are allowed to be repeated, so 11 and 22 are OK. And those are our choices. We have exactly 1, 2, 3, 4, 5, 6, 7 numbers like that and that corresponds to answer D. So the correct answer is 7. Question number 15. Seven sticks shown here lie on top of each other. We know that stick 2 here is on the bottom of the pile and stick 6 over here is on the top of the pile. Which stick is in the middle? So the reason that stick 6 is on the top is when we look at it, we do not see any sticks lying on top of it, in fact 6 is clearly visible here on the picture. But stick 2 lies below 1, below 3, below 5, in fact every stick here lies on top of 2, so that one is on the bottom. So what we can do is we can say which stick is next below 6 and next above 2 and then continue like so and make a list or write down the order of the sticks. So on the top over here we have number 6 and on the bottom we have number 2. And what happens next? So next, looking at the top sticks, we see that here 6 lies on top of 4 immediately and 4 lies on top of 7 and on top of 5, but 7 lies on top of 5. So 4 is closer to the top than 7 and also closer than 5. So if we were to order these, we would say that 6 is at the very top followed by 4 and followed by 7. So let's write that down, 6 followed by 4 followed by 7. And now looking at the bottom of the pile with stick 2, we see that on top of 2, the closest stick here is 1 and then after that we have stick number 5, because 5 is on top of 1 and on top of 2 and those are the only two sticks that it is on top of. So 2 is at the very bottom, on top of 2 is stick number 1 and on top of 1 is stick number 5. And so that gives us 6 here and the missing number in between is 3. So 3 lines up exactly in the middle. We can probably decide that from looking at the picture, but it's very complicated. But looking at the top 3 and at the bottom 3 sticks, we can easily decide the order there and that leaves 3 as the middle one. So the answer here would be B, stick number 3 is in the middle of the pile. Question number 16, how many frogs did the 3 pelicans catch all together? So we have 3 pelicans here, each of them makes a statement. The one on the left is named Pelly and he says I caught at least 2 frogs. So we can make a list of all the possibilities. He could have caught exactly 2 or more. He could have caught 3, he could have caught 4 or he could have caught 5 or possibly we can continue the list, but at least he caught 2. Now the pelican in the middle says I caught more frogs than Pelly did. So at least 3 because that's the smallest catch Pelly could have had. So he caught at least 3, possibly 4, possibly 5, but then we also know that he caught less than Canu and now we have to figure out how many frogs Canu caught. He is here on the right and he says I caught no more than 4 frogs. So in his list, we have 4 is the maximum number, then 3, then 2 and then possibly he caught 1 frog. And now we can go back and compare the catches. The pelican in the middle caught at least 3, so that is a possibility, but the question is could he have caught 4? And the answer is no, he couldn't have caught 4 because he must have also caught less frogs than Canu and Canu caught no more than 4. So right away, we know that we have to stop the list over here and the pelican in the middle caught exactly 3 frogs, which is 1 more than 2, so the pelican on the left, Polly must have caught 2 frogs and 3 is also 1 less than 4, so Canu must have caught 4 frogs. And then we can add up these numbers all together. We have 2 plus 3 plus 4 and that gives us 9. So 9 frogs were caught all together and that is answer D.

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