Problem 13. A dragon has three heads. Every time a hero cuts off one head, three new heads grow. The hero cuts one head off, and then he cuts one head off again. How many heads does the dragon have now? So, to solve this problem, let's try to understand what's going on. Let's begin by drawing a dragon. As you can see, this dragon has three heads. Now, our hero comes by, and he chops off one of the heads, just like this. But, as the problem says, that means that three new heads grow back. And, the problem continues to say that our hero chops off one of these heads. So, in its place, we have three more heads that have grown back again. One, two, and three. So now, let's try to count how many heads we have. One, two, three, four, five, six, and seven. So, the correct answer is D. Problem 14. Stars, clovers, gifts, and trees repeat regularly on a game board. Some juice was spilled on the board. As a result, some of the pictures can't be seen. These are the white spaces in the picture. How many stars were on the board before the juice was spilled? Let's begin by figuring out how many stars there already are on the board. One, two, three, four, five, and six. Now, the problem says that the pictures that are on the board repeat regularly, and that they form a pattern, so that the star is followed by a clover, which is followed by a present, which is followed by a Christmas tree, which is then followed by another star. So, let's continue this pattern around the board. Here is a star, here will be a clover, here will be a present, here will be a Christmas tree, and here will be another star. Here will be a clover, here will be a present, here will be a Christmas tree, here will be another star. Here will be a clover, here will be a present, here will be a Christmas tree, and here will be another star. And then here will be a clover, a present, a Christmas tree, and then there already is a star right here. So, let's count how many stars we have. We have one, two, three, four, five, six, seven, eight, nine. We have nine stars, so the correct answer is D. Problem 15. Eve brings twelve pieces of candy, Alice brings nine pieces of candy, and Irene doesn't bring any candy. They put all the pieces of candy together on a table, and divide them equally among themselves. How many pieces of candy does each of the girls get? So in this problem, Eve brings twelve pieces of candy, Alice brings nine pieces of candy, and Irene brings no pieces of candy. Now, to figure out how much candy there is together, let's add up twelve plus nine, and then zero is zero, so that equals twenty-one pieces of candy altogether. And now to figure out how many pieces of candy each girl gets, we have to divide twenty-one by three, because there are three girls, and so you get seven. So each of the girls gets seven pieces of candy. Problem 16. Tim is looking at seven silk paintings on a wall. On the left, he sees the dragon, and on the right, the butterfly. Which animal is to the left of the tiger and the lion, and to the right of the peach? Now let's try to understand what this problem is asking us. There are seven silk paintings on a wall. On the left is the dragon, and on the right is the butterfly. Now, which animal is to the left of the tiger and lion? Here is the tiger, here is the lion, and we're looking for animals that are to the left of the tiger and lion. And which animal is to the right of the peach? The peach is here, and we're looking for an animal that is to the right of the peach. So really, we have two choices. There's a stork, and there's a rose. And a rose is a flower, so it's not an animal. So the stork is to the right of the peach, and to the left of the tiger and lion. So the correct answer is B. Problem 17. Winnie the Pooh bought four apple pies, and Eeyore bought six cheesecakes. They each paid the same amount of money, and together they paid $24. How many dollars does one cheesecake cost? We know that Winnie the Pooh bought four apple pies, and that Eeyore bought six cheesecakes. And we know that together they paid $24. We also know that both Eeyore and Pooh paid the same amount of money, which equals $24. So if they each paid the same amount, we have to divide 24 by 2 to figure out what they each paid, which is equal to $12. That means that four apple pies is equal to $12, and six cheesecakes equals $12. So to solve this equation, all we have to do is divide 12 by 6, and we find out that a cheesecake is equal to $2. So the correct answer is A. Problem 18. Sparrow Jack jumps on a fence from one post to another. Each jump takes him one second. He makes four jumps ahead, then one jump back, and again four jumps ahead, and one back, and so on. How many seconds does it take Jack to get from start to finish? Let's begin by drawing how Jack is jumping along this fence. Each jump he takes takes him one second, and he makes four jumps ahead, and then one jump back. So let's begin. Here is one jump, two jumps, three jumps, four jumps, and then he jumps back one, and then he goes one jump, two jump, three jump, four jumps, and then he jumps back one, and he goes one jump, two jump, three jump, and four jumps, and is done. Let's count how many jumps there are. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. There were 14 jumps, and since one jump is equal to one second, it took Jack 14 seconds to go from start to finish along the fence, and so the correct answer is E. Problem 19. Grandmother made 11 cookies. She decorated 5 cookies with raisins, and then 7 cookies with nuts. At least how many cookies were decorated with both raisins and nuts? So let's begin by drawing what's going on in this problem. Grandma made 11 cookies. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. She decorated 5 cookies with raisins. And then she decorated 7 cookies with nuts. 1, 2, 3, 4, 5, 6, 7. So there is one cookie that is decorated with both raisins and nuts, so the correct answer is A. Problem 20. At a school party, Dan, Jack, and Ben each received a bag with 10 pieces of candy. Each of the boys ate just one piece of candy and gave one piece of candy to the teacher. How many pieces of candy do they have left altogether? There are three boys. Dan, Jack, and Ben. And each received a bag with 10 pieces of candy. So Dan has 10 pieces, Jack has 10 pieces, and Ben has 10 pieces. Each of the boys ate one piece of candy, so they had one piece less, which equals 9. And then each of the boys gave one piece of candy to the teacher. So again, they each have one piece of candy less. So then each of them has 8 pieces of candy. Then if the question is how many pieces of candy they have altogether, they have 8 plus 8 plus 8, which is equal to 24. So the correct answer is C. Problem 21. What number is covered by the flower? Now let's begin by looking at the equation triangle plus triangle is equal to 4. Each triangle represents the same number. So you can write 2 times triangle is equal to 4. And if you divide both sides by 2, you get triangle is equal to 2. So let's look at the equation right above that one. Circle plus triangle is equal to 3. You can write it as circle plus 2 is equal to 3, since you know that triangle is equal to 2. And if you subtract 2 from both sides, you get circle is equal to 3 minus 2, which is equal to 1. So you can write circle is equal to 1. And then let's look at the third equation. Triangle plus square is equal to 5. You know triangle is equal to 2, so you can rewrite that equation as 2 plus square is equal to 5. And if you subtract 2 from both sides, you get square is equal to 5 minus 2, which is equal to 3. So you get square is equal to 3. So if you write out this last equation, you get circle, which is equal to 1, plus square, which is equal to 3, sorry about that, is equal to flower. So flower is equal to 4. So the answer is D. Problem 22. Anne has a lot of these tiles. How many of the following shapes can Anne make by gluing together two of these tiles? To begin, let's understand what this shape is. It's an L made up of three squares and then a one-squared leg, like this. And so to figure out which of the following shapes Anne can make by gluing two of these L's together, let's try to trace over and find the L's in each shape. So let's start here with this first shape. You see there's one L here, rotated, and that there is another L here, so that you can definitely make this first shape out of two L's. Now the second piece, again let's try to draw the three squares together. Now the second piece, again let's try to draw the three squares first, and then you see that here is one L, and here is a second L. So you can also make this shape. Now let's look at the next shape. Here again, let's draw the first three tiles of the long leg, and then you see the short leg appears. So you have one shape here, and then you quickly see that there's another L right below it, and so again you can definitely make this shape. And then finally, let's again try to draw the long leg of the L with the three squares. You see it right here, and then you draw the short leg, and you see there's an L right here, and then you very quickly see that there's also a rotated L right next to it. So all four shapes are possible to make by gluing two L's together, and so the correct answer is that you can make four shapes, and so the correct answer is E. Problem 23. In a box there are three boxes, and each one of these boxes contains three smaller boxes. How many boxes are there in total? So let's begin by trying to draw this problem out. We know that there is a box, and that inside this box are three more boxes, and that each of these boxes contains three smaller boxes. Three, and three, and finally three. So if we count these all together, we'll get how many boxes there are in total. So here we have one, two, three, four, five, six, seven, eight, nine, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, and thirteen. So there are a total of thirteen boxes. The correct answer is D. Problem 24. There are coins on a board. We want to have two coins in each column, and two coins in each row. How many coins need to be removed? On this board we want to have two coins in each column, and two coins in each row. And just so you know, rows are horizontal, and columns are vertical. So let's begin by checking whether each row and each column meet the conditions that we want. So let's try this row right here first. There are two coins in this row, so everything is okay. Now let's try this column. There are two coins in this column, so it's okay. Now let's move to this row. There are two coins in this column, so it's okay. Now let's move to this column. There are three coins, so we have to X one of them out, remove it, and then we have two coins. So now it's okay. Now let's move down to this row, and we see there are two coins, and so we know that that is okay now. And let's move up to this column here, and look down, and we see that there are two coins, and that's okay. And we move to this row, and we see that there are three coins in this row, so we have to remove one of them. And now we know that it's okay. And then we move up to this column, and we confirm that there are only two coins, so it's okay. So if we count the X's, the coins that we've removed, we see that we've removed two coins from this board, and so the correct answer is C.

Dragon Puzzle

Candy Division

Silk Painting

Cheesecake Cost

Box Within Boxes