Problem number nine. Roberto makes designs using tiles like this. How many of the five designs is it possible to make? A, 1. B, 2. C, 3. D, 4. E, 5. Each of the tiles Roberto uses has one star in one circle, so any design that we make needs to have the same number of stars and circles in it. First, we're going to figure out if, based on that, there are any designs that we definitely cannot make, and then we will go through the remaining designs and make sure that we can indeed make them. On the first design, there are 1, 2, 3, 4, 5 circles and 1, 2, 3, 4, 5 stars, so that's okay. We can make that one. On the second, there are 1, 2, 3, 4, 5, 6 circles, and now let's count the stars. 1, 2, 3, 4, 5, 6, 7, 8, so 8 stars, so we definitely cannot make this one. Next design, as you can easily see, has 1, 2, 3 circles and also 1, 2, 3 stars, so that's okay. The fourth design has 1, 2, 3, 4, 5 circles and 1, 2, 3, 4, 5 stars. That one's okay. And the last one has 1, 2, 3, 4 circles and 1, 2, 3, 4 stars. So, so far, we know that we cannot make all five designs. Let me clean up our board a little, and we can see which of the four remaining designs we can actually make. We said we cannot make this one. Looking at the first design, I'm going to outline each tile, and remember, we can rotate the tiles. One tile like this, next one here, one here vertically, this one here vertically, and this one horizontally. It's flipped over, but we can still use it. So the first design we can make. The next one that's available to us in the middle, we can have three vertical tiles, just like this. So that's not a problem. The next design, this needs to be a vertical tile, this one needs to be vertical, two more vertically, and the last one horizontally. That works. And then the last one there is kind of all laid around the center. So that one works. Now we see that the answer is 4D. There are four designs that it's possible to make with these tiles. Problem number 10. Albert fills the grid with these five figures. Rhino, shark, frog, ghost, and flower. Each figure appears exactly once in every column and every row. Which figure must Albert put in the cell of the question mark? A, ghost, B, shark, C, frog, D, rhino, or E, flower. We will need to figure out which figures are missing from the row and the column in which the question mark is, and then make further deductions to figure out which one of these goes in the spot of the question mark. The question mark is in the fourth column, it is in the fourth row. In the fourth column we have the flower, the frog, and the shark, which means that we're missing the rhino and the ghost. In the fourth row we also have the frog, the shark, and the flower, which means we're also missing the rhino and the ghost. This means that the figure in the cell of the question mark will be either the rhino or the ghost, and will that be the flower, the frog, or the shark. If we were to put the rhino in place of the question mark, we would need to place the ghost in the other empty space in the fourth column, but we already have a ghost in that row, so this will not work. Now we know that we cannot place the rhino in the spot of the question mark. Let's try placing the ghost where the rhino is. We can put the rhino where we thought of putting the ghost, so that works. We can also place a rhino right next to the ghost, and then we can place a shark in the last empty space. Solving the problem this way, we see that the answer is A. Albert will need to put the ghost where the question mark was. Problem number 11. Tom cuts two kinds of shapes out of grid paper as shown to the left. What is the smallest number of shapes that Tom needs in order to exactly cover the bullet in the picture? A, 5, B, 6, C, 7, D, 8, E, 9. Hint, we can rotate the pieces, but we cannot cut them. Think of the boat as made up of two parts. There's a top and a bottom. Let me divide them here. Because the top part has right angles, we can fill it in with squares. Here is one square, another square, and a third. The bottom part has angles like the trapezoid does, which are made when we cut across a small square diagonally. But we will need to rotate the trapezoids to fill the sides. Take one trapezoid, rotate it, and fill it on this side. And a second trapezoid, rotate it, and fill on the other side. Now we have a white part left that's exactly the shape of a trapezoid that's not rotated, so we'll put that one in. And we see that we used three squares and three trapezoids to fill in the shape of the boat. So our answer is B, 6. Problem number 12. The colors in this picture are switched. Then the picture is rotated. What does the picture look like now? The colors being switched simply means that the black background becomes yellow and the yellow dots become black. This is true in all the answers listed. To solve this problem, we will need to look at the shapes of the dots and count them, and then compare their placement. The original image has one dot with a hole in it, right here, one really small dot here, and then one, two, three, four, five medium-sized dots. Based on this, we can see right away that B will not be the answer because there is no dot with a hole in it. Also, without even counting too carefully, we see that D has too few of the medium dots and doesn't have a small one, so that is definitely not the answer. Let's count the dots now. In answer A, there's the dot with the circle in it, there's one small one, and one, two, three, four, five, six medium-sized ones. So A is also not the answer. In C, there is a dot with a circle in it, a small dot, and one, two, three, four, five medium dots. This is also true of answer E. Now we need to compare answer C and answer E to the original image. Notice that in the original image, there's a space between the dot with a hole in it and the closest medium dot. This is also true in E. It would be right here. In D, there's no such space here. There's a space on the other side, but there's no space on one of the sides. Therefore, C is not the answer. Our answer is E. Problem number 13. Peter Rabbit has 20 carrots. She eats two carrots every day. She ate the 12th carrot on Wednesday. On which day did she start eating the carrots? A. Monday, B. Tuesday, C. Wednesday, D. Thursday, E. Friday. To solve this problem, we will lay out a chart. In one column, we can put which number of carrots she's eating, and in the other column, we can put down the day of the week. For our purposes, it doesn't matter if Peter has 20 carrots. What is important is that she eats two carrots every day and that she ate the 12th carrot on Wednesday. So in our chart, we can have the first day when she eats the first and the second carrot. Then the next day, she eats the third and the fourth, and the fifth and the sixth, seventh and eighth, ninth and the tenth, and on the day that we get to, she eats the eleventh and twelfth. We know it's Wednesday. We can just now work backwards. If she ate the twelfth carrot on Wednesday, she ate the ninth and tenth carrots on Tuesday. The seventh and eighth would have been Monday. The fifth and sixth carrots were eaten on Sunday. The third and fourth carrots were eaten on Saturday, which means that the first and second carrots were eaten on Friday. That will be our answer. Friday is when she started eating the carrots. Problem number 14. Toby glues 10 cubes together to make the structure shown to the right. He paints the whole structure, even the bottom. How many cubes are painted on exactly four of How many cubes are painted on exactly four of their faces? All the exposed faces of the little cubes will be painted. So the only faces that will not be painted are those where a small cube touches another small cube. Each of the small cubes has six faces. Any place it touches another small cube, one face is covered. So to have four faces not covered, it needs to be touching two other cubes. Looking at the figure, we will start at the top left. We see here that the first cube on the top only has one face touching another cube. So there are five exposed faces on it. However, the next one has two cubes touching it, one on top and one on the bottom. So it has four faces exposed. The same is true for the very next one. The cube in the bottom left has one cube above it and one cube to the right. And those are all the cubes that are touching it. So it still has four faces exposed. The next cube is touching a cube on the left and a cube on the right. It leaves four faces exposed. Same is true of the next one. The following cube has a cube to the left and one behind it. So there are also four faces exposed. Going to the back, we see that the next cube has a cube in front of it and behind it, but the top, bottom, and sides are exposed. So there also are four faces exposed. The cube here in the very back also has two cubes that it's touching, one in front and one above. The other four faces, the one in the back, below, and on the sides are exposed. So there are four faces exposed. And coming to the next cube, it only touches the one cube, the one underneath. So there are five faces exposed. So we see that there are two cubes where there's a different number of faces exposed than four. That's the very last cube and the first cube. That leaves us with 10 minus two, which is eight cubes that have four faces exposed. Therefore, there are eight cubes that will have exactly four of their faces painted. The answer is C, eight. Problem number 15. There are eight flowers on a rosebush. Some butterflies and some dragonflies are sitting on the flowers. There is no more than one insect on each flower. More than half of the flowers have an insect on them. The number of butterflies on the flowers is twice the number of dragonflies on the flowers. How many butterflies are sitting on the flowers? A, two, B, three, C, four, D, five, or E, six. To solve this problem, we will figure out the least possible number of insects and the greatest possible number of insects. And then we will guess and check to find the number of butterflies. There are eight flowers. We know that most are eight insects. This most number is found by noting that more than half of the flowers have an insect on them. Half of eight would be four, but more than half the flowers have an insect on them. So there are at least five insects. So there are at least five and at most eight insects. We also know that the number of butterflies is twice the number of dragonflies. So we can think of a number of dragonflies, figure out how many butterflies that would make and how many insects total that would make. This will help us make some educated guesses. So for example, if there was one dragonfly, there would be two butterflies, that's twice as many, for a total of three insects. That obviously is not right because three is less than five. If there were two dragonflies, there would be two times two is four, four butterflies, for a total of six insects, which could work. So I'll just mark that. If there were three dragonflies, there would be three times two is six butterflies, for a total of nine insects, which is too much. Because we're counting insects, they need to be counted by whole numbers. So we don't have a choice for a number of dragonflies that is more than two and less than three or less than two but more than three. So we don't have a choice for less than two but more than one. Therefore, we already found the only option that works, which is that there are two dragonflies and four butterflies. So the answer is C, there are four butterflies sitting on the flowers. Problem number 16. Captain Cook wants to sail from the island called Easter through every island on the map and back to Easter. The total journey is 100 kilometers long. The direct distance between desert and lake is the same as the distance between Easter and Flower via Volcano. How far is it directly from Easter to lake? A, 17 kilometers. B, 15 kilometers. C, 20 kilometers. A, 17 kilometers. B, 23 kilometers. C, 26 kilometers. D, 33 kilometers. E, 35 kilometers. We are looking for the distance between Easter and lake. The only other distance we do not know is from Volcano to Flower. However, we do know that going from Easter to Flower via Volcano, that is this whole distance, is the same as going from desert to lake, which is 26 kilometers. So we can ignore the 17 and mark the total distance here as 26 kilometers. Now we know the total distance, which is 100 kilometers, and we know all except the distance that we are looking for, which is from Easter to lake. To solve, we are going to add the distances that we know and then subtract them from the total of 100. Adding 26, 15, and again 26 together, that's 26 from Easter to Flower, 15 from Flower to desert, and 26 from desert to lake, we will get 6 plus 6 is 12, plus 5 more gives us 17. We do 1. 1 plus 2 is 3, plus 1 more is 4, plus 2 more is 6. We get 67. Now we can just subtract the 67 from 100. You do some borrowing. The 10 becomes a 9. 0 becomes a 10. 10 minus 7 is 3. 9 minus 6 is 3. We come up with 33 kilometers being the distance from Easter to lake we were looking for. So the answer is D, 33 kilometers.

mathematical problems

problem-solving

logic puzzle

3D cube analysis

map distance