 Welcome back everyone, just in time for a review to exam 2 for Math 1030 Contemporary Mathematics for Students at Southern Intel University. As usual, I'll be your professor today, Dr. Andrew Misseldine. So exam 2 is now upon us, which is probably why you're watching this video. And so let's talk about what we're going to see on exam 2. Now first of all, the time, place, and manner of exams in this course will change often from semester to semester. So those specific details you're not going to find in this review video, I would ask you to consult the exam 2 syllabus for all of that information. This review video for the exam is just going to focus on the structure and topics that will be covered on this exam. So for this exam, there's going to be 15 questions total. You're going to have questions in the multiple choice section. There's 10 questions there with five points each. Just like exam 1, the multiple choice questions are all or nothing. If you select the correct response, you get five points. If you don't, then you don't get those five points. Then with the five free response questions, those ones, you do have to show all of the supporting work to get the full credit. A final answer might not be enough for many of these questions. They're also worth more points. And then if you don't get it perfect, you still can get partial credit for partially correct work. Most of these questions are with 10 points. So half of the exam is multiple choice. Half of the exam is free response. We saw a similar structure for that on exam 1. Now what's different, of course, about exam 2 is the topics in play here. Exam 2 is going to cover the topics from lessons 9 through lesson 16. Now as a reminder, lesson 9, 10, 11, 12 was about the topic of scheduling. So these were topics where we had projects with various tasks, and we had to assign processors to finish those tasks in the most efficient way possible, being constrained by precedence relations. And it turns out that we use digraphs in this unit. And so it's really a continuation of the graph theory we saw on exam 1, just very specialized to this problem of scheduling. So that was lessons 9 through 12. Now lessons 13 through 16 there, 13, 14, 15, 16, it gave us the idea of fair division. Now fair division is a branch of financial mathematics for which we have certain individuals which have equal access, equal ownership of some collection of assets. And then we have to decide how to fairly distribute the assets so that every player receives at the very least their fair share or more of the assets. And I'm of course speaking of financially, right? You know, if you have two people who share an asset that's worth $1,000, then everyone deserves at least $500 worth of assets. And so we have that financial math problem to do with fair division. And so both of these topics will be covered equally on this exam. So without further ado, let's talk about the specific types of questions we're going to see on this exam. We'll begin, of course, with question number one in the multiple choice section. This question from one is just going to be a question about digraphs in general. So no projects are going to be in play here. Just some vocabulary and terminology about digraphs. We saw similar questions to this on exam one. So on this graph right here, it's a digraph. It asks you to compute the in degree of a vertex. So that's the number of edges that are a number of arcs, you could say that are entering a vertex. Conversely, you could talk about the out degree of a vertex, how many arcs are coming out of the vertex there. We could talk about adjacency. Like what does it mean for two edges or two vertices to be adjacent? You have to be a little bit more careful now because we don't usually use the word adjacent anymore because it can get a little bit confusing since direction matters. We typically use the word incident. So we could say that, you know, A is incidence, F. So be familiar with that digraph vocabulary arcs and in degree, out degree incidents. And question number one is going to ask you a question based upon something to do with those things. So that digraph terminology is just a basic digraph question. Question number two is going to be very similar in that this one. It gives you a digraph and ask which of the following are valid paths? Unlike the graphs that we'd seen previously on exam one, paths are a lot more restrictive because we have to go with the flow. If you go against an arc here, that's not a valid path. And so things like paths and circuits are much more restrictive. Question number two is going to ask you about that as well. So again, the topics for questions one and two are just going to be questions digraphs, and therefore you can learn some more about these. Go to the lesson 10 to see some of the vocabulary we because we introduced digraphs in that lesson as to help us understand the project digraph. So we'll go down to question number three now. This is going to be a question about fair division. On this question, you're going to see the value table of various players. And so you see how much they care about things. And you have to decide which of the following is a fair division. Of the assets. So you have to determine how much is a fair share. And therefore which of these give everyone a fair share. So we've learned lots of fair division methods like a loan divider, loan chooser, et cetera. You don't have to run any of those on question number three. It's just about, do you understand what a fair share is? Do you know what a fair division? So we, of course, did that in, that would be lesson 13, which we introduced ourselves to what a fair division is, what a fair share is. Question number four is actually related to that one. Question number four is going to ask you how much is a fair share? You don't even have to distribute anything. How much is a fair share going to be? Or how, how much do people value things? So as you're preparing for the test, again, question three and four, you can think of being drafted from lesson 13 about introduction to fair shares. What's a fair share? What's a fair division? Be prepared to answer some multiple choice questions based upon those topics right there, you know, based upon people's value systems. How much is a fair share and how much will they value this thing? All right, moving on to the next page here. Question number five, this is another question about fair division. But this one here is going to be about the loan divider method, the loan divider, which we introduced in lesson 14, if I remember correctly. So then we're given a value system, much like we did on question number three. But now, given this value system, run the loan divider procedure, like who was who was the who was the divider there? There's a loan divider, then there's choosers. So like on this question and ask, what would Chase bid for doing the loan divider here? What might other you could ask? What are other players going to bid for? Like who would David bid for? He doesn't care. He was the divider, obviously. But like Connie, Chase, Clarence, who would they? Well, what what shares would they bid for using the loan divider? I might also ask you, like if there was a standoff, what what what shares is there a standoff over? Maybe maybe not. Remember, standoff is when people value the same like the same. They bid for the same shares, but there's not enough. There's not enough options to distribute them. Like if if two people only bid on one share, there's a standoff. So question number five is going to ask you something about the loan divider method. Because these these these fair division methods are like games playing. I don't ask you necessarily to go through the whole game, at least on the multiple choice section, but I am like, Oh, pause the video. We're watching someone play the game. Pause the video. I could ask you, what's the next step? What are they going to do next based upon the rules of the game? So question number five is going to do that, but it's going to ask about the loan divider game. So then question number six, this is a question about scheduling and projects here. So we see the following. We see a project diagram illustrated here. All the precedents and processing times are labeled and you're asked to find the critical time, find the critical time for the whole project, whose digraph is illustrated to the right. So you can run through the backflow algorithm, but this this graph you're going to see right here is small enough that you might not need to do everything, but at the very last least go through the backflow algorithm and compute this. This is exactly what we talked about in lesson 12, finding the critical time here. What's the longest path on the graph? I do want to point out that there will be a more complicated project digraph that you see in the free response section for which you do have to go through all the steps of the backflow algorithm. Question number six is multiple choice. So you don't have to show all your work. So you might be able to skip some steps there to find the correct answer. OK, question number seven, this is going to be a question about the method of markers, method of markers. So we have five players in this situation, although that number could change and we have 20 items. Again, that number could be changed, which are lined up in order. So you see one, two, three, four, five, et cetera. I do apologize that the inverted colors make some of these a little bit harder to see on the actual test. You have a white background. This will be a lot easier to read in that situation. My apologies for that. But using the method of markers, you would then you see everyone's bid. So they've done the bids already. Now they're revealed. So you have to then answer some questions like, oh, what assets would player A get? What assets would player B get? What assets would player C get? Or they might ask you, what were the extra assets that didn't get assigned to anyone? Because that does happen. There's often extra items with regard to the method of markers. It's kind of a nice method because typically everyone gets more than a fair share as opposed to some of our divider and chooser methods, which dividers rarely get more than their fair share. This one kind of is nice because people usually get more than a fair share. But this only works if we can line up the objects, these discrete objects in this manner, regardless. Question number seven is going to be about the method of markers. If I remember correctly, that should be less than 15. I hope you don't quote me on that one. I'm pretty sure it was less than 15 that we did the method of markers because it was the last method we did before we started talking about sealed bids. If I remember correctly, if I'm not, I apologize. If it's not four, if it's not 15, then it's 14. It's one of those two. All right, let's go to the next page. We're going to have a question about scheduling here. So this is the thing is this test is about scheduling. It's also about fair division. So we're going to jump back and forth between scheduling problems and fair division problems. Question number eight here is going to give you like the skin that the priority list is already given to you. It's right here. You're going to see it illustrated on the screen. So the tasks with their processing times are listed here. So A cost 10, B cost 7, C cost 3, etc. We don't know the we don't know the precedence relations, but I want you to interpret as we're building the schedule. What's going on here? So we have it'll tell you how many processors you have. So let's see processor one. It began a task at A at time zero, processor three is doing some stuff. So give you information, processor two just finished a task. We're at time seven right now. So it's like we're in the middle of building a schedule. You've seen this in the lessons where I'm drawing the schedule on the board and we have these decisions have to be made. So what we see right now is given this priority list. Processor one looks like it's working on something. Processor two looks like it's working on something. After all, processor one was working on a processor. Three is working on I something like that. And processor processor two was working on B, but it just finished it. So then it starts working on F that again, this it gives you a story on what's happening and then it wants to know what happens next. Just like the fair division games, scheduling is kind of like a game. We have to follow the rules of the game. And so we're telling you, OK, pause the game right now. What's the next thing that's going to happen? So how do we interpret these things, right? This one would represent a task that's ineligible. This would represent a task that is ready. This is a task that's in execution and this is a task that's completed. So using information about the priority list with this notation, we then have to decide what's the next thing that's going to happen. So like this one here, when is the next time that a processor needs to be assigned a task? Because a processor one and processor three already had a task. Processor two just got a task. So when's the next time someone's going to finish a job and have to be assigned a task? That's what this one's asking about. But you might also be asked, OK, someone so just finished their job. What's the next job they're going to do? And so using this information, how do we make those decisions? So what's the next step of the game? So these priority list, excuse me, priority list scheduling. We started doing in less than 11. We particularly learned about the decreasing time algorithm in that lesson, but you don't need to know that here. Question number eight here is going to ask you to build the next step of building a schedule using priority list. A prior list will be given. And that was the main topic from less than 11. Moving on to question number nine. I mentioned the decreasing time algorithm right here. So a scheduler is building a schedule for a project with six tasks. OK, and two processors. You're supposed to compute the decreasing time priority list. So you have these tasks, A, B, C, D, E, F. They're not in order of their decreasing time. So you then put them in order based upon their their decreasing time. The decreasing time, excuse me, the decreasing time algorithm is fairly simple and so much that you just put the highest processing time first, then the next highest processing time second, et cetera, et cetera, et cetera. This is also what we talked about in lesson 11 right here. So construct the decreasing time priority list given all of the task with their processing time. So you don't need to know any precedence relations to do the decreasing time priority list that's in contrast to the critical time. You'll want the digraph to do that one. OK, question number 10. We're here to the end of the multiple choice section. There's one last one here. This is going to be another fair division question, but there's really two things you can see here. I want to you're going to see questions either involving the moving knife or the last diminisher. So right now, in either case, they're going to be cutting a cake, right? But in this question, it's framed as a last diminisher question for which, remember, this is the one where we can either when someone's cutting the cake, we can either pass or we can trim off a piece of it last diminisher. Moving knife is a little bit simpler, but also related to this. The moving knife method. Question number 10 will ask you to play the game of either last diminisher or moving knife, in which case, what's the next move? So like there's these six players in play. There's a cake that's worth 24. Some people have already done some things like C already got a cake, a cut, then B trimmed it. What's D going to do? You know, something like that. So it's going to ask you, what's the next step that's going to happen? What's player D going to do? Or in this case, on player E, what's player E going to do? Based upon what's happened already. So what's the next step of the game? We're in the middle of the game. You're going to tell me the next thing that happens here. The moving knife we introduced in lesson 13. Right after we did the the divider-chooser method. Because remember, divider-chooser works great for two players. How do you do it with three players or more? The moving knife was our first generalization of doing that. The last diminisher, I do believe, showed up in lesson 14. So go to those lessons if you want some more practice on those. And of course, you can look at the company and assignments attached to those lessons as well. That ends the multiple choice section. Moving on to question number 11. On this question, I'm going to want you to draw a graph. So in this case, it actually gives you some vertices and some arches. So you're going to draw the corresponding digraph. But I might also give you I might give you a project with task and precedence relations for which you could then draw the project digraph. So I want you to draw a digraph. It could just be an arbitrary digraph or it could be a project digraph. And so this is things we did that would have been in lesson 10 that we introduced digraph. So I want you to draw a picture. That's what question 11 is going to do. It's going to be worth eight points. Be aware that there's a lot of freedom in how you draw a digraph. But the only thing I care about is you have the right number of vertices, the right number of arches, and all of the arches have to go in the right directions. The directions of those things matter significantly for a digraph. Moving on to the next page, we then get to question 12, which is worth 10 points here. And question number 12, you are going to use the backflow algorithm to compute all of the critical times of every task in a project. And the project might be a little bit more complicated than what we saw in the multiple choice section. Remember, there was this question. What was that question again? Question number six was about computing the critical time of a project. A much smaller digraph. You didn't have to show any work. Question number 10 is going to ask you to do more than that. You have to compute the critical times of each and every one of these tasks. Now, to show the work, all you have to do is write the numbers in each of those blanks right there. Because honestly, it just comes down to certain arithmetic. At one point, you might do something like 7 plus 15 or whatever. I don't need you to show the work when it comes to 7 plus 15. Just write each of the numbers in the brackets. And if you fill in all the brackets, that would then be showing your work for this question right here. The backflow algorithm was introduced in lesson 12. It's really just Dijkstra's algorithm applied to a project digraph. But now we're looking for longest path, the critical path, as opposed to the shortest path. But because of that small modification is all that it is, it's basically Dijkstra's algorithm. Be aware of that as you're working on number 12. Number 13 is another scheduling problem. And as you can probably guess from what you see on the screen right now, I want you to build a schedule. OK, so the digraph is given to you. The project digraph is given to you right here. The priority list is given to you right here. Don't worry where the priority list came from. I don't care. We're not worried about like a decreasing time priority list or a critical time priority list. The idea is we have a priority list, construct the schedule. So be like, OK, processor one is going to do this job. And then it's going to processor two is going to do this job. And then processor one is going to do this job. I want you to fill it in. Now, if you want to bring in like colored pencils or crayons or something can make this be a whole lot easier to grade, also a lot easier to read. I'm not going to require that. If you just want to use a pen or probably not a pen, don't use a pen on math test, just use a pencil. That's not such a big deal. You just be like, draw a box and then label like, oh, this is task A and then draw another box. This is task B, draw another box. This is task C, whatever you come up with. You can just label it doesn't have to have multiple colors. Although I promise you, these are a lot easier to read when you have lots of beautiful colors here. That's question number 10. Priority scheduling, we introduce. Let's see, that would have been in less than 11, I believe. We introduced how you can build a schedule from a priority list and not all priority lists are currently equal. But nonetheless, you are going to use the priority list that's listed right here for question number 13. It's worth 10 points. Question number 14. This one is going to ask you to do a divider-chooser scenario. So you have two people playing the game. In this case, you have Brad and Angelina who want to cut a cake. Half chocolate, half strawberry. The cake is valued at $36 and it describes their value system. So the different players might value things differently. In this fictional situation, Angelina values, let's see, chocolate twice as much as strawberry. Brad, on the other hand, let's see. Brad, what does it say? He values the two flavors the same. So it's going to ask you things like, OK, how much does and then here's a cut. You can see illustrated right here. So Brad cuts the cake and so you then ask, OK, how much does Brad value the two slices? How much does Angelina value the two slices? And therefore, how is Brangelina going to divide up the cake? That's that's what you need to decide for this one. So there's a story here. You're going to explain all the things, plenty of things to see there. If you're not sure what to put, look at the solution on the practice test to see what goes there. I want to hear the story. What's going to happen? How much does Brad value this slice and this slice? How much does Angelina value this slice and this slice? What slice is Angelina going to choose? Clearly, she's going to choose the one she values the most. So how much does she value her slice? How much does Brad value his slice go from there based upon the value systems that they have here? There's going to be some algebra necessary to compute how much they value these. This question is worth 10 points for that reason. There's a lot going on. All right, so I should mention before I move on here, unlike many of the other fair division problems where you just have to do the next step of the game with number 14, with the vital chooser, you're going to go through the whole game. The players give you your value system and then you describe what happens, how are things allocated? And then question number 15. This is the last question on the exam. This is the SEAL BIDS distribution, fair distribution here. So this is the main topic from lesson 16. 16 was basically all about this. You're going to have some number of players in this case. We have Anna, Bell and Chloe for which they have some assets. They have four assets right here, a dress, a desk, a vanity, and a tapestry for which these are their SEAL BIDS for the various different the various different assets that they co-own here. And so then you need to fill out the table. Who's going to get what? What is a fair share? How much do they pay everyone? Because oftentimes you bid for it too high, then you can do all the calculations and decide what's going to go here. So you'd be like, OK, who's going to get the dresser? Or maybe it's an actual dress. It says furniture, it's probably supposed to be a dresser there. So be like, OK, Bell gets the dresser because Bell made the highest bid. Who's going to get the desk? The desk goes to Anna because she bid the most there. So you have to just tell me what assets go to whom and then who pays the estate and how much do they pay the estate? How much do they get back in the end? So what happens here? So I could be like, Anna gets a desk and she gets maybe something else. And then how much money does she either pay to the estate or get from the estate? I want you to solve the SEAL BIDS problem for question number 15. And that will then be the last question on this exam. So like I said, this exam covers everything we've learned about from fair division and scheduling, and you'll be asked to solve various parts of the scheduling or fair division games that bring us to the end of this review. Hopefully you found this helpful, and at the very least, you hopefully will find nothing surprising on the actual exam because this video plus the syllabus should explain everything what you should see on this exam. Now, of course, as you naturally will probably have questions before you take the exam, please reach out to me. If you have any questions whatsoever, I'd be glad to help you out here. Utilize the resources materials that are on canvas, work together with classmates or with tutors and working together. I think we can prepare ourselves to do very well on this exam. Best of luck, everyone. I'll see you next time.