 One way to approach the apportionment problem is to consider it as a process rather than an end goal. So instead of focusing on determining how much of each resource a group will get, focus instead on how we'll assign the resources in the first place. And we'll begin as follows. Everyone gets one and the next item will be given to provide the greatest reduction in ball apportionment. So for example, suppose there are three schools with 525, 174, and 250 students, we want to assign eight counselors based on population. And for contrast, let's use Jefferson's method to decide an apportionment. So we find that there's 949 students, so there should be 118.625 students per counselor. Now using that as our divisor won't allow us to allocate all of the counselors, so we'll use Jefferson's method and choose a different divisor. And after a little bit of work, we find a divisor of 105. We'll give the assignment 512. Now, let's measure the malapportionment by computing these students per counselor. The lower this number, the more time an average counselor can devote to individual students. And so our goal should be to make these numbers as low as we possibly can given our resource constraint. So let's try our problem again, and this time we're going to assign eight counselors one at a time, assigning each in a way that provides the greatest reduction in the students per counselor. So we begin by assigning one counselor to each school, and now we compute the students per counselor for each school, and how much this would be reduced by if assigned one more. So we have our schools, we have our students, we have our initial assignment of one counselor, and we compute the students per counselor. Now if we assign one more counselor to school one, it now has two counselors, and the students per counselor will now be, and so the students per counselor has decreased by, similarly, if we assign one more counselor to school two, it now has two counselors, and the number of students per counselor is now, and so the students per counselor has decreased by. And we'll do a similar computation for school three. If it has one more counselor, it now has two counselors, and our students per counselor, and so the reduction will be. Now we look at our reductions. Since the reduction is greatest, if we assign the counselor to school one, we assign the fourth counselor there. Since we haven't changed the number of counselors in school two or three, we don't need to recompute their values. We can use the ones we've already created. But since we have added a counselor to school one, we'll need to recompute the values in the first column. Let's focus on school one for a moment. It now has two counselors, and we've already computed the number of students per counselor. Since school one has two counselors, if we give it a third, its students per counselor will be, and the students per counselor will decrease by. Now we've only assigned four of the total eight counselors, so we now choose a school to assign the next counselor, and we look at our reduction. And the greatest reduction will occur if we assign the next counselor to school three. So again, we only changed the number of counselors at school three, so schools one and two, those values will stay the same, but we do need to recompute the third column. School three now has two counselors, so it has a different number of students per counselor, and if we give it a third counselor, one more, then it will have 83 and a third students per counselor, which will round to 83.3, and the students per counselor will decrease by 41.7. Now the greatest reduction will occur if we assign the next counselor to school one. Again, the values for school two and three don't change, but we'll need to recompute the values for school one now that it has three counselors. School one now has three counselors and 175 students per counselor, and if we give them one more counselor, a fourth counselor, it will now have, and that's a reduction of, and again we look for the school that will have the greatest reduction that will receive the most benefit from an additional counselor, and that's school two. Again, school one and school three haven't received a new counselor, so their situation remains unchanged, but we need to recompute the values for school two, which now has two counselors. So running the scenario where they get another counselor, we find, so there's eight counselors we need to distribute, and so far we've distributed seven, so we have one more counselor to assign, and the greatest reduction will occur if we assign the next counselor to school one. Now while we don't need it, let's go ahead and recompute the values for school one anyway. So school one has four counselors, and if we were to receive another counselor, we could find how school one would benefit, and so our final assignment, school one gets four, school two gets two, and school three also gets two, and note that this is actually different from the Jefferson apportionment, which assigned the counselors five, one, two. So this seems like it's a lot more work, but there's a few reasons this is better. The most important is that if we got another counselor, we could rapidly determine which school they get assigned to, or if we get another school, we can also determine how many counselors they should receive. So in an unprecedented display of generosity, the state agrees to fund one more counselor, which schools receive them. And remember, we left off our table as follows, and we can see that greatest reduction is going to occur if we assign that counselor to school three, or suppose we add another school with $300 in students, how many additional counselors should be added. So we'll update our table with the new school, and we might proceed as follows. We might add counselors until the next counselor added would go to one of the existing schools. So we'll start off by giving school four one counselor initially, and we'll go through our computations. If they get a second counselor, their students per counselor will be giving them a reduction of 155.5, and we see that the greatest reduction can be produced by giving school four a second counselor. So we'll recompute its values. School four now has two counselors, and if we give school four a third counselor, and again the greatest reduction can be produced by giving school for another counselor that brings it up to three, which means we'll need to update its values again. We'll update their information if they get one more counselor, and again we see where that next counselor would do the most benefit, and this means that the next school to receive a counselor will be school three. So we can stop assigning counselors when school four gets three counselors altogether.