 So let's put our ideas and apportionment together. In the Dean method of apportionment, we want to minimize the absolute differences. Suppose we have two states with populations A and B, and that they currently have M and N representatives. If we want to assign one more representative, which state should it go to? And we can decide using a priority value. We'll assume that the extra representative goes to the first state and see what that would require. So suppose our states have populations A and B, and currently have M and N representatives. The values A over M and B over N are the number of persons per representative, and we want these values to be small. So let's think about this. If we give the extra representative to the first state, we'd expect a few things. So first of all, suppose A over M is less than B over N. What this means is that the first state already has fewer persons per representative, so it doesn't actually make sense to give it an extra representative. Since we are giving the extra representative to the first state, we'd want A over M to be larger than B over N. Now with that extra representative, the first state now has A divided by M plus 1 persons per representative, and if this inequality still holds, then the state still has more persons per representative, which justifies giving them the extra congress run. So we'd expect that A M plus 1 should be greater than B Nth. Now the problematic case will occur if we end up with a smaller value. So now the state would have fewer persons per representative. Now again, since we did give them the extra representative, what that means is that our absolute differences must have dropped. So let's consider that our absolute difference will be B Nth minus A over M plus 1. Remember, this is supposed to be a positive number, and we need to subtract them this way to get a positive result. Now here's the important idea. We had to assign that extra representative to one of the states. And if we gave the extra representative to the second state, the absolute difference would be A Mth, the number of persons per representative in the first state that didn't get the extra representative, minus B over N plus 1, the persons per representative in the second state that did get the extra representatives. Now because we gave the extra representative to the first state, we want this situation to have the smaller absolute difference. In other words, we want the absolute difference, which we get by assigning the representative to the first state, to be smaller than the absolute difference we get by assigning the representative to the second state. And so that means the requirement for giving the extra representative to the first state is going to be this inequality is satisfied. Now we can do a little bit of algebra, and here's the important thing. If a state with population A and M representatives is going to be assigned one more representative, then this quantity must be greater than the corresponding value for any other state. And this means we can compute the priority values all at once. So again, if we go back to our school examples, we can compute the dean priority values and then assign the eight counselors to the schools. So for school one, we'll compute this population times 2M plus 1 divided by M times M plus 1, and we'll let M equal 1, 2, 3, 4, and so on. And if we do that, we find, similarly, for school two, we can compute our priority values, 174 times 2M plus 1 divided by M times M plus 1, and again for M equal 1, 2, 3, and 4, and likewise for school three. Now remember we began by assigning one counselor to each school, so we need to assign five more, and we can do this by picking the five highest priority values. And those are these, which give us our final apportionment. School one will get three more counselors for a total of four, while schools two and three both get one more counselor for two apiece. What if we wanted to minimize relative difference? Again, suppose our state populations are A and B, and they currently have M and N representatives. So again, if we give the extra representative to the first state, we'd expect that A over M must be greater than B over N, that is the persons per representative in the first state is larger than in the second state. If it's still true, then we were correct in our decision, or... Well, remember by assumption, giving the representative to the first state would reduce the relative difference. But here's the problem, relative to what? And there are several possibilities, but we'll compare the relative difference to the smaller of the quantities. So again, we'll start with the Dean method, which requires this inequality, if we're going to assign that extra representative to the first state. Now because of the way that we've set these up, we know that the smaller quantity is going to be the amount that we're subtracting. So we know the smaller quantity on the left and on the right. So if we want to produce a smaller relative difference we require, and if we go through the algebra we get... So if a state with population A and M representatives is to be assigned one more representative, this value must be greater than the corresponding value for any other state. And these values are the Huntington Hill priority values. So we'll compute our priority values. So again for school one, we'll compute the value of population divided by square root M times M plus 1 for M equal 1, 2, 3, 4, and so on. We do the same thing for school two and three. And again, since three counselors have already been assigned, we assign the remaining five to the schools with the highest priority values. And those are these. And once again, school one gets three more counselors, and schools two and three get one more apiece.