google this: "discrete structure and computing apportionement" the first link that pops up should be a pdf that says "Applications of Discrete mathematics- chapter 1"

1. Suppose a fictitious country with four states with populations 2000, 3000, 5000, and 8000 has a house with eight seats. Use each of the apportionment methods listed in Tables 1 and 2 to apportion the house seats to the states.

2. Consider the same country as in Exercise 1, but with nine seats in the house.

3. The Third Congress had 105 seats in the House. For apportioning the Third Congress, the population figures in Table 1 must be modified by reducing North Carolina to 353,522.2 to reflect the formation of the South- west Territory (which eventually became Tennessee), reducing Virginia to 630,559.2 to reflect the formation of Kentucky, adding Kentucky at 68,705, and adding Vermont at 85,532.6 (there were no slaves in Vermont, but 16 reported as free colored were recorded as slave). Find the apportionment for the Third Congress according to each of the five methods listed in Tables 1 and 2. (This can be done with a hand calculator, but computer programs would save time.)

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Chapter 1 The Apportionment Problem 17

Suppose that n H. Which of the five apportionment methods assure at least one seat for each state?

a) Modify both the and Huntington sequential methods for the greatest divisors apportionment to assure at least 3 representatives per state.

b) Show that they produce the same result.

Use the algorithms constructed in Exercise 5 to apportion a 15-seat house for the fictitious country of Example 2.

Show that the and sequential algorithms for the smallest divisors appor- tionment produce the same result.

Show that the and sequential algorithms for the smallest divisors appor- tionment minimize maxi(pi/ai).

a) Show that smallest divisors can violate lower quota. b) Can it violate upper quota?

a) Construct an example illustrating the Alabama paradox. (Recall that this can only happen under the largest fractions apportionment.)

b) Can this happen if there are only two states?

How many apportionments are possible with 15 states and 105 seats if each state has at least one seat?

How many apportionments are possible with 50 states and 435 seats if each state has at least one seat?

Solve Exercise 3, but for the 1990 census with a House size of 435. (Visit http://www.census.gov/ for the 1990 census data. Computer programs are strongly recommended for this problem.) Which of the five methods would benefit your state the most?

(Calculus required) The denominators in the Huntington sequential method (labeled ri in Table 2) involve averages of the form

(.5(ati + (ai + 1)t))1/t.

(The choices t = 1, 0, and 1 provide the arithmetic, geometric, and har- monic means, respectively.)

a) Show limt0(.5(ati +(ai +1)t))1/t = ai(ai + 1) (the geometric mean). b) Show that limt(.5(ati + (ai + 1)t))1/t = ai + 1 (the maximum). c) Show that limt(.5(ati + (ai + 1)t))1/t = ai (the minimum).

18 Applications of Discrete Mathematics

Computer Projects

Write a computer program to apportion the House by the method of major fractions using the lambda method.

Write a computer program to apportion the House by the method of major fractions using the sequential (ri) method.

Write a program which will check all apportionments consistent with rank order of size (no state which is smaller than another state will receive more seats, but equality of seats is allowed).