**The Apportionment **

*Abstract*

There are many methods to divide congressional seats or representatives among states of the United States. The most popular methods are Hamilton’s method, Huntington-Hill, Webster’s, Jefferson’s, Adams’ and others. Most of the methods use the quota rule. The last one states that the number of seats can be round up or down the standard quota. The following indicator is standard divisor. It helps to calculate initial quotas.Every method has as advantages as drawbacks. For example, using Hamilton’s method the Alabama paradox appears. It occurs when the number of total seats increases but the state loses representatives. There are some another paradoxes like population or new states. Thus, choice of the optimal method for dividing number of seats by states is not easy task. There are some extra rules which appear in the calculations.Keywords: apportionment method, standard quota, Hamilton’s method, Huntington-Hill method, the Alabama paradox, quota rule, standard divisor.

*The Apportionment Problem*

The apportionment method is the act of dividing items between different groups according to some plan, especially to make a proportionate distribution. It can be possible to use several methods for solving problem. It should be noted that “citizenship confirms an individual’s relevance in (and to) the state through rights” (Greenhouse, 2011, p.262). The main task is the fair representation in the US House of Representatives. In this case, let’s compare different apportionment methods and its “process of reasoning” (Schwols, Kendall & Dempsey, 2012, p.8) and determine the most optimal.The first is Hamilton’s method. It was first used to decide the initial apportionment of the seats in the House of Representatives in 1790.

That apportionment was vetoed by George Washington and the House was reapportioned that year according to the Jefferson Method. The Hamilton Method did come back into use in 1850 and was then used until 1900.There are some steps in Hamilton’s method. Firstly, it needs to calculate divisor by dividing the total population of all states by the total numbers of representatives. It is 5321,88. Fig.1. Hamilton’s Method. Initial Results.Then, the quota with decimals is calculated. All calculations are performed in Excel. It can help to create “beautiful mathematics” (Diaconis & Graham, 2011, p.17). The results are shown in figure

1. The first intermediate result is lower quota. It is calculated by formula “int”. The total of lower quota is less the initial total by 5 seats. So, it needs to sort the column “Decimal” and add extra seats to some states. One should not forget that each state ends up at least one representative. It concerns the state 5. Despite the quota is less than 0,5, the final number of seats is 1. Thus, the state 4 still has 16 seats. The extra seats are added to the state 1, 8, 10 and 2. The final results are below (figure 2). Using the Hamilton’s method satisfies the quota rule. The final answers are the whole numbers. It means that the seats number round up or down.Fig.2. Hamilton’s Method. Final results.The next task …