## Distributing Seats for Congress

With Puerto Rico voting to start the process for formally becoming a state in the US yesterday (be prepared for decades of delay by the other 50), it is time to look at the process that America uses to represent itself and analyse how it will change. There are two houses, one which represents the states, while the other represents the people (in theory). The first is the Senate which has 100 representatives, 2 from each state. This one is easy to update for Puerto Rico, although having 102 Senators is slightly displeasing from the perspective of having round numbers.

However Congress is a more complicated affair. Originally the rule was that each state got at least one representative and that no representative would be representing more than 30,000 citizens. This meant that more populous states would get more representatives. Unfortunately, exponential growth meant that the number of representatives was growing too fast and so in 1911 they got rid of the 30,000 citizen cap and fixed the number of representatives at 435. The average number of citizens for each representive is now about 750,000, which to put it another way, means that if the 30,000 citizen cap was still in place would mean that there would have to be close to 11,000 seats in Congress.

The method for assigning representatives is slightly complicated, but it is also quite elegant. It uses a system called the Huntington-Hill method which starts by assigning one seat to every state. After that it uses the formula:

Where n is the number of seats already claimed by that state and P is the population of that state. All of these numbers are compared, with the highest one gaining the next seat. Letâ€™s follow through the logic.

Every state gets a state to start with, so all start with n=1. This means that the denominator for each of them is root(2) and so the formula is just their population over root(2) for each state. California is the most populous state at so is awarded a second seat. Now n=2 for California and so its formula becomes population/root(6). The more seats you win, the less weighting your population carries for future consideration. California's population is around 37 million and root(6) is about 2.45. The next most populous state is Texas at 24 million, with root(2) lying around 1.41. Doing the divisions gives about 15 million for California and 17 million for Texas, so Texas wins the next seat.

This process continues until all 435 seats have been assigned. It is finely balanced between larger states getting more representation and all states getting some say. California ends up with 53 seats while Alaska, Delaware, Montana, North Dakota, South Dakota, Vermont and Wyoming all have only a single seat.

However by adding in Puerto Rico all of the numbers change. It gains 5 seats, which means that 5 seats are lost from elsewhere. Here's a table:

State | Seats Before | Seats After | Change |
---|---|---|---|

Calfornia | 53 | 52 | -1 |

Texas | 36 | 36 | 0 |

New York | 27 | 27 | 0 |

Florida | 27 | 27 | 0 |

Illinois | 18 | 18 | 0 |

Pennsylvania | 18 | 17 | -1 |

Ohio | 16 | 16 | 0 |

Georgia | 14 | 14 | 0 |

Michigan | 14 | 13 | -1 |

North Carolina | 13 | 13 | 0 |

New Jersey | 12 | 12 | 0 |

Virginia | 11 | 11 | 0 |

Washington | 10 | 9 | -1 |

Massachusetts | 9 | 9 | 0 |

Arizona | 9 | 9 | 0 |

Indiana | 9 | 9 | 0 |

Tennessee | 9 | 9 | 0 |

Missouri | 8 | 8 | 0 |

Maryland | 8 | 8 | 0 |

Wisconsin | 8 | 8 | 0 |

Minnesota | 8 | 7 | -1 |

Colorado | 7 | 7 | 0 |

Alabama | 7 | 7 | 0 |

South Carolina | 7 | 7 | 0 |

Louisiana | 6 | 6 | 0 |

Kentucky | 6 | 6 | 0 |

Oregon | 5 | 5 | 0 |

Oklahoma | 5 | 5 | 0 |

Connecticut | 5 | 5 | 0 |

Puerto Rico | 0 | 5 | +5 |

Iowa | 4 | 4 | 0 |

Mississippi | 4 | 4 | 0 |

Arkansas | 4 | 4 | 0 |

Utah | 4 | 4 | 0 |

Kansas | 4 | 4 | 0 |

Nevada | 4 | 4 | 0 |

New Mexico | 3 | 3 | 0 |

Nebraska | 3 | 3 | 0 |

West Virginia | 3 | 3 | 0 |

Idaho | 2 | 2 | 0 |

Hawaii | 2 | 2 | 0 |

Maine | 2 | 2 | 0 |

New Hampshire | 2 | 2 | 0 |

Rhode Island | 2 | 2 | 0 |

Montana | 1 | 1 | 0 |

Delaware | 1 | 1 | 0 |

South Dakota | 1 | 1 | 0 |

Alaska | 1 | 1 | 0 |

North Dakota | 1 | 1 | 0 |

Vermont | 1 | 1 | 0 |

Wyoming | 1 | 1 | 0 |

And for the sake of interest, here is my mess of a spread sheet that I used to work out the new distribution. I started with each state's population and divided it by each of the relevant roots going across the page. Then I copied all of the data points that had a chance of being large enough into a single column below the main table and then sorted them from largest to smallest. Since each state starts with a representative, that took out 51 straight away. That left 435-51=384 seats to distribute, so I looked for the 384th largest number in the list which was 736422.642. Then I conditionally formatted my original grid to highlight any numbers that were greater than or equal to 736422.642. Finally, I counted how many seats each had (including their free first one).

The data for the US states was from their 2013 census, while the data for Puerto Rico was from their 2016 estimates. The only other data point I could find for Puerto Rico was from 2010, but both figures gave the same distribution so I'm not going to worry about it. Of course, the populations of the various states are not all growing at the same rate and the next census data which will shift all of the seat numbers will be in 2020. Here's an article with some predictions if you are interested.

Finally, the much more pressing issue is what will become of the US flag if (when) Puerto Rico becomes a state?