What this calculator does
See how a set number of seats in a legislature gets shared out among states by population. Every ten years, after the census, the 435 seats in the U.S. House of Representatives are divided among the states this way. Seats come in whole numbers, so a perfectly exact split is impossible. The result depends on how you round. Since 1941, the rule has been the method of equal proportions, also called the Huntington–Hill method. This calculator applies it to any list of states and populations you enter.
Each state first gets its minimum seat. The rest are given out one at a time. Each one goes to the state with the highest priority value, a score for how much it needs another seat. That score is the state's population divided by the square root of n(n + 1), where n is the seats it already has. You see each state's seats and the priority values behind the last few seats given out. You also see which state would get the next seat if the House were one seat larger. The starting data is a made-up country with six states. A button loads the real 2020 census figures.
Where it's used
Why this math matters outside the classroom:
- Census-year apportionment. After each census, the 435 House seats are reassigned by ranking every state's priority value P ÷ √(n(n + 1)) for seats 51 to 435.
- Comparing rounding rules. A class compares methods on two states with shares of 1.45 and 8.55 seats: largest remainders gives 1 and 9, equal proportions gives 2 and 8.
- Student council seats. A school splits 20 council seats among grades of 412, 389, 350 and 301 students, giving each grade at least one seat.
- Estimating square roots. Students check that 1.45 seats rounds up because √2 ≈ 1.414 is the cut-off between 1 and 2 seats.
- Next seat in line. Analysts list the priority values just past seat 435 to see which states were closest to gaining or losing a seat.
Grade level and standards: Grades 8–9; Common Core State Standards for Mathematics 8.NS.A.2. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- List the states, one per line, each with a name and a population, such as "Northmere 4215300". Commas in the numbers are fine.
- Enter the House size, which is the total number of seats to divide. Then enter the minimum seats every state is guaranteed (the U.S. uses 1).
- Read the seats for each state. Then check the working for the priority values of the last seats given out and the next state in line.
- Use "Load 2020 census" above the form to recreate the official split of 435 seats among the 50 states.
What the inputs mean
- States and populations
- On each line, write the name first and the population last. Names can contain spaces. Populations must be whole numbers greater than zero, and each name can appear only once.
- House size (total seats)
- The total number of seats to share out. It must be at least the number of states times the minimum.
- Minimum seats per state
- The seats every state gets before any are given out by priority. The U.S. Constitution guarantees each state at least 1.
Formula and method
Start: every state gets m seats (m = minimum, usually 1) Priority value for a state with n seats: A(n) = P ÷ √(n × (n + 1)) Repeat until the House is full: give the next seat to the state with the largest A(n), then raise its n by 1 Exact share (quota) = P ÷ total population × House size
In these formulas, P is a state's population and n is the number of seats it has so far. A(n) is its priority value, and m is the minimum seats per state. The exact share, or quota, is the number of seats a state would get if seats could be split into fractions.
The priority value measures how strong a state's claim is to one more seat. It divides by √(n(n + 1)), which sits between n and n + 1. This number is called the geometric mean. In effect, a state's exact share is rounded up once it passes the geometric mean, rather than the halfway point. For example, a share of 1.45 seats rounds up to 2, because 1.45 is above √2 ≈ 1.414.
Giving seats out one at a time, in priority order, gives the same result as another approach. That approach finds one shared divisor and rounds each state's share at the geometric mean. The one-at-a-time way is easier to follow. It also produces the ranked list of seats used to say who is next in line.
The method gets its name because it keeps the differences between states as small as possible in relative terms, meaning as proportions rather than raw numbers. It does this for the average number of people per district, and for each state's share of representatives. A state with no seats has an infinitely high priority value. So the method guarantees every state at least one seat on its own.
Worked example
A made-up country has six states: Northmere 4,215,300, Eastbrook 2,987,450, Lakeland 1,604,820, Southvale 912,640, Westridge 655,110 and Highfield 348,900. Its House has 435 seats.
- Each state gets 1 seat: 6 × 1 = 6, leaving 429 to give out by priority.
- Seat 434 goes to Westridge, its 27th: 655,110 ÷ √(26 × 27) = 24,725.53.
- Seat 435 goes to Northmere, its 171st: 4,215,300 ÷ √(170 × 171) = 24,723.27.
- Next in line is Eastbrook. Its 122nd seat has priority 2,987,450 ÷ √(121 × 122) = 24,588.27.
Result: Northmere 171, Eastbrook 121, Lakeland 65, Southvale 37, Westridge 27, Highfield 14. Westridge's exact share is 26.57 seats. That's above √(26 × 27) ≈ 26.50, so it rounds up to 27.
Understanding the result
Compare each state's seats with its exact share in the table. Almost always, a state's seats are its share rounded up or down. The cut-off is the geometric mean of the two whole numbers on either side. Like every method of this type (a divisor method), it can, in rare cases, move a large state further from its share.
The 'Next in line' state is the one that would gain a seat from a larger House. The gap between the last priority value given a seat and the next one shows how close the final seat was.
The 'Priority for next seat' column ranks the states for any extra seats. The highest value would get the next one.
Assumptions and limitations
- Populations are the counts used for sharing out seats, as you enter them. The U.S. figures include federal workers and military stationed overseas, and leave out the District of Columbia.
- Every state gets the same minimum before seats are given out by priority.
- An exact tie in priority values goes to the state listed first. Real ties are extremely rare.
- Only the Huntington–Hill method is worked out. Other methods (Hamilton, Jefferson, Webster, Adams) can give different results for the same data.
- Drawing district lines within a state is a separate process, and it isn't included here.
- You can enter up to 100 states and a House of up to 10,000 seats.
Frequently asked questions
Why not just round each state's exact share?
Ordinary rounding rarely gives the right total. The rounded shares can add up to more or fewer seats than the House has. Divisor methods like Huntington–Hill pick a rounding point that always fills the House exactly. This one rounds at the geometric mean. So states are compared by relative differences, not absolute ones.
Does the calculator reproduce the real 2020 apportionment?
Yes. Load the 2020 census populations of the 50 states and use 435 seats. You get the official count for every state, from California's 52 down to the six states with 1. The last seat, number 435, goes to Minnesota, and New York is next in line.
What is a priority value?
It's a state's population divided by √(n(n + 1)), where n is the seats it already holds. Work out every state's priority values for its 2nd, 3rd, 4th seat and so on. Then list them from highest to lowest. That gives the order in which seats 51 to 435 are given out.
Why does a small state sometimes get an extra seat over a big one?
Seats are compared by relative need, not absolute need. Picture a small state at 1.45 of a seat and a large state at 8.55. The small state gains more, in proportion, from a second seat than the large one gains from a ninth. So equal proportions gives the small state its second seat first.
Sources
- U.S. Census Bureau: 2020 Census Apportionment Results, Table 1 (checked 2026-10-19)
- U.S. Census Bureau: 2020 Census Apportionment Data (checked 2026-10-19)
How this calculator is checked
Method: Method of equal proportions (Huntington–Hill) as described for U.S. apportionment (2 U.S.C. § 2a; U.S. Census Bureau priority-value method). Tests reproduce a hand-worked example and the official 2020 seat counts for all 50 states from the census apportionment populations. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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