# Congressional Apportionment Mathematics > [!note] Disambiguation > This page covers the division of House seats among the states. For the budget-execution sense of the term — the process by which OMB releases appropriated funds to agencies — see [[Apportionment]]. Every ten years the census produces state populations, and 435 House seats must be divided among them. Each state's *quota* — its exact proportional share — is almost never a whole number. Ohio is entitled to 15.34 seats and cannot have them. Every apportionment method is a rule for resolving that fractional remainder, and **the choice of rule changes who holds power.** This is not a hypothetical. Apportionment method disputes have altered the composition of Congress and, through the [[Electoral College]], the outcome of at least one presidential election. > [!info] The pipeline around the formula > [[How the Census Apportions Congress]] maps the process this math sits inside — who is counted and where, the statutory deadlines, and the steps at which courts can and cannot intervene. This page is the arithmetic; that one is the machinery. ## The methods, and the politics of each **Hamilton's method (largest remainders).** Give each state its quota rounded down; distribute leftover seats to the states with the largest fractional remainders. Intuitive and fair-seeming. Washington vetoed the first version — the first presidential veto in American history — and the method was later abandoned because of the paradoxes below. **Jefferson's method (greatest divisors).** Find a divisor such that when each state's population is divided by it and rounded *down*, the totals sum to 435. Systematically favors large states. Jefferson was from Virginia, then the largest state. Used 1792–1840. **Webster's method.** Same divisor procedure, but round to the nearest whole number. Nearly unbiased between large and small states. **Huntington-Hill (equal proportions).** The current method, adopted in 1941. Same divisor family, but the rounding threshold is the *geometric* mean of the two adjacent integers rather than the arithmetic mean. Because the geometric mean of 1 and 2 is about 1.414 while the arithmetic mean is 1.5, the method rounds up slightly more readily at small numbers — a mild small-state tilt. **The 1876 election turned on this.** Under one apportionment method Tilden would have won; under another Hayes did. The seat allocation in force was not the one the standing statute clearly required, and the resulting electoral-vote math delivered the presidency to Hayes — followed by the Compromise of 1877 and the end of Reconstruction. ## The paradoxes **The Alabama paradox.** Under Hamilton's method, *increasing the total number of seats can cause a state to lose one.* Discovered during the 1880 census computations: at a House size of 299 Alabama received 8 seats; at 300 it received 7. This is not a rounding error — it is a structural property of largest-remainder methods. **The population paradox.** A state whose population is growing faster than another's can lose a seat to it. **The new-states paradox.** Admitting a new state with its own fairly-computed allocation can change the allocation among existing states. Observed when Oklahoma was admitted in 1907. These paradoxes are why Hamilton's method — the most intuitive one — was abandoned. ## The impossibility result Michel Balinski and H. Peyton Young proved in 1982 that this is not a matter of insufficient cleverness. **No apportionment method can simultaneously (a) always stay within quota — giving every state either its quota rounded up or rounded down — and (b) avoid the population paradox.** You must give up one. Divisor methods including Huntington-Hill avoid the paradoxes and accept occasional quota violations. Largest-remainder methods stay within quota and accept the paradoxes. There is no third option. This puts apportionment in the same family as the other social-choice impossibility results — Arrow's theorem, Gibbard-Satterthwaite. **A recurring finding of twentieth-century mathematics is that our intuitive fairness criteria are jointly unsatisfiable, and that every real democratic institution has therefore already made a choice about which fairness property to abandon — usually without anyone having voted on it.** ## Why it is live again Two current pressure points: **The 435 cap is arbitrary and consequential.** The House has been frozen at 435 since 1929, when the population was about 122 million. Average district population has roughly tripled. Because every state gets at least one seat regardless of population, freezing the total amplifies small-state overrepresentation — which propagates directly into the [[Electoral College]], since electoral votes equal House seats plus two. The cap is a statute, not a constitutional requirement, and it is one of the largest single contributors to structural [[Minority Rule]]. **Who gets counted.** Apportionment uses total resident population. Proposals to exclude non-citizens from the apportionment base would shift seats from high-immigration states to low-immigration ones. This is a fight over the *input* to a settled formula rather than over the formula — the recurring pattern in this branch, where the contested politics migrates to wherever it is least visible. ## Related [[The Mathematics of Redistricting]] · [[Electoral College]] · [[Minority Rule]] · [[Apportionment]] · [[Mathematics and Power Hub]]