The Alabama Paradox

In 1880, a clerk in the United States Census Bureau was working through the apportionment of House seats for the following decade. Running the calculation at different total sizes, he noticed something that should not have been possible: as the hypothetical House grew from 299 seats to 300, Alabama's share fell from eight seats to seven. Adding a seat to the chamber cost a state a seat. The discovery gave the phenomenon its name.

The culprit was the largest-remainder method — known in the United States as the Hamilton method after Alexander Hamilton, whose proposal Congress had actually enacted. The procedure is straightforward: divide each state's population by a common quota, award every state its whole-number result, then distribute any leftover seats one by one to the states with the largest fractional remainders. The final step is where the trouble lives. Remainders do not scale neatly when the total seat count changes. A larger House means a smaller quota, which reshuffles every state's remainder simultaneously. A state that held a large remainder at 299 seats may find its remainder has dropped just below a competitor's at 300, losing its claim on a bonus seat even though the pool has grown.

The effect is not a rounding error or a data mistake. It is a structural property of any method that works by distributing leftovers after a quota cut. The numbers can be arranged so that the paradox appears repeatedly across a range of House sizes — and for more than one state at a time.

Congress was sufficiently alarmed to appoint a committee. The committee confirmed the arithmetic, and Congress responded by fixing the House size at a number chosen to avoid the worst outcomes — an improvisation that could not hold indefinitely. Two further anomalies eventually joined the list: the population paradox, in which a faster-growing state loses a seat to a slower-growing one, and the new-states paradox, in which admitting an additional state (with a correspondingly enlarged House) disturbs the apportionment of states already seated. All three are symptoms of the same underlying instability in remainder-based allocation.

The final break came with work by mathematicians Michel Balinski and H. Peyton Young in the 1970s and 1980s, who proved that all three paradoxes are unavoidable under any quota method. Their analysis pointed toward divisor methods — which set seats by successive division and rounding rather than by distributing leftovers — as the only family free of these instabilities. The United States switched to the Huntington-Hill divisor method in 1941, and the Alabama paradox has remained a historical curiosity ever since: a clerk's arithmetic exercise that ended a whole approach to democratic arithmetic.