Divide a fixed number of seats among states by population and you hit a wall: you can’t give fractional seats, and every rounding rule breaks fairness somewhere. The cruelest example — the Alabama Paradox — is real: INCREASING the total number of seats can make a state LOSE one, populations unchanged. It nearly broke the US Census in 1880. Slide the house size and watch a state get robbed.
Apportionment splits a fixed house of seats among states in proportion to population, but seats are integers — and Balinski & Young proved NO method avoids every paradox. Hamilton’s method (largest-remainder) suffers the ALABAMA PARADOX: with populations fixed, raising the total seat count can DECREASE a state’s seats, because the fractional remainders re-sort. Here, populations (6, 6, 2): at 10 seats the split is [4, 4, 2]; at 11 seats it becomes [5, 5, 1] — the small state loses a seat while the house GREW. A fail-loud self-check throws unless increasing the total causes some state to lose a seat. ◆ real apportionment theory, node-verified.
Hamilton’s method on a chosen population profile (the exact largest-remainder paradox); divisor methods (Huntington-Hill, used by the US since 1941) avoid Alabama but violate quota instead — the no-perfect-method impossibility is exact (Balinski–Young).