Reference

The Rules, Explained

Six ways of turning populations into whole numbers of seats, which one India has actually used, why they disagree, and why a popular alternative can take a seat away from a state when the house gets bigger.

1. The Apportionment Problem

You have a house of H seats and a set of states with populations p₁ … pₙ. You want each state's share of the seats to match its share of the population. The difficulty is that seats are whole numbers and shares are not.

State i's exact entitlement — its quota — is

qᵢ = H × pᵢ / P   where P is the total population

If every qᵢ happened to be a whole number the problem would not exist. They never are. Tamil Nadu's exact entitlement at a house of 543 on the 2011 Census is 32.35 seats. It cannot have 0.35 of an MP, so somebody has to decide whether it gets 32 or 33 — and whoever gains the seat Tamil Nadu does not get is a different state entirely.

Every method below is a different answer to "who gets the fraction". That is the whole of it. They are not competing approximations to a single right answer, because there is no single right answer; they are competing definitions of what fairness means when the thing you are dividing cannot be divided.

2. Quota, and the Two Families

A method satisfies quota if every state always receives either ⌊qᵢ⌋ or ⌈qᵢ⌉ — the whole number just below its exact entitlement, or the one just above. Nothing else. Tamil Nadu at 32.35 gets 32 or 33, never 31 and never 34.

That sounds like an obvious requirement. It is not free, as §8 shows.

Apportionment rules fall into two families:

3. Largest Remainder (Hamilton's Method, Hare Quota)

The most familiar way to build a quick projection. It is not the method India's Delimitation Commission used (see §4), but it is worth understanding because projections built this way circulate widely.

1. Work out the quota: total population ÷ house size. This is how many people one seat "costs".
2. Divide each state's population by the quota. That is its exact entitlement.
3. Give every state the whole-number part, rounded down.
4. You will now be a few seats short. Hand those out, one each, to the states with the largest fractional remainders.

It is transparent, it is easy to check by hand, and it never violates quota. It is also the only rule here that can produce the paradox in §6.

4. The Divisor Methods

A different idea entirely. Instead of computing entitlements and rounding, hand out the seats one at a time. Before each seat, ask every state: what is your claim? Give the seat to the strongest claim, then ask again.

A state's claim, when it already holds n seats, is

priority = pᵢ / d(n)

Every divisor method is the same algorithm with a different d(n). That one function is the entire difference between them, and it is the axis that decides how much small states are favoured.

D'Hondt

Divide by 1, 2, 3, 4… A state holding 10 seats has its population divided by 11 to compete for its eleventh. Because the divisor grows slowly relative to the seat count, a large state's claim stays strong for longer, so D'Hondt systematically favours large states. It is equivalent to rounding entitlements down and adjusting the quota until the total fits.

Sainte-Laguë

Divide by 1, 3, 5, 7… The odd divisors correspond to rounding at the halfway point: a state with an entitlement of 4.5 rounds to 5, one with 4.4 rounds to 4. This makes it the least biased of the divisor methods by most measures, and it is the usual recommendation of people who study the problem for a living.

India's method is close to Sainte-Laguë, but it is not the same. The Third Delimitation Commission (1972–76) set aside seats for the union territories and for states of six million people or fewer, which Article 81 exempts from the population ratio. It divided the rest of the population by the remaining seats, rounded the result to a national quotient of 10,44,000 people per seat, and divided each state's population by that quotient to find its seats. People per seat ranged only from 10.31 lakh in Rajasthan to 10.67 lakh in Kerala.

Two things separate this from Sainte-Laguë. The quotient was fixed, not tuned. Sainte-Laguë keeps adjusting its divisor until the rounded seats add up exactly. The Commission applied one quotient once, and in 1976 the rounded seats happened to total the 507 available. Applied to today's populations the same procedure often misses: at a house of 815 on the 2011 Census it produces 816 seats. And not every unit was on the quotient. Union territories and small states kept seats set outside it. Delhi, with 1.68 crore people in 2011, would therefore stay at 7 seats however large the house grew, where Sainte-Laguë over every unit gives it 11 at 815.

Sources: The India Forum and the Centre for Policy Studies. Neither records how individual states' fractions were rounded. Rounding to the nearest seat against 10,44,000 reproduces the 1976 seats of all fifteen major states exactly from the 1971 Census (as recast to today's boundaries). So do Sainte-Laguë, largest remainder and Huntington-Hill, because no state's fraction fell near the rounding line, which means the 1976 figures cannot tell those rules apart. The main page's default is the Commission's procedure applied literally, with its total shown as computed.

Huntington-Hill

Divide by the geometric mean √(n(n+1)) rather than an arithmetic one. Used for the United States House of Representatives since 1941.

Note that d(0) = 0, which would make the first seat's priority infinite, so every state is seeded with one seat before the method starts. That is not a quirk — it encodes a constitutional requirement that every state gets at least one representative, and it is why the method is described as favouring small states.

The geometric mean means the method equalises relative differences in representation rather than absolute ones. Moving a seat is justified only if it reduces the percentage difference between two states' people-per-seat.

5. Cube Root, and Base-Plus-Proportional

Cube Root of Population

Not a rounding rule at all: a decision that seats should be proportional to the cube root of population rather than to population. It comes from the observation that assembly sizes around the world track roughly the cube root of national population — the "cube root law".

It compresses the range hard. A state four times larger than another gets about 1.6 times the seats, not four times. It violates quota almost everywhere, which is expected: it is not trying to be proportional to population, so the quota it violates is not the quota it is aiming at.

Base Seats Plus Proportional

Every state gets a fixed base — two seats, say — and the rest of the house is distributed in proportion to population. This is the shape of the Cambridge Compromise, proposed for the European Parliament, where it is described as "degressive proportionality": smaller states are deliberately over-represented, and the base controls by how much.

The base is the whole argument. A base of one is nearly pure proportionality; a base of ten would give Sikkim and Uttar Pradesh far more similar weight than population alone ever would.

Uniform Scaling not a rule

Included on the main page because it was reportedly offered during the April 2026 debate, not because it is an apportionment method. It allocates in proportion to each state's current seats, so every state's share is preserved exactly and the house simply gets bigger. It resolves nothing about representation — which is precisely the point worth being able to see rather than being told.

6. The Alabama Paradox

Under largest remainder, a state can lose a seat when the house grows. Nothing about that state changes. No population moves. The house simply gets bigger, and the state gets smaller.

It is named for what happened in the United States in 1881. The Census Bureau computed apportionments for every house size from 275 to 350 and noticed that Alabama got 8 seats at a house of 299 and 7 at a house of 300. The same thing hit Maine in 1901, oscillating between 3 and 4 seats across a range of house sizes, which the chairman of the committee used to considerable rhetorical effect.

Why It Happens

Watch what growing the house does to each state's exact entitlement. When H increases by one, state i's entitlement increases by

Δqᵢ = pᵢ / P

Large states' entitlements grow faster than small states'. That is not controversial — it is what proportionality means. But largest remainder does not allocate on entitlements, it allocates on the fractional part of entitlements, and fractional parts wrap around.

A large state's fraction climbs quickly, crosses 1, and resets to near zero. A small state's fraction creeps up slowly. Most of the time this is harmless. But at the moment two large states' fractions both climb past a small state's slowly-creeping fraction, the small state loses its place in the remainder queue — and with it, the seat it was holding only because it was near the front of that queue.

Watch It Happen

Three states. A and B have 6 units of population each; C has 2. Drag the house size between 10 and 11 and watch C.

At a house of 10 the quota is 1.4. C's entitlement is 1.43, a fraction of 0.43, which is the largest in the room — so C takes the single leftover seat and finishes with 2.

At a house of 11 the quota falls to 1.27. Now A and B are each entitled to 4.71, a fraction of 0.71, while C is entitled to 1.57, a fraction of 0.57. There are two leftover seats and A and B are ahead of C for both. C drops to 1. The house grew by a seat and C lost one.

What It Means for India

India's own method is immune. With one national quotient and each state rounded to the nearest seat, a larger house only lowers the quotient, so no state's figure can fall. A Delimitation Commission working the way the last one did cannot take a seat from a state by enlarging the house.

The risk sits in the analysis around it. Largest remainder is a common way to build projections, and on India's 2011 Census it produces 32 instances of a state losing a seat as the house grows between 543 and 900. Uttarakhand falls from 5 seats to 4 at a house of 548, the total of the unchanged-strength column in the PRS annexure, and back to 5 at 549. Odisha falls from 21 to 20 at 595. A projection built that way can show a loss that India's method would never produce.

The main page lists every instance, recomputed live from whichever population series you have selected.

This is verified independently in exact integer arithmetic — BigInt division with no floating point anywhere — so it is a property of the rule and the data, not an artefact of the code. Cube root and base-plus-proportional exhibit it too, because both distribute leftovers by largest remainder. The divisor methods produce none at all, ever: awarding seats one at a time in priority order means a larger house can only ever add to what a state already has.

7. The Other Two Paradoxes

The Population Paradox

State A grows faster than state B — and A loses a seat to B. Largest remainder can do this for the same reason it produces the Alabama paradox: the remainder queue can reorder in ways that do not track the direction anyone moved in.

For a delimitation exercise this is arguably the more alarming of the two, because population growth rates between Indian states differ enormously and are the entire subject of the argument.

The New States Paradox

Add a new state, give it exactly the number of seats its own population warrants, expand the house by exactly that many — and the allocation between the existing states changes. This happened when Oklahoma joined the United States in 1907: Oklahoma took its 5 seats, and New York lost one to Maine.

India creates and reorganises states regularly. Telangana in 2014, the Jammu & Kashmir reorganisation in 2019, three new states in 2000.

8. Why You Cannot Simply Pick a Better Rule

Balinski and Young, 1982. No apportionment method can both satisfy quota and be free of the population paradox. Any method that always stays within quota must sometimes produce a paradox; any method free of paradoxes must sometimes fall outside quota.

This is a theorem, not an engineering limitation. It is not waiting for a cleverer algorithm. The two properties are genuinely incompatible for any method dealing with four or more states.

So the choice between the families is not between a right answer and a wrong one. It is a choice about which failure you would rather live with:

• Largest remainder guarantees that no state is ever more than one seat away from its exact entitlement — and in exchange, accepts that the allocation can move in directions nobody intended.
• Divisor methods guarantee that growth is always monotone and that nobody is punished for a change they did not make — and in exchange, accept that a state can occasionally end up further from its exact entitlement than rounding would allow.

Balinski and Young's own recommendation, on the evidence in Fair Representation, is Sainte-Laguë: of the divisor methods it is the least biased by size, and quota violations under it are rare.

9. What This Means for the Indian Debate

Published treatments of delimitation usually report one set of numbers as though it were the number of seats each state would get. Each rests on choices that are rarely spelled out: the census year, the house size, the exemptions, and the rounding rule.

Three consequences worth carrying into any argument about this:

1. The rounding rule matters less than the frame around it. Among the textbook proportional methods no state moves by more than a seat. What moves dozens of seats is whether to allocate by population at all, which census counts, how big the house is, and which states and territories sit outside the formula.
2. India's method avoids a trap that projections can fall into. Largest remainder takes a seat from Uttarakhand at a house of 548, the total of the PRS unchanged-strength column, though it holds that seat at 547 and 549. The national quotient method never does this.
3. Constraints decide more than rules do. Cap every state at two seats of movement at a house of 600 and the south ends up with more seats than it holds today. The argument about method may matter less than the argument about transition.

10. Further Reading

• Michel Balinski and H. Peyton Young, Fair Representation: Meeting the Ideal of One Man, One Vote (Yale, 1982; 2nd ed. Brookings, 2001). The standard work. The impossibility theorem in §8 is theirs, and the historical material on Alabama and Oklahoma comes from it.
• The Cambridge Compromise (2011), the report of the mathematicians' meeting convened for the European Parliament, which sets out the base-plus-proportional approach and the idea of degressive proportionality.
• Report of the Technical Group on Population Projections, National Commission on Population, 2020 — the source of the projected series on the main page.
• PRS Legislative Research, the Delimitation Bills of 2026 — legislative summaries and the projection table used here as a validation fixture.

← Back to the visualiser · Methodology · Sources