Generator

Hamilton's method on 27 seats and 5 regions

A generator in the applied library, called 48 times across 8 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

apportion is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

Hamilton's method on 27 seats and 5 regions. A worksheet of populations, exact quotas, floors, remainders and the seats Hamilton's method awards to 5 regions.

What each rule is answering

Which regions each rule favours

Five rules, one dial

Two out of three, and never all three

Webster's method on 27 seats and 5 regions

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

district 1 gets exactly the seats it is due ×4
district 1's total survives the rounding ×3
in a 2 × 2 × 2 box every table of halves rounds inside quota ×3
party 1 gets exactly the seats it is due ×3
the quotient in district 1, party A rounds to its seats without a tie ×3
the quotient in district 1, party B rounds to its seats without a tie ×3
the quotient in district 1, party C rounds to its seats without a tie ×3
the quotient in district 1, party D rounds to its seats without a tie ×3
the dial at δ = 0/20 is Adams's own answer ×2
the dial at δ = 10/20 is Webster's own answer ×2
the dial at δ = 20/20 is Jefferson's own answer ×2
the house size is a whole number between 2 and 200 ×2
4 to 16 half cells ×1
a column with a fractional cell has a second one ×1
a divisor above the interval awards fewer than the house size ×1
a divisor strictly inside the interval awards exactly the house size ×1
a quota violation is a whole number of seats away from the band ×1
a rational is a whole numerator over a non-zero whole denominator ×1
a rational is never divided by zero ×1
a region outside quota does not sit exactly on its quota ×1
a row with a fractional cell has a second one ×1
a table inside every cell's quota exists ×1
a table meeting every district's target and every party's target exists ×1
a table of 2 to 4 districts by 2 to 4 parties with positive whole vote counts ×1
a trial divisor awards no more seats than the search allows for ×1
Adams does the opposite ×1
Adams is found outside quota ×1
Adams is never found failing house or population monotonicity, which is a theorem about every divisor method ×1
Adams's awarded seats sum to the house size ×1
almost every instance drawn is usable ×1
alternate halves give every row exactly one seat ×1
and every column exactly one ×1
and exactly one table does it at least cost ×1
and Hamilton is found losing a seat as the house grows ×1
and Hill's is the one no transfer improves for the relative difference ×1
and losing a seat to a region that grew more slowly ×1
and the best of them puts exactly one seat where the fair share is 0 ×1
and the smallest never gains one ×1
and Webster's bias for the largest region is smaller than either of theirs ×1
apportioning each district on its own gets at least one party's total wrong ×1
both ends of the interval land inside the drawn divisor axis ×1
each entry of the populations is a whole number between 1 and 10000000 ×1
every cell ends at the floor or the ceiling of its share ×1
every column holds none or two halves ×1
every half shares one line in each direction with another half ×1
every house size swept has a seat for every region ×1
every line holds no halves or exactly two ×1
every method searched fails at least one of the three properties ×1
every row holds none or two halves ×1
floor division is taken of a whole number by a positive whole number ×1
Hamilton gives every region the floor or the ceiling of its own quota ×1
Hamilton is never found outside quota, which is a theorem about it ×1
Hamilton's awarded seats sum to the house size ×1
Hamilton's seats sum to the house at every size swept ×1
Hill seats every region before any second seat ×1
Hill's awarded seats sum to the house size ×1
in a 3 × 3 × 3 box some cannot, the smallest with sixteen halves ×1
Jefferson gives the largest region more than its quota on average and the smallest less ×1
Jefferson is found outside quota ×1
Jefferson is never found failing house or population monotonicity, which is a theorem about every divisor method ×1
Jefferson's apportionment is improvable under every one of the three ×1
Jefferson's awarded seats sum to the house size ×1
multipliers exist that round the votes to exactly these seats ×1
no fair share is a whole number, so its quota is two seats wide ×1
no region's seat count falls anywhere in the swept range ×1
one seat target per district ×1
one seat target per party ×1
one seat total per district ×1
one seat total per party ×1
party A's total survives the rounding ×1
party B's total survives the rounding ×1
party C's total survives the rounding ×1
party D's total survives the rounding ×1
some allocation is stable for the district measure ×1
some allocation is stable for the relative measure ×1
some allocation is stable for the share measure ×1
some table meets every district's seats and every party's seats ×1
some whole-number table meets every line total ×1
the allocations stable for the district measure are all the same allocation ×1
the allocations stable for the relative measure are all the same allocation ×1
the allocations stable for the share measure are all the same allocation ×1
the awarded seats sum to exactly the house size ×1
the biproportional table is among the within-quota tables exactly when it breaks no quota ×1
the closed end of the interval awards exactly the house size ×1
the cycle alternates rows and columns and has an even number of cells ×1
the cycle through the halves has an even number of cells ×1
the dial changes the answer at all ×1
the dial has a step exactly at one half ×1
the districts' and parties' seats fill one house ×1
the districts' seats and the parties' seats add to the same house ×1
the divisor and the priority ranking award the same seats ×1
the divisor method is one of jefferson, webster, adams ×1
the exact arithmetic stays inside the safe integer range ×1
the exact floor of a quota and its decimal floor agree ×1
the exact quotas sum to exactly the house size ×1
the exact quotas sum to the house size in both censuses ×1
the fall the sweep found is by a single seat ×1
the family contains a table whose biproportional seats break a quota ×1
the family of second censuses is small enough to walk in full ×1
the faster-growing region holds exactly one seat fewer after the census ×1
the fitted shares meet every district's and party's total ×1
the floors leave between none and one seat per region over ×1
the growth step is a whole number between 1 and 100000 ×1
the halves contain a cycle of seven, and none shorter that is odd ×1
the house has at least one seat for every region ×1
the house has at most 24 seats, so every table can be visited ×1
the house is large enough for every seat the method gives away for free ×1
the house is large enough for the seats the method gives away ×1
the interval of divisors is not empty ×1
the largest house swept is a whole number between 2 and 200 ×1
the largest region never loses a seat as the dial turns toward Jefferson ×1
the marked region really holds fewer seats in the larger house ×1
the method is one of hamilton ×1
the method is one of hamilton, jefferson, webster, hill, adams ×1
the method is one of jefferson, webster, adams ×1
the mode is one of quota, alabama, divisor, population, family, bias, impossible, unfair, biproportional, quota2d, fit2d, certificate2d, cycle2d, rarity2d, halves3d, graph3d, slices2d, nearest3d, search3d ×1
the named quota violation is a real comparison of the seats against the exact quota ×1
the number of growth steps is a whole number between 1 and 12 ×1
the number of instances generated is a whole number between 50 and 1200 ×1
the number of instances searched is a whole number between 20 and 400 ×1
the number of random tables is a whole number between 50 and 400 ×1
the number of regions is a whole number between 3 and 6 ×1
the number of steps along the dial is a whole number between 8 and 40 ×1
the open end of the interval awards more than the house size ×1
the populations is a list of 2 to 8 numbers ×1
the populations together stay inside the exact range ×1
the priority list reaches past the house size, so the interval has both ends ×1
the region that lost a seat grew by the larger exact ratio ×1
the rounds drawn bring the wrong totals down by at least a factor of a million ×1
the seats awarded along the dial sum to the house size ×1
the second census fills exactly the same house ×1
the second census is larger than the first ×1
the second census totals what it says it does ×1
the seed is a whole number between 1 and 1000000 ×1
the signpost offset is a fraction between zero and one ×1
the slower-growing region holds exactly one seat more after the census ×1
the smallest house swept is a whole number between 2 and 200 ×1
the sweep is short enough to draw one house size at a time ×1
the sweep runs over at least two house sizes ×1
trying to mark alternate halves as a seat and not a seat runs into two neighbours marked alike ×1
Webster is found outside quota ×1
Webster is never found failing house or population monotonicity, which is a theorem about every divisor method ×1
Webster's apportionment is the one no transfer improves for the difference in seats per person ×1
Webster's awarded seats sum to the house size ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Applied

Choosing what unfair means

Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.

Applied

Five rules and one dial

Adams, Webster and Jefferson are usually taught as three rules for rounding a share. They are one rule with a number in it, and turning that number from nought to one moves seats from the smallest region to the largest, one at a time.

Applied

Seats to parties and places at once

Seats can be given to regions in proportion to one list of populations, and no rule does it perfectly. Ask for seats to regions and to parties simultaneously and the object stops being a list — and the impossibility that closed the subject does not apply.

Applied

The rule with no favourites

Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.

Applied

The seat that vanishes when the house grows

Twenty-seven whole seats have to be divided between five regions whose exact shares are 15.417, 7.209, 1.755, 1.431 and 1.188. Every rule for rounding those five numbers breaks something, and the instance drawn here breaks all three of the classical ways at once.

Applied

The table inside every quota

Give seats to districts and parties at once, and every cell of the table has a fair share it ought to round from. A table rounding every cell to its floor or its ceiling, with every total exact, always exists. The biproportional method does not always choose one: here it gives a party 2 seats where its fair share is 3.088.

Applied

Two out of three, and never all three

Stay inside every region's quota, never take a seat away when the house grows, never take one from a region that grew faster. Each pair is achievable. All three together are not, and the proof is that no rule anywhere manages it.

Applied

Where the rounding runs out

In two dimensions a table of seats inside every fair share always exists. Add a third family of totals — every district and party split between groups — and it need not. Sixteen halves in a three-by-three-by-three table meet every total, and no whole table does it without a seat where the fair share is nothing, because the halves close a loop of seven.

The whole library · What the figures prove