Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

The Constitution and Paradoxes

January 2002

par·a·dox 1. A seemingly contradictory statement that may nonetheless be true: the paradox that standing is more tiring than walking. 2. One exhibiting inexplicable or contradictory aspects: "You have the paradox of a Celt being the smooth Oxonian" (Anthony Burgess). 3. An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises. 4. A statement contrary to received opinion.

The American Heritage Dictionary

A paradox is a situation in which, alongside one opinion or interpretation, there is another, mutually exclusive one.

From The Words of Mathematics,
Steven Schwartzman
MAA, 1994

Representatives and direct taxes shall be apportioned among the several States which may be included within this Union, according to their respective numbers, ... The number of Representatives shall not exceed one for every thirty thousand, but each State shall have at least one Representative ..."

So states Article 1, Section 2 of the Constitution of the United States (1787). The rich history of attempts to comply with the direction of the founding fathers supplies an unequivocal proof that some things are easier said than done. If anything, the controversy surrounding this particular constitutional item moved past its bicentennial anniversary in a stride with no signs of abating.

The problem of actual assignment of the whole number of seats in the United States Congress among necessarily fractional numbers of relative state populations is known as the problem of Apportionment. The Constitution gave Congress three years to come up with a practical solution. Two were submitted right away. One by Alexander Hamilton, the other by Thomas Jefferson. After heated deliberations, Congress opted for the former, but in the very first exercise of the veto power by President of the United States, President Washington blocked the measure. Facing a dangerous stalemate, Congress then adopted Jefferson's method, but with a different number of seats. (It is said that Washington was motivated by the fact that Jefferson's method would allocate one additional seat to his home state of Virginia.)

Other apportionment methods were submitted by Rep. William Lowndes (1822), Rep. (and, by that time, former President) John Quincy Adams (1832) and Senator Daniel Webster (1832). The latter was eventually adopted in 1842, only to be replaced by Hamilton's method in 1852.

In 1872, Congress was considering the idea of expanding to 283 seats, because of the agreement of Hamilton's and Webster's methods for this number. As it happened, Congress added 9 additional seats without officially adopting any method. In 1876 Rutherford B. Hayes became President of the United States based on the apportionment (which some say was unconstitutional) of 1872. A hundred years later, M. L. Balinski and H. P. Young showed that had the 1876 elections followed Hamilton's method, Samuel Tilden would have been elected instead. (Nonetheless, as we shall see shortly, President Hayes has all the reasons to rest in peace.)

In 1880, to everyone's surprise a flaw was discovered in Hamilton's method that is now known as the Alabama paradox. More cracks showed up later. Close to 1900 Hamilton's method was shown to lead to the Population paradox and in 1907, on the occasion of Oklahoma joining the Union, the New-States paradox was also detected.

Hamilton's method was replaced by Webster's in 1901, which stayed put until 1941, when Huntington-Hill's method was signed into law by President Roosevelt. Huntington-Hill's method has been used since, although there is much activism going on with the goal of replacing it back with Webster's method.

In addition to being subject to the irregularities traditionally known as paradoxes (Alabama, population, new-states), three other points set Hamilton's method aside from other apportionment methods mentioned above:

  1. Hamilton's method is the only one that requires fixing the number of seats. All other methods work as well when what is fixed is the ratio of population per seat. Indeed, up until 1842, when Hamilton's method has been adopted by Congress, Jefferson's method was used with such a fixed ratio.

  2. All apportionment methods, but Hamilton's, violate the Quota Rule if used with the number of seats fixed. However, by the tradition established after 1842, Congress fixes the number of seats up front, with 435 seats being the norm since 1931.

The result known as Balinski and Young's (1982, see [Tannenbaum, p. 140], or in a little different form [Hoffman, p. 270]) impossibility theorem states that

Any apportionment method that does not violate the quota rule must produce paradoxes, and any apportionment method that does not produce paradoxes must violate the Quota Rule.

However, the theorem must be qualified by a rather obvious remark that the latter is only true when the number of seats to be apportioned is fixed up front. It could be argued that none of the methods at hand violates the Quota Rule when used with a fixed ratio of population per seat, in which case the total number of seats becomes a calculated quantity.

For the completeness sake, let me mention that, due to the population growth and incorporation of new states, the number of Representatives in the House has kept changing: 65 (1787), 105 (1791), 240 (1832), 223 (1842), 234 (1852), 292 (1872), 325 (1882), 386 (1901), 433 (1911), 435 (1931 and since, except when Alaska and Hawaii became states, there was a temporary addition of one seat for each until the apportionment following the 1960 census). An anecdotal event took place in 1901 when The Bureau of the Census submitted to Congress tables showing apportionment based on Hamilton's method for all size Houses between 350 and 400. In another demonstration of the Alabama paradox, for all sizes between 350 and 400, but 357, the state of Colorado would get three seats, with only two seats in the exceptional case of 357. The opportunity was seized by Rep. Albert Hopkins (IL), at the time the chairman of the House Committee on Apportionment, who submitted a bill to fix the House size at 357 seats. In a display of rationality, the bill was defeated.

Let's step a little back. I just mentioned that three points set Hamilton's method aside from other apportionment methods. The third one is in fact the most fundamental:

  1. It's well known that, when the number of seats is fixed in advance, all other methods may occasionally fail. Hamilton's method never fails.

This feature that on the first glance might have placed Hamilton's method way ahead of the pack, should have sealed its fate from the very beginning: paradoxically, the method is unsuitable for apportionment of the House seats. I think the method violates the basic tenet of the Constitution: ... according to their respective numbers ... Strict proportionality of the representation could be hardly achieved or expected, and hence meant. However, no one would ever question the principle of the equality of the representation: states with equal populations must have equal representations. When other methods can't achieve that they fail. Not so Hamilton's method. The latter will easily distribute 3 seats between 2 states with equal populations. One will get two seats, the other will get only one. An example of that phenomenon is below.

(How to use the applet.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

The deficiency is obvious and should have disqualified Hamilton's method from the outset as unconstitutional. It's a great surprise that it did not. As seen from the examples below, the same feature of the method that permits absurd apportionments, like the above, also leads to other "paradoxes."

Balinski and Young's theorem might be reformulated as

Provided the number of seats is fixed in advance of the census, all Constitution compliant apportionment methods violate the Quota Rule.

(Under these circumstances, it's paradoxical to have the size of the House determined arbitrarily prior to the census.)

Alabama Paradox

(How to use the applet.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

The Alabama paradox occurs when an increase in the total number of seats causes a state to lose one of its seats. To see how this may happen, increase the total number of seats from 24 to 25.

Population Paradox

(How to use the applet.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

The population paradox occurs when a state with a higher grows rate loses a seat to a state with a lower growth rate, when the apportionment is recalculated on the basis of new figures. (Compare the starting position above with that shown in the Alabama Paradox section. If you played with applets and caused the starting position to change, reload the page.)

New-States Paradox

(How to use the applet.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

The new-states paradox occurs when addition of a new state with a parallel increase in a fair amount of seats affects apportionment of other states. (Compare the starting position above with that shown in the Alabama Paradox section. If you played with applets and caused the starting position to change, reload the page.)

Quota Rule

All apportionment methods include calculation of the state's standard quota

State's Standard Quota = (State's population)·(Total number of seats)/(Total population)

The Lower Quota is then computed as the integral (floor) part of the standard quota. The Upper Quota is the ceiling (the lower quota plus one) of the standard quota.

The Quota Rule stipulates that any fair apportionment should assign to every state either its lower or upper quota.

Fair Division

It could be argued that none of the methods at hand violates the Quota Rule unless the number of seats is fixed. Which is quite true, too. But, after a short introduction, I shall only apply Webster's and Huntington-Hill's methods to the real 2000 Census data.

In a well known puzzle, a father willed to his three sons 17 camels with the proviso that 1/2 of the inheritance should go to the oldest among them, with 1/3 being due to the middle one and 1/9 to the youngest. Shortly after the father's death, a wise man riding on his camel through the village noticed the three brothers in quandary. He added his camel to 17, thus getting a herd of 18 animals. He gave 1/2 of these (i. e. 9 camels) to the oldest brother, 6 (= 18/3) to the middle one, and 2 (= 18/9) to the youngest. 1 camel remained (1 = 18 - 9 - 6 - 1), which he climbed up and rode away. To the great satisfaction of all brothers, each of them received more than was willed by their father.

The above situation is very common in problems of fair division (see [Tannenbaum, Ch. 3] or [Steinhaus, p. 67] when, even after the "goods" has been divided among several "players" to their satisfaction, some goods have been left over. See, for example, the Method of Sealed Bids or Method of Markers.

Let Pi, i = 1, ..., 50, denote ith state's population, P = P1 + ... + P50, and let S be the desired number of House Representatives. According to Webster's (unmodified) method, state #i should receive Ni seats, where

Ni = round(S·Pi/P).

Let N = N1 + ... + N50. The perfect match S = N is plausible. In any event, the difference |S - N| could not be large. If N < S, and no further action is taken, then, as in the above story, each state receives no less than it might have been expecting had the starting number of seats been N, and all thus must be satisfied. If N > S, (S = 425, N = 426 and S = 429, N = 430 below) the solution is to try another S, and, as the table below demonstrates, there's no fear that a suitable S may be hard to find.

Using the 2000 Census data (xls or html), it's a trivial matter to verify that for S = 435, N = 433. The nice feature of 433 is that N = 433 even when S = 433. So that 433 seems to be a more reasonable number than 435. If S = 433 were adopted, California and North Carolina would lose 1 seat each. The apportionment of all other states would not change.

Here's a short table of pairs S, N:


There exists a group of numbers (426, 427, 428, 430, 431, 432, 433, 439) around 435 for the size of the House that, in a sense, would satisfy the letter of the Constitution better than 435. After a long gap, the next number with N = S is 459. As the population changes, the calculations are likely to produce different results.

For Huntington-Hill's method the numbers differ. The equality N = S, as well as the inequality N > S, appears more frequently:


The list of the preferred House sizes for the same range is now longer: 427, 428, 432, 433, 436, 437, 441, 443, 444, 445.

The conclusion that the best way to do what the Constitution requires is to let the number of seats be calculated and not fixed up front seems very natural. The combination of this approach with Webster's or Huntington-Hill's method could never cause the paradoxes nor violate the Quota Rule. The argument is rational. However, the political considerations that drive the decision making process in the House are not necessarily so.

(Years after writing this piece I came across a 1-page article in the 1939 American Mathematical Monthly relevant to the thoughts expressed above.)


  1. P. Hoffman, Archimedes' Revenge, W. W. Norton & Company, 1988
  2. H. Steinhaus, Mathematical Snapshots, umpteen edition, Dover, 1999
  3. G. Szpiro, Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present, Princeton University Press, 2010.
  4. P. Tannebaum & R. Arnold, Excursions In Modern Mathematics, 4th edition, Prentice Hall, 2001

Hamilton's Method

The given total number of seats is to be apportioned between several states proportionally to their populations. To accomplish that task according to A. Hamilton,

  1. Compute the divisor D = (Total population)/(Number of seats)
  2. Find and round down state quotas {(State population)/D}. The leftover fractional parts add up to a whole number of seats.
  3. Distribute the surplus seats, one per state, starting with the largest leftover fractional part, then proceeding to the next largest, and so on, until all the surplus seats have been dealt with.

How to use the applets

Bold numbers could be clicked upon. To increase the number, click to the right of its vertical center line. To decrease it click to the left of the line. Dragging the mouse near the center line will accomplish the same task, but faster.

Related material

  • Student's Social Choice
  • Adams' Apportionment Method
  • Banzhaf Power Index Calculator
  • Fair Division: Method of Lone Divider
  • Fair Division: Method of Markers
  • Fair Division: Method of Sealed Bids
  • Fair Division: Method of Sealed Bids II
  • Fast Power Indices
  • Five Methods of Apportionment
  • Four Voting Methods
  • Hamilton's Apportionment Method
  • Huntington-Hill Apportionment Method
  • Jefferson's Apportionment Method
  • Method of Markers II
  • Shapley-Shubik Power Index Calculator
  • Voting Methods and Social Choice
  • Webster's Apportionment Method
  • Weighted Voting and Power Indices
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