11.5), and then round to the nearest even integer. Using this algorithm,
34.8 is rounded to 35, and 34.5 would be rounded to 34. This is an emi-
nently reasonable algorithm for rounding for the purposes of calculation,
but it encounters a problem when rounding for representation in the
House of Representatives.
Suppose that the United States consisted of the original thirteen colo-
nies. Twelve of these colonies each have 8 percent of the population; the
remaining colony has only 4 percent of the population. According to the
above calculation, each of the Big 12 is entitled to 34.8 representatives,
which is rounded by the elementary-school algorithm to 35 representa-
tives. The small colony gets only 17.4 representatives, which is rounded
down to 17. This procedure designates a total of 12 35 17 437 repre-
sentatives to a House that has room for only 435.
This might seem like a minor problem in number juggling, but the
presidential election of 1876 turned on the method by which these
numbers were juggled!^3 In that year, Rutherford B. Hayes won the elec-
tion by 185 electoral votes to 184 for his opponent, Samuel Tilden (who,
incidentally, won the popular vote by a convincing margin). Had the
rounding method used been different, a state that supported Hayes
would have received one less electoral vote, and a state that supported
Tilden would have received one more. This difference would have swung
the election.
The Alabama Paradox^4
The Founding Fathers recognized the importance of determining the
number of electoral votes that each state should receive; indeed, the
first presidential veto ever recorded occurred when George Washing-
ton vetoed an apportionment method recommended by Alexander
Hamilton. (Congress responded by passing a bill to utilize a rounding
method proposed by Thomas Jefferson.) Nevertheless, when the dis-
puted election of 1876 occurred, the Hamilton method, also known as
the method of largest fractions, was the one being used, having been
adopted in 1852.
To illustrate the Hamilton method, we’ll start by assuming that we are
going to assign representatives to a country that has four states, for a rep-
resentative body that has thirty-seven members. The following table gives
the percentage of the population residing in each of the four states, and
also the quota for each state, which is the exact number of representatives
to which the state is proportionally entitled.
228 How Math Explains the World