To apportion is to divide into parts according to some rule, the rule typically being one of proportion. Certain quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot—only whole numbers will do. In the latter case, there is an inherent tension between our desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values. This results, at times, in unintuitive observations, or paradoxes.

Several paradoxes related to apportionment, also called fair division, have been identified. In some, such as the case of the dying farmer and his horses (one of the horse paradoxes), the resolution is simple—the farmer didn't apportion all of his horses. Others, such as those relating to the United States House of Representatives, boggle the mind and shake our faith in mathematics, which, we have been brought up to believe, always tells the truth. The Alabama paradox was discovered in 1880, when it was found that increasing the number of seats would decrease Alabama's share from 8 to 7. There was more to come—in the 1900s, Virginia lost a seat to Maine as a result of its population growing faster than Maine's. When Oklahoma became a new state in 1907, a recomputation of apportionment showed that the number of seats due to other states would be affected even though Oklahoma would be given no more or no less than its fair share of seats and the total number of seats increased by that amount.

The method for apportionment used during this period (put forth by Hamilton) was as follows: first, the fair share of each state is computed (the fair share is the share that the state would get if fractional values are allowed). Next the share of each state is rounded down; this results in some "left over" seats. These seats are allocated to the states whose fair share exceeds the rounded-down number by the highest amount.

One might expect that the abundance of paradoxes is perhaps due to some deficiency of Hamilton's method. Indeed, a number of schemes have been proposed and four different methods signed into law (five counting repetitions). Amusingly, this vacillation has had less to do with mathematical than political considerations, such as the total number of seats that each party would be alloted by a given method. No method, however, has been found perfectly satisfactory in practice. It should therefore come as no surprise that in 1982, two mathematicians Balinski and Young developed a theory showing that any method of apportionment will result in paradoxes.

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy