Writing at History News Network, George Szpiro, author of Numbers Rule, first gives us the bad news:

Invariably, the number of representatives that a state would be due according to the census, is a fractional number. So, should a state that is due, say, 7.38 representatives, send 7 or 8 congresspersons to the Capitol? Rounding the numbers up or down, according to whether the fractions are above or below 0.5, won’t do since, depending on how many states get rounded up or down, the total could be 434, 436 or some other number.

Why is this bad news? Well, further on he explains:

After a while, however, small states began to feel cheated, one reason being—there are others—that being rounded down from 3.5 hurts much more than being rounded down from 30.5 to 30.

So, what’s the solution? According to Szpiro:

The above problem and paradoxes, which have kept politicians and courts busy for over two hundred years, stem from the annoying fact that rational numbers (such as when a state’s population is divided by the total population, and the result is multiplied by 435) are nearly never integers (whole numbers). So let us think outside of the box! My suggestion: send fractional congresspersons to the Capitol.

Click over to read this light-hearted history of apparently impossible mathematics of apportionment.