Riddler Nation has many, many districts, each with the same population. Each district elects one representative (from one of the two major political parties) to the Riddler Nation Legislative Body by majority vote within the district.
However, after recent decisions by the highest court in the land, the redistricting process has resulted in districts that appear to favor one party over the other.
What is the greatest percentage of Riddler Nation’s total population that can vote for one of the parties while resulting in that party not having a majority of seats in the Legislative Body?
Let X and Y represent the two political parties.
In order to maximize Party X's population:
The following table provides some examples.
| # of Districts | Favoring X | Favoring Y | Population of X | Population of Y |
|---|---|---|---|---|
| 2 | 1 | 1 | ~75% | ~25% |
| 4 | 2 | 2 | ~75% | ~25% |
| 6 | 3 | 3 | ~75% | ~25% |
| ... | ... | ... | ... | ... |
| $2n$ | $n$ | $n$ | $\dfrac{n + \dfrac{n}{2}}{2n} = \dfrac{3n}{4n}$ | $\dfrac{\dfrac{n}{2}}{2n} = \dfrac{n}{4n}$ |
Note that Y can win the majority and the population distribution approaches the same value.
| # of Districts | Favoring X | Favoring Y | Population of X | Population of Y |
|---|---|---|---|---|
| 3 | 1 | 2 | ~66.67% | ~33.33% |
| 5 | 2 | 3 | ~70% | ~30% |
| 7 | 3 | 4 | ~71.43% | ~28.57% |
| ... | ... | ... | ... | ... |
| $2n+1$ | $n$ | $n+1$ | $\dfrac{n + \dfrac{n+1}{2}}{2n+1} = \dfrac{3n+1}{4n+2}$ | $\dfrac{\dfrac{n+1}{2}}{2n+1} = \dfrac{n+1}{4n+2}$ |
The greatest percentage is slightly less than 75%. That is, slightly less than 75% of the Riddler Nation population can vote X and still lose the majority, given the above districting criteria.