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# Math of Voting

Just count the votes and see who has the most, right? Voting systems can actually be quite complex, and the puzzling mathematical paradoxes that arise from them may surprise you!

Three states are being allocated 62 seats in a legislative body. State A has 17% of the population, State B has 37%, and State C has 46%.

Applying these percentages to the 62 seats and rounding to the nearest integer means State A gets 11 seats, State B gets 23 seats, and State C gets 29 seats.

What’s wrong with this approach?

For example, if the voting regions were drawn like this, they would win 1:

A political score \(P\) is a number between 0 and 100 where \(P=0\) represents the extreme left-wing and \(P=100\) represents the extreme right-wing. A candidate's positions are also assigned a score, and each voter will vote for whichever candidate has positions with a score closest to their own.

In a two-candidate race where each candidate wants to get as many votes as possible, what position(s) will they take?

**Note:** In the case where two candidates are equidistant from a voter, assume the voter decides randomly.

There are four candidates in the beginning of an election: 1, 2, 3, and 4. Given the preferences of the four voters below, the only head-to-head race that **will not** end in a tie is between candidates numbered \(A\) and \(B.\) What is \(A \times B?\)

\[\begin{align}
\text{Alexa:} \ 1 &> 2 > 3 > 4 \\
\text{Benny:} \ 2 &> 1> 4> 3 \\

\text{Celina:} \ 3 &> 4 > 2 > 1 \\

\text{Dakota:} \ 4&>1>3>2 \\
\end{align}
\]

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