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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!

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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?

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In the map above, the Blue party wins none of the five voting regions. If voting regions must contain 5 connected squares, what is the **most** voting regions the Blue party could win?

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

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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.

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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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