In a political system, four seats are elected in single-member districts with a plurality. There are 44 voters: 33 Dots and 11 Stars. The number of seats that Stars win depends on the district structure. The only restriction is that every district must have exactly 11 voters. You have five options: check if such an option is possible or impossible.

Question

In a political system, four seats are elected in single-member districts with a plurality.  There are 44 voters: 33 Dots and 11 Stars.  The number of seats that Stars win depends on the district structure.  The only restriction is that every district must have exactly  11 voters.  You have five options: check if such an option is possible or impossible.

BlackTom AI · Accepted Answer

The problem has 4 single-member districts, each with 11 voters, and there are 33 Dots and 11 Stars in total. In a district, the winner is the candidate with more votes; with 11 voters, a winner must have at least 6 votes in that district.

Option 1: Dots can win exactly four seats.
- If Dots win all four districts, Stars win zero districts. This is possible because Stars could concentrate their 11 votes in a way that they do not secure a majority in any district, for example distributing Stars as (3,3,3,2) across the four districts. In each district, Dots would have the remaining votes (8,8,8,9 respectively), ensuring Dots win every district. Therefore, four Dots seats is possible.

Option 2: Dots can win exactly zero seats.
- This would mean Stars win all four districts. To win a district, Stars must have at least 6 votes in that district. Since there are 4 districts, Stars would need at least 6 × 4 = 24 Stars in total, which far exceeds the available 11 Stars. Hence, Stars cannot win every district, and Dots cannot win zero seats. This scenario is impossible.

Option 3: Dots can win exactly two seats.
- If Dots win two districts, Stars would have to win the other two. To win a district, Stars need at least 6 votes in that district, so Stars would need at least 6 × 2 = 12 Stars, which is already more than the total 11 Stars available. Therefore, this distribution is impossible.

Option 4: Dots can win exactly one seat.
- Here, Stars would win three districts. Again, Stars would need at least 6 votes in each winning district, totaling at least 6 × 3 = 18 Stars, which is impossible with only 11 Stars. So this arrangement cannot occur.

Option 5: Dots can win exactly three seats.
- If Dots win three districts and Stars win one, Stars only need to win 1 district, which requires at least 6 Stars in that district. The remaining three districts can be won by Dots with distributions like (6–5) or (5–6) in the respective districts to allocate the remaining votes. A concrete example is: one district with Stars 6, Dots 5 (Stars wins), and the other three districts with Stars totaling 5 distributed as (2,2,1) so Dots have (9,9,10) respectively, giving Dots three wins total. This is consistent with the total counts (Stars: 11, Dots: 33) and thus is possible.

Summary of feasibility per option:
- Dots win exactly four seats: possible.
- Dots win exactly zero seats: impossible.
- Dots win exactly two seats: impossible.
- Dots win exactly one seat: impossible.
- Dots win exactly three seats: possible.

VPM Spring 2025 VPM Practice Final

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