Consider an election with 421 votes

a) If there are 4 candidates, what is the smallest number of first-place votes a candidate could win with under the Plurality method?

To find the smallest number of first-place votes a candidate could win with under the **Plurality method**, we assume the worst-case scenario for the distribution of votes among the 4 candidates.

In the **Plurality method**, the winner is the candidate with the most first-place votes, even if they do not have an absolute majority.

### Solution:
To minimize the number of first-place votes a winning candidate could get, we distribute the votes among the 4 candidates as evenly as possible. 

#### Total votes: 421
We want to distribute these votes such that the winner has the fewest first-place votes.

1. Assume three candidates (Candidate 2, 3, and 4) receive as many votes as possible, just under the maximum the winner can have.
   - If we divide 421 votes among 4 candidates as evenly as possible:  
     \[
     \frac{421}{4} \approx 105.25
     \]
   - So, to minimize the winner's votes, we give three candidates 105 votes each.
     - This totals \( 105 + 105 + 105 = 315 \) votes.
2. The remaining votes for the fourth candidate (Candidate 1, the winner) will be:
   \[
   421 - 315 = 106
   \]

Thus, **106** is the smallest number of first-place votes a candidate could win with under the Plurality method.

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Do you want more details on this solution or have any further questions?

Here are some related questions to consider:
1. How does the Plurality method compare with other voting methods like Instant Runoff or Borda Count?
2. What is the smallest number of votes required to win in an election with 5 candidates?
3. How would the result change if there were more than 421 votes?
4. What would be the result if one candidate had 100 votes and the others had an equal share of the remaining votes?
5. How could a tie occur in a Plurality system with 4 candidates?

**Tip:** In plurality voting, even a small advantage in votes can be enough to win, so strategic distribution of votes matters a lot.

If there are 4 candidates, what is the smallest number of first-place votes a candidate could win with under the Plurality method?

Discover how to calculate the smallest number of first-place votes a candidate could win with in a 4-candidate election under the Plurality method. This problem illustrates vote distribution and election theory, ideal for Grades 9-12 students.

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