1. **State the problem:** We have a preference table with three columns representing vote counts 8, ?, and 4. The candidates are A, B, and C ranked in each column as follows: | Votes | First Choice | Second Choice | Third Choice | |-------|--------------|---------------|--------------| | 8 | A | B | C | | ? | B | A | C | | 4 | A | C | B | We want to find the minimum missing vote count $x$ (the "?" column) such that candidate A wins by the Borda count method, but the majority criterion is violated. The majority criterion is violated if candidate B has more than 12 first-place votes (i.e., $x > 12$). 2. **Recall the Borda count method:** - Each voter ranks candidates. - For 3 candidates, points are assigned as follows: - 1st place: 2 points - 2nd place: 1 point - 3rd place: 0 points 3. **Calculate Borda points for each candidate:** Let $x$ be the missing vote count. - Candidate A: - From 8 votes: 1st place → $8 \times 2 = 16$ - From $x$ votes: 2nd place → $x \times 1 = x$ - From 4 votes: 1st place → $4 \times 2 = 8$ - Total: $16 + x + 8 = 24 + x$ - Candidate B: - From 8 votes: 2nd place → $8 \times 1 = 8$ - From $x$ votes: 1st place → $x \times 2 = 2x$ - From 4 votes: 3rd place → $4 \times 0 = 0$ - Total: $8 + 2x + 0 = 8 + 2x$ - Candidate C: - From 8 votes: 3rd place → $8 \times 0 = 0$ - From $x$ votes: 3rd place → $x \times 0 = 0$ - From 4 votes: 2nd place → $4 \times 1 = 4$ - Total: $0 + 0 + 4 = 4$ 4. **Condition for A to win:** Candidate A must have more points than B and C: $$24 + x > 8 + 2x$$ Simplify: $$24 + x > 8 + 2x$$ $$24 - 8 > 2x - x$$ $$16 > x$$ So, $x < 16$. 5. **Condition for majority criterion violation:** Candidate B has more than 12 first-place votes: $$x > 12$$ 6. **Combine conditions:** $$12 < x < 16$$ The minimum integer $x$ satisfying both is $x = 13$. 7. **Verify with $x=13$:** - A's points: $24 + 13 = 37$ - B's points: $8 + 2 \times 13 = 8 + 26 = 34$ - C's points: $4$ A wins with 37 points. B has $13$ first-place votes, which is more than 12, so the majority criterion is violated. **Final answer:** The minimum missing vote count is $\boxed{13}$.