In a round-robin tournament with 6 players, each player plays every other player exactly once. A win is worth 2 points, a draw is 1 point each, and a loss is 0 points. After the tournament: Player A has 10 points, Player B has 8 points, and Player C has 6 points. Players D, E, and F each have the same score. What score do D, E, and F each have? Show your reasoning.

Judge: Correct answer of 2 points each with elegant reasoning: calculates total games via C(6,2)=15, derives total points as 30, subtracts known scores (24), and divides remainder equally. Clean, complete, no gaps.

Judge: Correct answer (2 points each) with elegant, complete mathematical reasoning. Correctly calculates total games (15), total points (30), subtracts known scores (24), and divides remaining (6) by 3. Clean step-by-step presentation that's easy to follow.

Judge: Correct answer (2 points each) with elegant mathematical reasoning. Correctly computes total games (C(6,2)=15), total points (30), sets up the equation 24+3x=30, solves for x=2. Then goes further to verify feasibility by constructing a valid tournament outcome (A beats all, B beats all except A, C beats D/E/F, then D/E/F play among themselves). This verification step elevates the answer from good to excellent.

Judge: Correct answer of 2 points each with elegant reasoning. First establishes total points (15 games x 2 = 30), then subtracts known scores to get D+E+F = 6, dividing by 3. Then goes further to verify the construction: A won all 5, B won 4 (lost to A), C won 3 (lost to A and B), and D/E/F drew all their matches among themselves. Complete and rigorous.

Judge: Correct answer (2 points each) with elegant reasoning. Correctly computes total games (C(6,2)=15), total points (30), then solves the simple equation. Verification step confirms the answer is within valid bounds. Clean mathematical presentation. Shows reasoning as requested.

Judge: Correct answer (2 points each) with elegant reasoning. Properly calculates total games (15), total points (30), sets up the equation, solves cleanly. Includes verification step. The mathematical presentation is crisp and well-structured.

Judge: Correct answer (2 points each) with elegant reasoning. Cleanly calculates total games (C(6,2)=15), total points (15x2=30), subtracts known scores (24), and divides remainder (6) by 3 players. The mathematical reasoning is airtight and clearly presented with proper notation.

Judge: Correct answer (2 points each) with elegant reasoning. Immediately identifies the key insight: each game distributes exactly 2 total points, 15 games = 30 total points, solve for x. The algebra is clean and the verification step adds confidence. Also checks that the score is achievable (1 win + 3 losses or 2 draws + 3 losses), though the response is truncated before completing that check.

Judge: Response is truncated before reaching an answer. Sets up the problem correctly (15 games, known scores sum to 24) but never completes the calculation. The reasoning starts going off track in Step 5-6 with confused discussion of 'remaining points' and incorrect assertion that Player A 'must have won all his matches' (10 points from 5 games = 5 wins is correct, but the logic trail is abandoned mid-thought).