Determine the maximum number of players across teams who can win all their games in a round-robin tournament | Step-by-Step Solution

Understanding the Shogi Tournament Problem

What We're Solving: We need to find the maximum number of players who can have a perfect record (win every single game they play) in a team tournament where N teams each have M players, and teams play against each other in a round-robin format.

The Approach: This is a beautiful logic puzzle! We need to think about what it means for a player to have a "perfect record" and then figure out the constraints. The key insight is understanding that when two teams play each other, not every player from each team can win - there have to be some losers too!

Step-by-Step Solution:

Step 1: Understand the tournament structure

• We have N teams, each with M players
• Teams play each other once (round-robin), so each team plays N-1 other teams
• When Team A plays Team B, players from Team A face players from Team B
• Each player plays exactly N-1 games (one against each other team)

Step 2: Think about what "perfect record" means

• A player with a perfect record wins ALL their games
• So they win against players from ALL other teams they face

Step 3: Crucial Insight When two teams play each other:

• Some players from Team A will win their matches
• Some players from Team B will win their matches
• But here's the key: NOT ALL players from both teams can win!

Step 4: Find the constraint In any match between two teams, at most M players total can win all their individual games in that match.

When Team A plays Team B:

• At most M players (from one team or split between teams) can win in this inter-team match
• For a player to have a PERFECT record, they must win against ALL other teams

Step 5: The key realization For a player to maintain a perfect record throughout the tournament, their team essentially needs to "win" against every other team they play. Since each team plays N-1 other teams, and we want to maximize perfect records:

The maximum occurs when exactly one team wins against all others, and all M players from that winning team can achieve perfect records.

The Answer: The maximum number of players who can achieve a perfect record is M (all players from one team).

Memory Tip: Think of it like this: "In any tournament, there can be at most one undefeated team, and all players from that team can be undefeated!" This connects to the familiar idea that in sports, typically only one team can go undefeated in a season.

Great job working through this logic puzzle! The key was recognizing that perfect individual records are tied to team performance in this format. Does this reasoning make sense to you?