Problem 5 · 2004 AMC 8
Easy
Logic & Word Problems
elimination-bracket
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
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Answer: D — 15 games.
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Hint 1 of 2
Don't try to draw the bracket round by round. Flip the question: instead of 'how many games?', ask 'how many teams must disappear?' Each game removes exactly one team.
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Hint 2 of 2
This is the single-elimination principle: to crown one champion out of N teams, exactly N − 1 teams must lose — and each game produces exactly one loser, so games = N − 1. The bracket's shape never matters.
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Approach: count losers, not games
- Every game knocks out exactly one team. The tournament ends with 1 champion, so 16 − 1 = 15 teams had to be eliminated.
- One game per elimination ⇒ 15 games — no bracket-drawing needed.
- Worth keeping: any 'games to find a single winner in single-elimination' = (teams − 1). A 64-team bracket takes 63 games; a 128-team draw takes 127. The answer never depends on byes or how the rounds line up.
Another way — add up the rounds (the long way):
- 16 teams → 8 games → 8 teams → 4 games → 4 teams → 2 games → 2 teams → 1 game.
- 8 + 4 + 2 + 1 = 15. Same answer — and notice it equals 16 − 1, confirming the shortcut.
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