A tennis tournament involves 1,024 players in a single-elimination format (the winner advances, the loser is eliminated). How many games do you need to play to determine the champion? Don't use formulas, think about the logic of the problem.
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The elimination tournament
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- In a knockout tournament: Each match eliminates exactly 1 player.
- In a knockout tournament: To determine 1 champion out of 1024 players, we must eliminate 1023 players.
- Total: 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 1023
Solution
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Answer: 1023 matches.
Elegant reasoning:
In a knockout tournament:
- Each match eliminates exactly 1 player
- To determine 1 champion from 1024 players, we must eliminate 1023 players
- Therefore, we need exactly 1023 matches
Verification by levels:
- Round 1: 512 matches (512 left)
- Round 2: 256 matches (256 left)
- Round 3: 128 matches (128 left)
- Round 4: 64 matches (64 left)
- Round 5: 32 matches (32 left)
- Round 6: 16 matches (16 left)
- Round 7: 8 matches (8 left)
- Round 8: 4 matches (4 left)
- Round 9: 2 matches (2 left)
- Round 10: 1 match (1 left)
Total: $512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 1023$
Or simply: $1024 - 1 = 1023$
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