A knockout tournament with 15 players

Question

Fifteen players enter a single-elimination tennis tournament: lose once and you are out, and the last player standing is the champion. Because 15 is not a power of two, some players will need byes in the early rounds. How many matches will be played in total? Find a way to answer without designing the draw.

Accepted Answer

14 matches. Count eliminations. Each match has exactly one loser, who is eliminated, and nobody is eliminated without playing a match. To get from 15 players to 1 champion, 14 players must be eliminated, so 14 matches are needed, regardless of how the byes are arranged. Why the draw does not matter. You could arrange the tournament as 7 first-round matches with one bye (8 remaining), then 4, 2, 1: 7 + 4 + 2 + 1 = 14. Or 3 first-round matches with 9 byes (12 remain), then 6, 3, then 1 match with a bye, then the final: 3 + 6 + 3 + 1 + 1 = 14. Any legal arrangement gives the same total, because …

A knockout tournament with 15 players

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