Question

Consider a knock-out women's badminton singles tournament where there are no ties. The loser in each game is eliminated from the tournament. Every player plays until she is defeated or remains the last undefeated player. The last undefeated player is declared the winner of the tournament. If there are 64 players in the beginning of the tournament, how many games should be played in total to declare the winner of the tournament?

• A. 127
• B. 64
• C. 63
• D. 32

Hint:

In a knock-out event with no ties, exactly one player is eliminated per game, and all but the champion must eventually be eliminated.

Answer 1

The Correct Option is C

Think of the knock-out draw as a complete binary tree with 64 players placed at the leaves. Each match corresponds to one internal node of the tree, where two players, or two sub-tournament winners, meet and one advances upward while the other is removed from the tree.

Building the tree from the bottom up, each match merges two nodes into one, reducing the total node count by exactly 1 per match, until only the single root, the champion, remains. Starting from 64 separate players and ending at 1 champion is a net reduction of $64-1=63$, and since every match causes a reduction of exactly 1, exactly $63$ matches must be played.

This matches the direct elimination-counting argument: the winner never loses, so all other $63$ players must lose precisely once, and each match produces precisely one loss, giving $63$ matches in total.