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

Step 1: Understand what each game does.
This is a single-elimination tournament with no ties, so every game played produces exactly one loser, who is immediately eliminated. No game eliminates more than one player, and no game ends without eliminating someone.
Step 2: Count how many players must be eliminated.
The tournament starts with 64 players and ends with exactly 1 winner, the last undefeated player. So the number of players who must be eliminated is \(64 - 1 = 63\).
Step 3: Relate eliminations to games played.
Since each game eliminates exactly one player and every eliminated player is eliminated by exactly one game, the total number of games played equals the total number of players eliminated, which is 63.
Step 4: Cross-check with a round-by-round count.
Round 1 has \(64/2=32\) games, round 2 has 16, round 3 has 8, round 4 has 4, round 5 has 2, and round 6 (the final) has 1. Adding these: \(32+16+8+4+2+1=63\), matching Step 3.
Step 5: Eliminate the wrong options.
64 is simply the number of players, not the number of games. 32 only counts the first round and ignores every later round. 127 does not correspond to any natural count for this 64-player knock-out format.
\[ \boxed{63} \]