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?
Show Hint
Each game eliminates exactly one player, and the tournament ends with exactly one player left undefeated.
Step 1: Set up the rounds.
With 64 players, the first round pairs them into $64/2=32$ matches, so 32 players survive to round 2.
Step 2: Continue halving round by round.
Round 2 has $32/2=16$ matches. Round 3 has $16/2=8$ matches. Round 4 has $8/2=4$ matches. Round 5 has $4/2=2$ matches. Round 6, the final, has $2/2=1$ match.
Step 3: Add up every round's matches.
\[ 32+16+8+4+2+1 \]
Adding these step by step: $32+16=48$, then $48+8=56$, then $56+4=60$, then $60+2=62$, then $62+1=63$.
Step 4: Cross check with a shortcut.
A single elimination bracket always needs one fewer game than the number of starting players, because every game removes exactly one player and only one player, the champion, is never removed. So the total is $64-1=63$, matching the round by round sum.
Step 5: Rule out the distractors.
64 would mean an extra pointless game is played after a winner is already found. 32 is just the first round alone. 127 does not match a simple knock-out format at all.
