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Chapter 6: Problem 72

Determine the number of matches played in a single-elimination tournament with n players, where for each game between two players the winner goes on, but the loser is eliminated.

Short Answer

Expert verified
The number of matches is \(n-1\).

Step by step solution

01

- Understand the Single-Elimination Concept

In a single-elimination tournament, each match eliminates one player, with the winner advancing to the next round.
02

- Identify the Number of Matches in the Final Round

In the final round, there are two players, and one match determines the winner. So, there is 1 match in the final round.
03

- Determine Matches Leading Up to the Final

Every round before the final must also eliminate half of the remaining players until only one player (the winner) remains. Each round contains \(\frac{n}{2}\) matches if \(n \) is even.
04

- Generalize to Any Number of Players

Consider that to find the tournament champion, every player except the winner must lose exactly once. Therefore, with \(n \) players, there will be \(n-1 \) eliminations, which means \(n-1 \) matches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Match Elimination
In a single-elimination tournament, the concept of match elimination is straightforward: each match results in one player advancing and one player being eliminated. The winner of the match moves on to the next round, while the loser is out of the tournament. This process continues until only one player remains, who is declared the champion. Each match effectively reduces the number of participants by one, making it easy to determine how many matches are needed overall. For instance, with 8 players, the total number of eliminations required to determine a winner is 7, because one player will lose in each match leading up to the final.
Tournament Rounds
A single-elimination tournament is organized in rounds. Each round halves the number of participants until a final round determines the champion. The first round has the full number of participants, but only half advance to the next round. Let鈥檚 break it down:
  • With an even number of participants, say 8, the first round will have 4 matches, eliminating 4 players.
  • The second round then has 4 remaining players, which leads to 2 matches, eliminating 2 players.
  • The third round (often called the semifinal) then has 2 matches, with the winners going on to the final.
Thus, in each of these rounds, the number of matches equals half the number of participants at the beginning of that round. This continues until we reach a single final match.
Final Match
The final match in a single-elimination tournament is critical as it's the decider of the champion. At this point, the two remaining players compete against each other, and only one can emerge victorious. The final is essentially the culmination of all the previous eliminations. To understand its significance, here's a breakdown:
  • By the time players reach the final, they have each won several matches, proving their skill and determination.
  • The final match is often the most anticipated and watched game of the tournament since it determines the overall winner.
  • As there are only 2 players left, this round consists of just 1 match.
Winning this final match is the ultimate goal for every participant, as it crowns them the champion of the tournament. Thus, understanding how the final match operates and its place in the structure of the tournament helps appreciate the competitive journey of the participants.

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