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Chapter 1: Problem 44

Eight teams are competing in a volleyball tournament. Any team that loses a game is eliminated from the tournament. How many games must be played to determine the tournament winner?

Short Answer

Expert verified
The tournament must hold seven games to determine the winner. One team remains undefeated and seven teams are eliminated, each elimination happening after a game, thus seven games.

Step by step solution

01

Understanding the scenario

Eight teams are participating in a tournament. To proceed in the tournament, a team needs to win its match. So, whenever a match is played, one team advances to the next round, and one gets eliminated. In other words, the outcome of one game will eliminate exactly one team.
02

Calculating the number of games

We start with eight teams. After the first round of matches, half of the teams (four) would be eliminated, leaving only four teams. This ends the first round. In the second round, two more teams would be eliminated, leaving two teams. This concludes the second round. The final round sees the last two teams face off, and one more team is eliminated, leaving the champion. So three rounds of games are played.
03

Summarizing the total number of games

Each of the three rounds eliminates half of the remaining teams. So, for eight teams, seven matches must be played to decide the winner. At each stage, half of the teams are eliminated. So, the number of games would be the number of teams minus one (because one team remains undefeated).

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Key Concepts

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

Elimination Process
In any tournament that involves an elimination process, the objective is to systematically remove competitors until a single champion remains. In our volleyball tournament example, any team that loses a game is immediately out of contention. This format is quite common in sports and ensures that only the most consistent team progresses to win the tournament.
  • Each game results in exactly one team being eliminated.
  • The overall number of eliminations is determined by the number of teams minus one, as one will emerge as the winner.
This method is straightforward and allows the quick progression of a tournament, making it ideal for situations where time or participant availability is limited.
Volleyball Tournament
A volleyball tournament typically involves a series of matches between competing teams. When structured using the elimination process, each match acts as a crucial decider for the teams, putting considerable pressure on performance.
  • Teams must constantly win to stay in the running for the championship.
  • With every match, the stakes are higher because losing means the end of the journey.
Such tournaments not only test the physical prowess of the teams but also their mental fortitude, as they have to handle the stress of potential elimination.
Game Counting
Counting the number of games necessary in an elimination tournament is easier than it might seem at first glance. The key point is to remember that every match results in one team being knocked out of the tournament. Therefore, the number of games needed to determine a winner is directly related to the number of teams participating.
  • For any tournament, the formula is simple: Number of games = Number of teams - 1.
  • This is because only one team emerges undefeated at the end.
In our eight-team tournament, this means that seven games are required, since eight teams minus one leaves us with seven games.
Rounds Calculation
Calculating the number of rounds in an elimination-based tournament helps in understanding the progression of matches over time. Each round effectively halves the number of participants until only one winner remains.
  • In any given round, half of the teams that played are eliminated, while the others go on to the next round.
  • For a tournament starting with 8 teams, one might observe the following:
  • First round: 8 teams, 4 games, 4 eliminated, 4 advance.
  • Second round: 4 teams, 2 games, 2 eliminated, 2 advance.
  • Final round: 2 teams, 1 game, 1 eliminated, 1 winner.
Understanding the breakdown of rounds provides clarity on the overall structure and pace of the tournament.

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