Problem 132 - Algebra and Trigonometry

Question

In a round-robin chess tournament, each player is paired with every other player once. The formula $$ N=\frac{x^{2}-x}{2} $$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve. In a round-robin chess tournament, 36 games were played. How many players were entered in the tournament?

Step-by-Step Solution

Answer

The tournament had 9 players.

Step 1: Substitute the given number of games into the equation

To find the number of players, we need to substitute the given value of \(N\) into the equation. Therefore, \(36 = \frac{x^{2}-x}{2}\).

Step 2: Simplify the equation

To simplify the equation, we need to get rid of the fraction, which we can do by multiplying the entire equation by 2. Therefore, the simplified equation becomes \(2 * 36 = x^{2} - x\), or \(72 = x^{2} - x\).

Step 3: Rearrange the equation into a format suitable for applying the quadratic formula

The quadratic formula can be applied to equations in the format of \(ax^{2} + bx + c = 0\). To achieve this, we can subtract 72 from both sides to get \(x^{2} - x - 72 = 0\).

Step 4: Solve the quadratic equation

The quadratic equation \(x^{2} - x - 72 = 0\) can be factored into \((x - 9)(x + 8) = 0\). Setting each factor equal to zero gives the solutions \(x = 9\) and \(x = -8\). However, the number of players cannot be negative, so we discard \(x = -8\).

Key Concepts

Round-robin tournament

A round-robin tournament is a popular format where every participant faces every other participant exactly once. This type of tournament ensures that all contestants have the opportunity to compete and there is a clear winner. This is different from knockout tournaments where a single loss can eliminate a player from the competition. In a round-robin setting:

• Each player faces every other player once.
• The total games played depend on the number of participants.
• The formula \( N = \frac{x^2 - x}{2} \) helps determine the number of games \( N \) when you know the number of players \( x \).

For example, in a chess tournament with four players, labeled A, B, C, and D, the pairings would be AB, AC, AD, BC, BD, and CD--resulting in 6 games total.

Factoring

Factoring is a mathematical process used to simplify equations and solve problems. In the context of quadratic equations, factoring involves expressing the quadratic expression as a product of two binomials. This is particularly useful when the quadratic equation can be solved by setting each binomial to zero.
Consider the quadratic equation \( x^2 - x - 72 = 0 \). This can be factored into:

• The product \((x - 9)(x + 8) = 0\).

This process involves determining two numbers that multiply to the constant term (-72) and add to the linear coefficient (-1). Once factored, each binomial can be set to zero, leading to potential solutions for \( x \). These solutions are crucial for determining possible outcomes, such as the number of participants needed in a tournament.

Chess games

Chess is a strategic board game that has been played for centuries and often features in round-robin tournaments. Each chess game is a battle of wits, requiring players to think several moves ahead, anticipate opponents' strategies, and adapt to changing circumstances on the board.
In the context of a round-robin chess tournament:

• Each match contributes to the overall standings of the tournament.
• The formula used helps predict the total number of games, providing logistical details for organizing the event.

For example, with 9 players, the formula calculates precisely 36 matches, allowing the tournament organizer to schedule accordingly.

Quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The solutions for \( x \) can be found using the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula is particularly useful when factoring is difficult or impractical. By substituting values of \( a \), \( b \), and \( c \) (coefficients from the quadratic equation) into the formula, one can determine the roots efficiently.
In the exercise context, the equation \( x^2 - x - 72 = 0 \) could also have been solved using this method, confirming \( x = 9 \) as the valid solution. Utilizing this formula ensures accuracy and provides a systematic approach to finding answers.