Question

A tennis tournarnsnt has 342 players. A singlo match imolves 2 players. The winner of a match will play the winner of a match in the next round, wheress losers are eliminated from the toumament. The 2 players who have won all previous rounds play in the final game, and the wirner wins the tournament. What is the total number of matches needed to determine the winner? a. Here is one algorithm to answer this question. Compute \(342 / 2=171\) to get the number of pairs (matches) in the first round, which results in 171 winners to go on to the second round. Compute \(171 / 2=85\) with 1 left over, which results in 85 matches in the second round and 85 winners, plus the 1 left over, to go on to the third round. For the third round compute \(86 / 2=43,50\) the third round has 43 matches, and so on. The total number of matches is \(171+85+43+\ldots\) Finish this process to find the total number of matches. b. Here is another algorithm to solve this problem, Each match results in exactly one loser, so there must be the same number of matches as losers in the tournament. Compute the total number of losers in the entire tournament. (Hint: This isf't really a computation; it is a one-sentence argument.) c. What is your opinion on the relative clarity, elegance, and efficiency of the two algorithms?

Short answer

341 matches are needed to determine the winner. The second algorithm is more efficient.

Step-by-step solution

First Algorithm - Initial Calculation

For the first algorithm, the tournament starts with 342 players. The first round is calculated by dividing the number of players by 2, since each match involves two players. Thus, in the first round, there are \( \frac{342}{2} = 171 \) matches.

Second Round Matches

The first round produces 171 winners. The second round has these 171 players, so we calculate \( \frac{171}{2} \) to determine the number of matches. This gives 85 matches, with 1 player remaining without an opponent.

Third Round Matches

Adding the leftover player from the second round, we have 86 players moving to the third round. The number of matches is \( \frac{86}{2} = 43 \).

Fourth Round Matches

With 43 winners from the third round, 43 players proceed to the next round. Thus, \( \frac{43}{2} = 21 \) matches, with 1 player remaining.

Fifth Round Matches

Including the leftover player, 22 players compete in the fifth round, resulting in \( \frac{22}{2} = 11 \) matches.

Sixth Round Matches

11 winners progress to the sixth round, yielding \( \frac{11}{2} = 5 \) matches, and 1 player remaining.

Seventh Round Matches

Bringing the leftover player into the seventh round results in 6 players, leading to \( \frac{6}{2} = 3 \) matches.

Eighth Round Matches

The three winners from the seventh round compete in the next, resulting in \( \frac{3}{2} = 1 \) match, and 1 player left.

Ninth (Final) Round Matches

Including the leftover player, 2 players remain to compete in 1 final match. Thus, the total number of matches is calculated.

Total Number of Matches for First Algorithm

Add all the matches: \( 171 + 85 + 43 + 21 + 11 + 5 + 3 + 1 + 1 = 341 \). Thus, 341 matches are needed to ensure a single winner.

Second Algorithm Explanation

In the second approach, each match eliminates exactly one player. Since we start with 342 players and end with 1 winner, a total of 341 players must lose to leave one winner. Therefore, 341 matches are necessary.

Comparison of Algorithms

The first algorithm involves detailed step-by-step match calculations, while the second algorithm simply relates each match to an eliminated player. The second algorithm is more elegant and efficient as it provides a direct explanation for the number of matches based on the number of players to be eliminated.

Key concepts

Match Counting

When organizing a tournament, it becomes crucial to determine how many matches are needed to declare a winner. This is known as match counting, and it involves some straightforward logic. In our case, we start with 342 players.
Each match eliminates one player, meaning to crown a single winner, we need 341 players to lose. Therefore, a total of 341 matches are necessary. This approach provides clarity, as it immediately links the number of matches to the number of players being eliminated. It simplifies the process of counting matches by considering the core objective: eliminating players until only one remains.

Algorithm Efficiency

The elegance and speed at which an algorithm solves a problem are key indicators of its efficiency. In the context of our tournament problem, there are two proposed algorithms.

• The first algorithm calculates the number of matches per round until one player is left, requiring detailed step-by-step calculations.
• The second algorithm provides a more efficient solution by recognizing that each match results in a single player being eliminated.

This second algorithm shines in efficiency by connecting directly to the key insight that 341 matches eliminate 341 players, effortlessly yielding a single winner. Its simplicity enables quicker resolution and understanding without redundant operations, exemplifying a refined problem-solving approach.

Problem Solving

Effective problem solving in algorithms often involves identifying and simplifying underlying patterns. This skill is showcased in the tournament issue, where we see two methods tackling the challenge.
The first method breaks down the problem into rounds, computing matches iteratively. Though detailed, this method can be intricate and time-consuming.
The second approach demonstrates a deeper understanding by using logic to streamline the calculation. Recognizing that every match contributes to reducing the player pool by one leverages the natural structure of elimination tournaments to count matches with ease.
Both methods solve the problem, but the second method exemplifies how simplifying assumptions can lead to more powerful solutions in algorithmic problem solving, drawing a direct line from the problem to its efficient resolution.