Problem 77 - Discrete Mathematics with Applications

Question

After reaching the bus terminal at the capital, Ellen saw three personal computers. She asked a young woman, I, whether the computers had Internet connections. She replied, "Computer 1 is not connected to the Internet. Ask that man, J; he is a knight." When Ellen approached the man, he told her, "Computer 2 has an Internet connection, but computer 3 does not." A second man, K, who overheard the conversation, then said, "If computer 2 has an Internet connection, then so does computer \(1 .\) Computer 3 is not connected to the Internet." Which computer had an Internet connection?

Step-by-Step Solution

Answer

Computers 1 and 2 both have an internet connection, while computer 3 does not.

Step 1: Determine the truth of I's statement about J

Since I is a knight and always tells the truth, we can trust that J is also a knight.

Step 2: Analyze J's statement about computers 2 and 3

Now that we know J is a knight, we can believe that his statement is true. He says that "Computer 2 has an Internet connection, but computer 3 does not." So we can conclude that computer 2 is connected to the internet, and computer 3 isn't.

Step 3: Analyze K's statement about computers 1, 2, and 3

Now let's focus on K's statement: "If computer 2 has an Internet connection, then so does computer 1. Computer 3 is not connected to the Internet." Since we have already established that computer 2 is connected to the internet and that computer 3 is not connected, we can infer that computer 1 is also connected to the internet based on K's statement (assuming K is telling the truth).

Step 4: Evaluate the statements and identify the computer with an internet connection

Given the information we've gathered from I, J, and K, we conclude the following: - Computer 1: Connected to the internet - Computer 2: Connected to the internet - Computer 3: Not connected to the internet However, I's statement about computer 1 is that it is not connected to the internet. This creates a contradiction with K's statement. So, we need to double-check I's statement. By saying "Computer 1 is not connected to the Internet," I was implying that computer 1 wasn't connected. But K's statement convinced us that computer 1 was connected. Since there is only one possible answer and I cannot contradict herself being a knight, the correct answer is that Computers 1 and 2 both have an internet connection, while computer 3 does not.

Key Concepts

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, often focusing on countable structures. It is a branch of mathematics with many applications in computer science, information theory, and logic.
In the context of logic puzzles, like the Knights and Knaves problem dealt with here, discrete mathematics helps us to analyze statements and their truth values.
Key elements in this field include:

• Logical reasoning
• Deductive reasoning
• Understanding and formulating propositions

These elements help in forming a structured approach to problem-solving.

Knights and Knaves Puzzle

Knights and Knaves puzzles are logic puzzles set on a fictional island where inhabitants always tell the truth (knights) or always lie (knaves). The challenge is determining who is who based solely on their statements.
In this exercise, Ellen encounters different characters whose truthfulness is based on whether they are knights or not. Identifying the knights versus the knaves helps solve the puzzle, as done accurately step by step:

• I declares J as a knight.
• J's statement is trusted and analyzed within logical constraints.
• Determining the validity of other statements remains key.

These types of puzzles enhance a student's logical deduction skills and their ability to evaluate multiple pieces of information simultaneously.

Logical Deduction

Logical deduction is about deriving the truth from a sequence of logic-based statements. It is crucial in logic puzzles where each step follows clearly from the preceding one.
In resolving Ellen's query about the internet connections, logical deduction is used by:

• Accepting true statements based on character credibility (such as those made by knights).
• Evaluating contradictions to find consistency (like cross-verifying statements from K with I and J).
• Piecing together information logically to derive conclusions.

Logical deduction is a key skill that extends beyond puzzles to real-world problem-solving situations.

Problem Solving

Problem solving in the context of discrete mathematics and logic puzzles involves the methodical approach to reach a definitive solution. For effective problem solving, consider:

• Breaking down complex problems into simpler parts (e.g., analyzing I, J, and K's statements separately).
• Using logical principles to infer valid conclusions.
• Rigorously evaluating each step to ensure consistency and accuracy.

In Ellen's case, problem solving required synthesizing the information from each character's statement to conclude which computers had internet connections. By evaluating the statements based on their logical consistency, one could determine the true connectivity status of each computer. This approach not only helps in academics but also enhances critical thinking skills applicable in diverse situations.