Understanding the Theft Case Scenario

This problem involves a theft case with four suspects: P, Q, R, and S. We are given specific statements made by each suspect and two crucial conditions:

• Condition 1: Exactly one of the four suspects is guilty.
• Condition 2: Innocent people always tell the truth, and the guilty person always lies.

Our goal is to use these statements and conditions to identify the guilty person.

Analyzing Suspect Statements and Conditions

We need to examine each suspect as the potential guilty person and check if the statements and conditions hold true. Let's analyze each possibility:

Case 1: Assuming P is Guilty

If P is the guilty person:

• P lies. P's statement is "Q is guilty." Since P is lying, this means Q is innocent.
• Since only one person is guilty, Q, R, and S must be innocent.
• Q (innocent) tells the truth. Q's statement is "R is guilty." This implies R is guilty.
• This leads to a contradiction: we assumed only P is guilty, but we deduced R is also guilty. Therefore, P cannot be the guilty person.

Case 2: Assuming Q is Guilty

If Q is the guilty person:

• Q lies. Q's statement is "R is guilty." Since Q is lying, this means R is innocent.
• Since only one person is guilty, P, R, and S must be innocent.
• P (innocent) tells the truth. P's statement is "Q is guilty." This matches our assumption that Q is guilty.
• R (innocent) tells the truth. R's statement is "P is innocent." This matches the deduction that P is innocent.
• S (innocent) tells the truth. S's statement is "I am innocent." This matches the deduction that S is innocent.
• All conditions are satisfied: Q is the single guilty person, Q lies, and P, R, and S tell the truth. This scenario is consistent.

Case 3: Assuming R is Guilty

If R is the guilty person:

• R lies. R's statement is "P is innocent." Since R is lying, this means P is guilty.
• This leads to a contradiction: we assumed only R is guilty, but we deduced P is also guilty. Therefore, R cannot be the guilty person.

Case 4: Assuming S is Guilty

If S is the guilty person:

• S lies. S's statement is "I am innocent." Since S is lying, this means S is guilty. This part is consistent with the assumption.
• Since only one person is guilty, P, Q, and R must be innocent.
• P (innocent) tells the truth. P's statement is "Q is guilty." This implies Q is guilty.
• This leads to a contradiction: we assumed only S is guilty, but we deduced Q is also guilty. Therefore, S cannot be the guilty person.

Conclusion: Determining the Guilty Party

By analyzing each possibility, we found that only the assumption 'Q is guilty' satisfies all the given conditions. In this case:

• Q is guilty and lies (Statement: "R is guilty" - False).
• P is innocent and tells the truth (Statement: "Q is guilty" - True).
• R is innocent and tells the truth (Statement: "P is innocent" - True).
• S is innocent and tells the truth (Statement: "I am innocent" - True).

Therefore, the guilty person is Q.