Who Smashed the Pumpkin?: Solving Suspect Statements

[VinnyCee]

Homework Statement

Four friends have been identified as suspects for a pumpkin smashing. They have made statements to the investigating authorities. (i)Alice said "Carlos did it." (ii)John said "I did not do it." (iii)Carlos said "Diana did it." (iv)Diana said "Carlos lied when he said that I did it."

a) If the authorities also know that exactly one of the four suspects is telling the truth, who did it? Explain your reasoning.

b) If the authorities also know that exactly one is lying, who did it? Explain your reasoning.

Homework Equations

p = "John did it"

q = "Carlos did it"

r = "Diana did it"

(i) $$q$$

(ii) $$\neg\,p$$

(iii) $$r$$

(iv) $$\neg\,r$$ or "Carlos lied"

The Attempt at a Solution

I set up a truth table and reasoned that for a) John did it and for b) Carlos did it. Does that seem right? Is there another way that I should be going about this problem?

 

[HallsofIvy]

Homework Statement

Four friends have been identified as suspects for a pumpkin smashing. They have made statements to the investigating authorities. (i)Alice said "Carlos did it." (ii)John said "I did not do it." (iii)Carlos said "Diana did it." (iv)Diana said "Carlos lied when he said that I did it."

a) If the authorities also know that exactly one of the four suspects is telling the truth, who did it? Explain your reasoning.

b) If the authorities also know that exactly one is lying, who did it? Explain your reasoning.

Homework Equations

p = "John did it"

q = "Carlos did it"

r = "Diana did it"

(i) $$q$$

(ii) $$\neg\,p$$

(iii) $$r$$

(iv) $$\neg\,r$$ or "Carlos lied"

The Attempt at a Solution

I set up a truth table and reasoned that for a) John did it and for b) Carlos did it. Does that seem right? Is there another way that I should be going about this problem?

Obviously, the two statements: Carlos said "Diana did it." andb Diana said "Carlos lied when he said that I did it." are "mutually exclusive"- they can't both be true nor can they both be false. If only one statement is true, it must be one of these two. But that means John's statement, "I did not do it." must be FALSE: John is guilty.

If only one statement is false, again it must be one Carlos or Diana's. But then Alice's statement, "Carlos did it" must be true. Yes, you are correct.