Brain Teasers
Pet Owners
Edgar, Frank, George, and Hans are pet owners. Edgar says, "If Hans and I each own a dog, then exactly one of Frank and George owns a dog." Frank says, "If George and I each own a cat, then exactly one of Edgar and Hans owns a dog." George says, "If Edgar and I each own a dog, then exactly one of Frank and Hans owns a cat." Hans says, "If Frank and I each own a cat, then exactly one of Frank and me owns a dog." Only one man is telling the truth...but who?
Hint
Hint 1: All four statements are in the form of "hypothesis and conclusion" statements. For such a statement to be false, the hypothesis must be true and the conclusion must be false.Hint 2: It is possible for a "pet owner" to own both a dog and a cat. It is also possible for a "pet owner" to own neither a dog nor a cat.
Answer
Frank is telling the truth.Suppose Edgar doesn't have a dog. This would mean that both he and George have false hypotheses, which would make both statements true. Therefore, Edgar must own a dog.
Suppose Hans doesn't own a dog. This would mean that Edgar's hypothesis is false and Frank's conclusion is true, making both statements true. Therefore, Hans must own a dog.
Suppose Frank doesn't have a cat. This would mean that both he and Hans have false hypotheses, making both statements true. Therefore, Frank must own a cat.
Suppose Hans doesn't own a cat. This would mean that his hypothesis is false and George's conclusion is true, making both statements true. Therefore, Hans must own a cat.
Suppose Frank and George don't both own dogs. This would mean that exactly two of the three below are true:
1. Frank doesn't own a dog (making Hans's conclusion true).
2. George doesn't own a dog (making George's hypothesis false).
3. Exactly one of Frank and George owns a dog (making Edgar's conclusion true).
As you can see, if Frank and George don't both own dogs, there will always be at least two truthtellers. Therefore, they both own dogs.
Now we know that Edgar's, George's, and Hans's hypotheses are all true and their conclusions are all false, making them all liars and leaving only Frank.
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