In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
- A)
- B)
- C)
- D)
- E)
Answer
D
Key insight
Chris and LeRoy accuse each other, so exactly one is a toad; a toad Mike makes Brian's statement impossible, so Mike and Brian are frogs.
Solution
Chris and LeRoy. If Chris is a toad, his statement is true, so LeRoy is a frog; LeRoy's claim "Chris is a frog" is then false, as a frog's must be. If Chris is a frog, his statement is false, so LeRoy is a toad, and LeRoy's true claim matches. Either way, exactly one of Chris and LeRoy is a toad.
Mike. Suppose Mike is a toad. Then consider Brian. If Brian is a toad, he and Mike are the same species, so his statement is false, contradiction. If Brian is a frog, he and Mike really are different species, so his statement is true, again a contradiction. Hence Mike is a frog.
Consequences. Mike's statement is false, so fewer than two are toads. One toad already sits among Chris and LeRoy, so Brian must be a frog. Check Brian: he and Mike are both frogs, so "we are different species" is false, as required.
The frogs are Brian, Mike, and one of Chris or LeRoy: three frogs.
The answer is .
Why this works
Two creatures who call each other liars are always one truth-teller and one liar, which fixes their combined count without identifying who is who. After that, a statement of the form "X and I are different" can never be made by anyone when X is a truth-teller, a small paradox that instantly decides X's status. Look for pairs of statements that constrain each other before brute-forcing all sixteen assignments.
The trap
Forgetting that a frog's statement must be false, not merely unverified, when testing Brian's claim.
Common mistakes
- Forgetting that a frog's statement must be false, not merely unverified, when testing Brian's claim.
- Trying to determine whether Chris or LeRoy is the toad; the problem only asks for a count, and that is undetermined.
Techniques
Split into exhaustive cases and handle each