Question

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 four amphibians are frogs?

$0$

$1$

$2$

$3$

$4$

Answer: D

Concepts:truth-tellers and liars logical deduction

Difficulty rating: 1540

Solution:

Chris and LeRoy are of opposite species: if Chris is a toad, his statement makes LeRoy a frog; if Chris is a frog, his false statement makes LeRoy a toad. Thus exactly one of them is a toad.

If Brian is a toad, his true statement makes Mike a frog. If Brian is a frog, his false statement says that he and Mike are the same species, again making Mike a frog. Therefore Mike is always a frog.

Mike's statement is therefore false, so there are fewer than two toads. Chris and LeRoy already supply exactly one toad, forcing Brian to be a frog. Hence there is one toad and $3$ frogs.

Thus, D is the correct answer.

2010 AMC 12A Problem 12

Answer: D

Difficulty rating: 1540

Solution:

Problem 12 in Other Years