AMC12 2010 A

AMC12 2010 A · Q12

AMC12 2010 A · Q12. It mainly tests Logic puzzles.

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?

在一个神奇的沼泽中有两种会说话的两栖动物：蟾蜍，它们的陈述总是真实的；青蛙，它们的陈述总是假的。四只两栖动物 Brian、Chris、LeRoy 和 Mike 一起生活在这个沼泽中，它们做出了以下陈述。 Brian：“Mike 和我是不同物种。” Chris：“LeRoy 是青蛙。” LeRoy：“Chris 是青蛙。” Mike：“我们四个中至少有两个是蟾蜍。” 这四只两栖动物中有多少是青蛙？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (D)

正确答案：（D）

Solution

Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog. As Mike is a frog, his statement is false, hence there is at most one toad. As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the other one tells the truth, and therefore is a toad. Hence we must have one toad and $\boxed{\textbf{(D)}\ 3}$ frogs.

从 Brian 开始。如果他是蟾蜍，他说真话，因此 Mike 是青蛙。如果 Brian 是青蛙，他说谎，因此 Mike 也是青蛙。于是 Mike 必定是青蛙。由于 Mike 是青蛙，他的陈述为假，因此蟾蜍至多只有一只。既然蟾蜍至多只有一只，那么 Chris 和 LeRoy 中至少有一只是青蛙。但这样另一只就说了真话，因此它是蟾蜍。因此必有一只蟾蜍和 $\boxed{\textbf{(D)}\ 3}$ 只青蛙。

Topics

LM.001 Logic puzzles