Question

在一片神奇的沼泽中，有两种会说话的两栖动物：蟾蜍总是说真话，青蛙总是说假话。Brian、Chris、LeRoy 和 Mike 四只两栖动物一起住在这片沼泽中，并作出以下陈述。

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只是青蛙？

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?

$0$

$1$

$2$

$3$

$4$

答案：D

知识点：说真话者与说谎者逻辑推理

难度评级：1540

解答：

如果 Brian 是青蛙，那么他在说谎，所以 Mike 必须也是青蛙。

如果 Brian 是蟾蜍，那么他在说真话，所以 Mike 仍然是青蛙。

因此 Mike 是青蛙，也就是说 Mike 在说谎。于是四者中至多有一只是蟾蜍。 Chris 和 LeRoy 的陈述互相指认对方是青蛙，因此他们中至少有一只是青蛙，另一只说真话，是蟾蜍。因此青蛙共有 $3$ 只。

所以正确答案是 D。

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 第 12 题

答案：D

难度评级：1540

解答：

其他年份的第 12 题