AMC10 2010 A

AMC10 2010 A · Q15

AMC10 2010 A · Q15. 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 four 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。有四只两栖动物，Brian、Chris、LeRoy 和 Mike 一起生活在这个沼泽，它们做了以下陈述。Brian：“Mike 和我是不同种。” 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

Answer (D): LeRoy and Chris cannot both be frogs, because their statements would be true and frogs lie. Also LeRoy and Chris cannot both be toads, because then their statements would be false, and toads tell the truth. Hence between LeRoy and Chris, exactly one must be a toad. If Brian is a toad, then Mike must be a frog, but this is a contradiction as Mike’s statement would then be true. Hence Brian is a frog, so Brian’s statement must be false, and Mike must be a frog. Altogether there are 3 frogs: Brian, Mike, and either LeRoy or Chris.

答案（D）：LeRoy 和 Chris 不可能同时都是青蛙，因为那样他们的陈述就会为真，而青蛙会说谎。同样，LeRoy 和 Chris 也不可能同时都是蟾蜍，因为那样他们的陈述就会为假，而蟾蜍说真话。因此，在 LeRoy 和 Chris 之间，恰好有一个必须是蟾蜍。如果 Brian 是蟾蜍，那么 Mike 就必须是青蛙，但这会导致矛盾，因为那样 Mike 的陈述将为真。因此 Brian 是青蛙，所以 Brian 的陈述必为假，而 Mike 必须是青蛙。总共有 3 只青蛙：Brian、Mike，以及 LeRoy 或 Chris 中的一个。

Topics

LM.001 Logic puzzles