Question

万圣节时， $31$ 个孩子走进校长办公室要糖果。他们可以分为三类：有些总是说谎；有些总是说真话；有些交替说谎和说真话。交替者可以任意选择第一次回答是谎话还是真话，但之后每句话的真假都与前一句相反。校长按以下顺序问每个人相同的三个问题。

“你是说真话者吗？”校长给每个回答“是”的 $22$ 个孩子一颗糖。

“你是交替者吗？”校长给每个回答“是”的 $15$ 个孩子一颗糖。

“你是说谎者吗？”校长给每个回答“是”的 $9$ 个孩子一颗糖。

校长一共给总是说真话的孩子发了多少颗糖？

On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

$7$

$12$

$21$

$27$

$31$

答案：A

知识点：说真话者与说谎者逻辑推理分类讨论

难度评级：1530

解答：

对“你是说真话者吗？”这个问题，说真话者和说谎者都会回答“是”，而交替者中只有这次说谎的人会回答“是”。对“你是交替者吗？”这个问题，说谎者回答“是”，交替者中只有这次说真话的人回答“是”。对“你是说谎者吗？”这个问题，只有这次说谎的交替者回答“是”。

按第一次回答拆分交替者。先说谎的交替者回答模式为（谎话，真话，谎话），所以三个问题都回答“是”；先说真话的交替者回答模式为（真话，谎话，真话），所以三个问题都不回答“是”。最后一个问题的 $9$ 个“是”正好都是先说谎的交替者，因此他们有 $9$ 人。

第二个问题的 $15$ 个“是”来自说谎者和这 $9$ 个交替者，所以说谎者有 $6$ 人。第一个问题的 $22$ 个“是”来自说真话者、说谎者以及这 $9$ 个交替者，所以说真话者有 $22-6-9=7$ 人。

说真话者只会在第一个问题回答“是”，每人得到一颗糖，共 $7\cdot1=7$ 颗。

因此，正确答案是 A。

To "Are you a truth-teller?" the truth-tellers and liars both answer yes, and only alternaters who lie on this question answer yes. To "Are you an alternater?" the liars answer yes, and among alternaters only those telling the truth on this question answer yes. To "Are you a liar?" only alternaters lying on this question answer yes.

Split the alternaters by first response. Those starting with a lie answer (lie, truth, lie), so they say yes to all three questions; those starting truthful answer (truth, lie, truth) and say yes to none of the three. The $9$ yeses on the last question are exactly the lie-first alternaters, so there are $9$ of them.

The second question's $15$ yeses are the liars plus these $9,$ so there are $6$ liars. The first question's $22$ yeses are truth-tellers plus liars plus the $9,$ so the truth-tellers number $22-6-9=7.$

Truth-tellers answer yes only to the first question, receiving one candy each, for $7\cdot1=7$ pieces.

Thus, the correct answer is A.

2022 AMC 12A 第 9 题

答案：A

难度评级：1530

解答：

其他年份的第 9 题