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| − | ==Problem==
| + | #redirect [[2022 AMC 10A Problems/Problem 12]] |
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| − | On Halloween 31 children walked into the principal's office asking for candy. They
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| − | can be classified into three types: Some always lie; some always tell the truth; and
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| − | some alternately lie and tell the truth. The alternaters arbitrarily choose their first
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| − | response, either a lie or the truth, but each subsequent statement has the opposite
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| − | truth value from its predecessor. The principal asked everyone the same three
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| − | questions in this order.
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| − | "Are you a truth-teller?" The principal gave a piece of candy to each of the 22
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| − | children who answered yes.
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| − | "Are you an alternater?" The principal gave a piece of candy to each of the 15
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| − | children who answered yes.
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| − | "Are you a liar?" The principal gave a piece of candy to each of the 9 children who
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| − | answered yes.
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| − | How many pieces of candy in all did the principal give to the children who always
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| − | tell the truth?
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| − | ==Solution==
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| − | Consider when the principal asks "Are you a liar?": The truth tellers truthfully say no, and the liars lie and say no. This leaves only alternaters who lie on this question to answer yes. Thus, all 9 children that answered yes are alternaters that falsely answer question 1 and 3, and truthfully answer question 2. The rest of the alternaters, however many there are, have the opposite behavior.
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| − | Consider the second question, "Are you an alternater?": The truth tellers again answer no, the liars falsely answer yes, and alternaters that truthfully answer also say yes. From the previous part, we know that 9 alternaters truthfully answer here. Because only liars and 9 alternaters answer yes, we can deduce that there are <math>15-9=6</math> liars.
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| − | Consider the first question, "Are you a truth teller?": Truth tellers say yes, liars also say yes, and alternaters that lie on this question also say yes. From the first part, we know that 9 alternaters lie here. From the previous part, we know that there are 6 liars. Because only the number of truth tellers is unknown here, we can deduce that there are <math>22-9-6=7</math> truth tellers.
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| − | The final question is how many pieces of candy did the principal give to truth tellers. Because truth tellers only answer yes on the first question, we know that all 7 of them said yes once, resulting in <math>(A) 7</math> pieces of candy.
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