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counting and probability senior
Problem
On Halloween 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 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth?
(A)
(B)
(C)
(D)
Solution
Note that: Truth-tellers would answer yes-no-no to the three questions in this order. Liars would answer yes-yes-no to the three questions in this order. Alternaters who responded truth-lie-truth would answer no-no-no to the three questions in this order. Alternaters who responded lie-truth-lie would answer yes-yes-yes to the three questions in this order. Suppose that there are truth-tellers, liars, and alternaters who responded lie-truth-lie. The conditions of the first two questions imply that Subtracting the second equation from the first, we have Remark The condition of the third question is extraneous. However, we know that there are alternaters who responded lie-truth-lie, liars, and alternaters who responded truth-lie-truth from this condition.
Final answer
A