Suppose I Have Three Bags, One with Two Peaches, Another

question 95

Multiple Choice

Suppose I have three bags, one with two peaches, another with two plums, and a third mixed bag with one peach and one plum. Now I give the bags (in mixed-up order) to Alice, Annie, and Nancy. I tell the three to look into their bags and that I want each to make a statement about the contents of her bag, but I want them to lie. Here is what they say: ​
Alice : I have two peaches.
Annie : I have two plums.
Nancy : I have one peach and one plum.
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Now here is the game. I want you to develop a strategy of asking one of the three to reach into their bag and pull out one fruit and show it to you. The fruit is then returned to the bag and you ask another to do the same thing. Continue until you can deduce which bag is the mixed bag. What is the minimum possible number of necessary moves?
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• A) Only one question is necessary: Ask Nancy (who falsely claimed to have the mixed bag) to pull out one fruit. Suppose she pulls out a peach; this means she has the bag containing two peaches. Then, Alice, who falsely claimed two peaches, must have two plums. This leaves Annie with the mixed bag. Suppose she pulls out a plum; this means she has the bag containing two plums. Then, Annie, who falsely claimed two plums, must have two peaches. This leaves Alice with the mixed bag.
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• B) Only one question is necessary: Ask Annie (who falsely claimed to have the mixed bag) to pull out one fruit. Suppose she pulls out a peach; this means she has the bag containing two peaches. Then, Alice, who falsely claimed two peaches, must have two plums. This leaves Nancy with the mixed bag. Suppose she pulls out a plum; this means she has the bag containing two plums. Then, Nancy, who falsely claimed two plums, must have two peaches. This leaves Alice with the mixed bag.
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• C) Only one question is necessary: Ask Nancy (who falsely claimed to have the mixed bag) to pull out one fruit. Suppose she pulls out a peach; this means she has the bag containing two peaches. Then, Annie, who falsely claimed two peaches, must have two plums. This leaves Alice with the mixed bag. Suppose she pulls out a plum; this means she has the bag containing two plums. Then, Alice, who falsely claimed two plums, must have two peaches. This leaves Annie with the mixed bag.
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