One morning you wake up with a huge craving for orange marmalade. Alas, there is none in the pantry. But you are not a person that is easily thwarted. You don your rain slicker (this is Port Orchard after all) and head to the local Quickie Food Mart to buy a crate of oranges so that you can whip up a batch of marmalade before breakfast.
When you arrive at QFM (get it?) they are busily restocking the produce section and there are no oranges in site. You look about in desperation and discover three crates on the top shelf. One is labeled "Oranges", one "Apples" and one "Apples and Oranges". You ask a stock boy to hand you the crate of oranges. He says, "Ok, but all the labels are wrong. One contains only oranges, one contains only apples and one contains both apples and oranges, but I don't know which is which. We had a trainee last night who got them all mixed up." Undaunted he reaches up, pulls out one piece of fruit from one crate, puts it back and promptly hands you the crate of oranges. How did he do it?
Solution
What we know
One box contains nothing but apples.
One box contains nothing but oranges.
One box contains both apples and oranges.
None of the boxes are labeled correctly, but there is one box labeled "Apples", one labeled "Oranges" and one labeled "Apples & Oranges"
The Reasoning
The first decision to make is which box to draw from. Suppose that we draw from the box labeled "Oranges" and we draw an apple. The box labeled "Apples" may either be mixed or contain oranges. So we don't know for sure what is in all three boxes. So this approach requires withdrawing more than one piece of fruit from just one box.
Suppose instead that we draw an orange from the box labeled "Apples & Oranges" we then know for certain that box contains oranges (remember each box is mislabeled). Therefore, the box labeled "Apples" must contain apples and oranges which leaves only the box labeled "Oranges" which must contain apples. If it were otherwise then one box would be correctly labeled.
By similar reasoning we know if we first drew an apple then the box labeled "Apples & Oranges" must contain apples and the box labeled "Oranges" must contain apples and oranges and the box labeled "Apples" must contain oranges.
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