## Apples and Oranges

While waiting to do an improv set, Chrysteena told me that she and Jim had three cartons: one with apples, one with oranges, and a third with apples and oranges. Unfortunately, each carton was mislabeled, and they wanted to move the labels to the correct cartons. What is the minimum number of pieces of fruit one would one need to take out of the cartons to correct the labels, and what would be the strategy?

The thing I like about this puzzle is that there’s information encoded in the wording of the problem, and changing one sentence changes the puzzle significantly. “Each carton was mislabeled” indicates that no box contains the correct label, which eliminates 4 out of 6 permutations of labels. If we replace the phrase “each carton was” with “some cartons were” in that sentence, the sentence only eliminates 1 out of 6, and it uncovers a potential ambiguity in how the final sentence of the puzzle is worded, but regardless of how one interprets that sentence, its solution is different from the “each carton was” case. If “each carton was” changes to “the cartons may have been” then the sentence doesn’t eliminate any of the permutations, and depending on how one interprets the wording of the final sentence of the puzzle, it may or may not have the same solution as the “some cartons were” case.