Apples and Oranges

Question

Three boxes are labelled as in the picture, but a mischievous greengrocer imp has swapped the labels so that they don’t match what’s in the box. You can’t see inside the boxes or smell their contents, but you are allowed to pick out one piece of fruit from one box. Which box should you choose? Can you explain why?

Courtesy of Alex Bellos (Can You Solve My Problems?)

If the piece of fruit you pick from the ‘apples and oranges’ box is an apple, you know the box must contain only apples (if it contained both, it would not be mis-labelled). This leave the ‘apples’ box and the ‘oranges’ box. The box labelled ‘oranges’ cannot contain oranges, so that must be the one with both fruits, and the one labelled ‘apples’ must contain the oranges. Similarly, you can work out which box is which if the if the one piece you pick out is an orange – just swap the fruits in the explanation.

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