The Puzzle:

I have three boxes of fruit. One box contains apples only, another box contains oranges only, and the third box contains a mixture of apples and oranges. Each box is labeled telling what is in that box. The problem is that all three of the boxes are labeled incorrectly; that is, the box labeled ’Apples’ does not contain only apples but rather either oranges only or a mixture of apples and oranges; the box labeled ’Oranges’ does not contain only oranges but rather either apples only or a mixture of apples and oranges; and the box labeled ’Apples and Oranges’ does not contain a mixture of apples and oranges but rather either apples only or oranges only.

I want to be able to reach into one box (any one but only one), take out one (and only one) piece for fruit, and correctly label the three boxes. How should I do it? Which box should I reach into and choose only one piece of fruit and how will I then be able to label each box correctly?

The Solution:

We know that all labels on all boxes are incorrect.

We reach into the box labeled ’Apples and Oranges’ and take out one piece of fruit.

1. If the piece of fruit we remove from the box is an apple, we know that the box contains apples only. Since the other boxes are also labeled incorrectly, we then know that the box labeled ’Oranges’ must in fact be apples and oranges and the box labeled ’Apples’ must be oranges only.

2. If the piece of fruit we remove from the box is an orange, we know that the box contains oranges only. Since the other boxes are also labeled incorrectly, we then know that the box labeled ’Apples’ must in fact be apples and oranges and the box labeled ’Oranges’ must be apples only.

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