$\textbf{Label the Boxes}$
There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes are incorrectly labeled such that no label identifies the actual contents inside the box. Is it possible to correct all the labels by just randomly retrieving one fruit from one box? You cannot look inside the chosen box.
$\textbf{Solution}$
Yes, it is possible. Go to the box with label "apple + orange" and take out one fruit.
- If it is an apple, then re-label this box as "apple". The one currently labelled as orange should be labelled as "apple+orange" and the other box should be re-labelled as "orange".
- If it is an orange, then re-label this box as "orange". The one currently labelled as apple should be re-labelled as "apple+orange" and the other should be re-labelled as "apple".
$\textbf{Analysis}$
With many uncertainties, the key to solve this problem is to find something with certainty. If a box is labelled as apple, then it can be either orange only or orange + apple. In this case, if the retrieved fruit is an orange, we will still be unsure. However, if the box is labelled with apple + orange, then it must contain a single type of fruit. In this case, we will be able to determine its contents by retrieving just one fruit from that box.