Recall the problem:

FRUITful Problem:

There are three boxes, one has apples, one has oranges, and the other has apples and oranges. The boxes are labeled wrong so that no label is correct. Sue opens just one box, and without looking in the box, takes out one piece of fruit. She looks at the fruit and immediately labels all the boxes correctly. Which box did she open and how did she know?

The write-up below is by Max Elgart, who was a student of Steven Miller in Calc III at Williams College in the spring of 2013. Miller has supplemented and expanded on his write-up.

First hint: When facing any problem it’s important to pick out the key pieces of information and clues that will help lead you to the solution. For this problem it is vital to note that all the boxes are labeled incorrectly, there are only three options for the contents, and Sue only sees one piece of fruit. There is also a notable asymmetry in this problem; opening the oranges box will yield the same information as opening the apples box, so we have to decide whether to open a box whose label says it contains only one kind of fruit, or the box whose label says it contains both kinds of fruit.

When there are alternatives such as these, one approach is to take two pieces of paper. Use a separate sheet for each option. Try and see how far you can push the analysis in each case. Frequently people are hesitant to try something if they don’t know it will work. Don’t be afraid to try and explore. There are essentially two cases here: try a box with a label of just one fruit, or try a box with a label of two fruits. The situation is somewhat similar to the analysis of tic-tac-toe. You might think there are nine possible first moves, as there are nine squares; however, by symmetry moving in any of the four corners is equivalent to moving in any other corner, while similarly moving in any of the four sides is equivalent to moving in any other side. Thus there are really just three possible first moves (a corner, a side, or the center); for more on this see the comments on the chess riddle. For example, the chart below shows what happens if in our original pick we choose the box labeled apple or the box labeled orange. See how similar the two cases are.   

Second Hint: My first instinct was to start analyzing what would happen if Sue picked an orange from the box labeled both. However, I soon realized that there must be a way of displaying all of the possible scenarios simply. So I abandoned my original guess-and-check strategy for a more organized and practical approach: a flow chart.

Now that we have our first possibilities, what are the respective scenarios that emerge from this first step? Here symmetry appears; one side of this problem will lead to the exact same solution as the other side, so there is no use in wasting your time and doing both.

Scenario 1: Sue picks an apple from a box that is incorrectly labeled oranges. This means that the box that she picked out of could either be the box containing only apples or the box containing both apples and oranges.

Let’s try labeling that box apples. We have two labels remaining (oranges, apples-oranges). The other two boxes are incorrectly labeled apples and apples-oranges. Thus we give the label apples-oranges to the box labeled apples and the label apples to the box labeled apples-oranges.

What if instead we try labeling the box she picked apples-oranges. In this case we have two labels remaining (apples, oranges) and two boxes incorrectly labeled apples and apples-oranges. We thus  take the box labeled apples and label it apples-oranges, and the box labeled apples-oranges is now labeled apples.

Unfortunately, when she picked an apple from the box labeled oranges we ran into problems. There are two different, but consistent, possible labelings. Seeing an apple picked from a box labeled oranges is not enough information. The situation would be different if she picked an orange from the box labeled oranges. In that case, the label for that box would have to be apples-oranges, and since each box is incorrectly labeled the box labeled apples would have to be oranges. While we can solve the problem in this case, we can’t assume we’ll be lucky enough to get an orange when we open the box labeled orange.

Thus, there is no way to figure out which one is which, so this leads to a dead end.

Scenario 2: Sue picks an apple from the box that is incorrectly labeled apples and oranges. This means that the box she just picked out of must be the box containing only apples! Therefore Sue can correctly label this box “apples”. This leaves two boxes each incorrectly labeled. One is labeled oranges, and the other is labeled apples. The contents of these boxes are either oranges or apples and oranges. The box labeled oranges cannot contain only oranges; that would violate the rules of the problem. Therefore it must contain apples and oranges. Sue can now label this box correctly. This leaves only one option for the label of the last box: oranges.

 

General Comment: This problem effectively introduces the concept: without loss of generality. When we open the mixed box we don’t know what kind of fruit we will get, but it doesn’t matter, the problem is handled the same way for both. Without loss of generality, we may assume we got an apple; the analysis is similar if we got an orange.

Similarly, we saw we didn’t need to analyze both the case of opening the box labeled oranges and the box labeled apples; the two analyses are similar, so without loss of generality we may assume that if we open a box labeled with just one fruit, we chose the one labeled oranges.