Three boxes of fruit are labeled Apples, Oranges, and Apples and Oranges. Each label is wrong. By selecting just one fruit from just one box, how can you determine the correct labeling of the boxes?

pick a fruit from the box labeled Apples and Oranges. Because all the labels are wrong, you know the box is either all apples or all oranges.

If you draw an apple, you know that the box is all apples. So, the box labeled Oranges must be not oranges, but apples and oranges, since it cannot be oranges (since its label is wrong), and you have an apple.

The remaining box, labeled Apples, must be all oranges.

reverse the logic if you draw an orange.

To determine the correct labeling of the boxes, follow these steps:

1. Pick a fruit from the box labeled "Apples and Oranges." Since all labels are wrong, this box cannot contain both apples and oranges.

2. Let's say you picked an apple from the box labeled "Apples and Oranges." This means the box labeled "Apples" must contain oranges, as its label is wrong. Similarly, the box labeled "Oranges" must contain apples.

3. Now, you can correct the labels based on the fruit you picked. The box that contained apples and oranges should be labeled "Oranges," as it actually contains apples. The box that was labeled "Apples" should be relabeled as "Apples and Oranges." The box with the label "Oranges" should be relabeled as "Apples."

After following these steps, the correct labeling of the boxes should be:
- The box originally labeled "Apples" should now be labeled "Apples and Oranges."
- The box originally labeled "Oranges" should now be labeled "Apples."
- The box originally labeled "Apples and Oranges" should now be labeled "Oranges."

To determine the correct labeling of the boxes, you can follow these steps:

1. Let's assume that the boxes are labeled A, B, and C, corresponding to Apples, Oranges, and Apples and Oranges, respectively.
2. Since the labels are all wrong, we can deduce that each box must have a mix of fruits.
3. To start, select a fruit from the box labeled "Apples and Oranges." Since we know this box is incorrectly labeled, it cannot contain both apples and oranges.
4. If the fruit you selected is an apple, then the box labeled "Apples" must be the box containing oranges. The remaining box must then be the one labeled "Oranges and Apples."
5. Now, take a fruit from the box you labeled as "Oranges and Apples" in the previous step. If the fruit is an orange, it means this box is incorrectly labeled as well. Therefore, this box should be labeled as "Oranges."
6. The last remaining box must be the one labeled "Apples," even if its content is a mix of apples and oranges.
7. By selecting just one fruit from just one box, you have determined the correct labeling of the boxes.

Keep in mind that the key is to target the boxes with incorrect labels and deduce the contents based on the fact that each label is wrong.