Shopping Shuffle

You have to start somehow. Open box A. You might strike it lucky and find it is the correctly labelled box, but we have to assume the worst case scenario. Suppose it contains one of the other fruits, say (without loss of generality) fruit B. Then we know that neither A nor B are labelled correctly.

So now open box C. If it contains fruit C it is the correctly labelled box and we are done. And if it does not contain fruit C we are done as well, because box D remains as the only box which is not incorrectly labelled - and so must be the correctly labelled box.

Note.

By 'without loss of generality' I mean that the other cases (fruit C or fruit D being found in box A) are treated by the same argument, except for a trivial permutation of the letters.

Opening 1 incorrectly labeled box will give you 2 info, that the one you did open is wrong, and another box labeling the fruit you found inside the opened one is also wrong. Opening 1 correctly labeled box will also give you 2 info, that the one you did open is right, and all the other 3 are wrong, but you would not know which way they rotate. So, it would not be enough to deduce 3 things (out of 4, but we were already told about 3 wrong and 1 correct boxes) from just 2 info. Another box need to be opened, and we can relabel all of them no matter what kind of box we'd encounter. Thus, minimum number of box to be checked is 2.

Let the apple box be opened. Suppose it contains bananas, then open banana box, suppose it contains dates. So we know apple,banana,date are in wrong box after opening 2. Thus carrot is in right. Now if the first or second box contained what was labelled, then also we achieved our goal.

Another case is after I open apples box to get bananas. And banana box contain apples. This case isnt possible as apple has bananas, banana has apples, so other 2 must be right or wrong simultaneously, a contradiction

If you open the box labeled apples and it's bananas, you know the box labeled bananas must be wrong. Open the carrot box. It must be carrots or the box labeled dates is right.