Fruits Puzzle

There are three boxes. One box contains only mangoes, one box contains only bananas, and the other box contains both mangoes and bananas. The boxes are labeled with fruit names ‘mangoes’, ‘bananas’ and ‘mangoes and bananas’. Unfortunately, a naughty child labeled all the boxes incorrectly.

If you are allowed to take one fruit from a box of your choice, will it be possible to fix the labels on all the boxes?

Take a piece of fruit from the ‘mangoes and bananas’ box. If it’s a mango then you know that is the ‘mangoes’ box since ALL THE BOXES ARE LABELED INCORRECTLY. Now that we’ve determined that the box marked as ‘mangoes and bananas’ only contains mangoes, we can figure out what the other boxes contain.

If we look at the box labeled as ‘bananas’, we know that since the label is incorrect, this box either has only mangoes in it or has both mangoes and bananas. Since we’ve already established that the box labeled ‘mangoes and bananas’ contains only mangoes , we know that the box labeled as ‘bananas’ contains both mangoes and bananas. Then by simple process of elimination, we know that the box labeled as ‘mangoes’ contains only bananas.

You can apply the same logic if you assume you picked a mango from the box labeled as ‘mangoes and bananas’.

Will you be able to fix the labels if you picked up the first fruit from box marked “mangoes” or the box marked “bananas”?

Let’s see. If we take a fruit from the box labeled as ‘bananas’, and if we get a banana, we know that since the label is incorrect, this box can only be ‘mangoes and bananas’. Now the box labeled ‘mangoes’ has to be ‘bananas’ and therefore the third box labeled ‘mangoes and bananas’ has to be ‘mangoes’. Same argument will hold true if we get a mango from the box labeled “mangoes”.

What we if get a mango from the box labeled as ‘bananas’, or a banana from the box labeled “mangoes”. Can we fix the labels?

No. As the box can we labeled ‘bananas’ or ‘mangoes and bananas’. We do not have enough information to determine by just picking up 1 fruit from the box.

So, only picking up fruit from ‘mangoes and bananas’ box guarantees a solution.