## Puzzle Statement

There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes have been incorrectly labelled such that no label identifies the actual contents of the box it labels. Opening just one box and without looking in the box, you take out one fruit.

By looking at the fruit, how can you immediately label all of the boxes correctly?

## Solution

It might look tricky but if we can think for a while and decide upon which box to open the solution really becomes simple. Let us first note down the facts we know for sure:

- The box with label APPLES does not contain apples.
- The box with label ORANGES does not contain oranges.
- The box with label MIXED does not contain the mixed fruits.

I can probably give you the solution directly and tell you which box to open. But let us do it in a Brute Force way and develop our thinking.

**Case 1: Opening the box with label APPLES**

We know for sure that the box does not contain apples. Now when we take out a fruit from the box we can get the following:

- An
**orange:**In this case we are not sure if this is an orange box or mixed box. Hence, we cannot identify the correct labels if we open the box with a label APPLE and get an orange out of it. - An
**apple:**In this case we can identify that the box we opened is a box with mixed fruits and hence we can label it MIXED. Also, we can accordingly fix the labels of the remaining two boxes.

As we cannot identify the correct labels in all sub cases, hence opening the box with label APPLES is not the right choice.

**Case 2: Opening the box with label ORANGES**

We know for sure that the box does not contain oranges. Now when we take out a fruit from the box we can get the following:

- An
**apple:**In this case we are not sure if this is an apple box or mixed box. Hence, we cannot identify the correct labels if we open the box with a label ORANGES and get an apple out of it. - An
**orange:**In this case we can identify that the box we opened is a box with mixed fruits and hence we can label it MIXED. Also, we can accordingly fix the labels of the remaining two boxes.

As we cannot identify the correct labels in all sub cases, hence opening the box with label APPLES is not the right choice.

**Case 3: Opening the box with label MIXED**

We know for sure that the box does not contain both the fruits. And so, the moment we take out a fruit we will definitely know the correct label of this box. Let us see what happens in the two sub cases:

- An
**apple**comes out**:**In this case we are sure that the box contains only apples. This is because we know this box can’t be a box of mixed fruits. Hence, we can identify the correct labels if we open the box with a label MIXED and get an apple out of it. The labels of the rest two boxes can be accordingly arranged. - An
**orange**comes out**:**In this case we can identify that the box we opened is a box with oranges only. This is because we know this box can’t be a box of mixed fruits. Hence, we can identify the correct labels if we open the box with a label MIXED and get an orange out of it. The labels of the rest two boxes can be accordingly arranged.

As we can identify the correct labels in all sub cases, hence opening the box with label MIXED is the right choice.

Hope you got this, in case it seems difficult to understand, please let me know in the comments. I will try to add some more images.