In front of you are three boxes. One contains only apples, one contains only oranges and one contains a mix of apples and oranges. Each box is labeled, like this:

- Box 1: “apples”
- Box 2: “oranges”
- Box 3: “apples & oranges”

Unfortunately, all three boxes are mislabeled. That’s where you come in – you’re going to help me fix those labels.

Here’s the challenge: You get to choose one, and only one, box. I will remove a randomly selected piece of fruit from your chosen box and show it to you (so you can tell if it’s an apple or an orange). After that, you will be able to accurately and definitively relabel all three boxes.

Choose wisely, grasshopper. If you manage to solve this puzzle, leave me a comment with your solution. Good luck and don’t forget to eat your fruit!

**Solution:** If you choose box 1 or 2, you may be in trouble. For example, suppose you choose box 1 (the box labeled “apples”) and you see an orange – then you won’t know if that box contains only oranges or a mix of apples and oranges. You face a similar dilemma if you choose box 2 (the box labeled “oranges”) and you see an apple.

Let’s see what happens if you choose box 3 (the box labeled “apples & oranges”). Let’s assume you see an apple drawn from box 3. You therefore know box 3 contains only apples. Why? It can’t contain a mix (because you already know it’s mislabeled) and it can’t contain only oranges (because you now know it contains at least one apple). There’s only one possibility left: box 3 contains only apples.

Now think about box 2, which is labeled “oranges”. Since we know it’s mislabeled, it can’t contain only oranges and we just discovered that box 3 contains only apples so box 2 must contain a mix of apples and oranges. By process of elimination, we then know that box 1 contains only oranges.

So, if the fruit removed from box 3 is an apple, then the correct labeling is as follows:

- Box 1: “oranges”
- Box 2: “apples & oranges”
- Box 3: “apples”

By a very similar process, the details of which I’ll leave as an exercise to the reader :), if the fruit removed from box 3 is an orange, then the correct labeling is as follows:

- Box 1: “apples & oranges”
- Box 2: “apples”
- Box 3: “oranges”

There you have it – if you wisely choose the box with the mix of apples and oranges for your free sample, everything falls into place. And if you worked that out for yourself, orange you glad I didn’t tell you the answer? Sorry, I couldn’t resist.

Hats off to Muzaffer, Maurice, Al, Simon and Kimberly for finding the answer!

easy this week :)

select from box 3;

if it is an apple then;

box1 : orange

box2 : mixed

box3 : apple

if it is an orange then;

box1 : mixed

box2 : apple

box3 : orange

All 3 boxes are mislabeled. Thus there are only 2 possibilities for each box. Take Box 3 as an example. If can’t be A&O, therefor it leaves 2 choices: If box 3 is A, box 2 must be A&O, Box 1 is therfore O. If box 3 is O, then box 1 must be A&O, therefore box 2 is A.

If you start with box 1 or box 2 you find the same 2 possibilities for the 3 boxes as the sequence above. And since we know box 3 mus be A or O-that’s the one to pick from.

Box1 (apples) == (! apples) == oranges || (apples + oranges)

Box2 (oranges) == (!oranges) == apples || (apples + oranges)

Box3 (app+oran) == (!apple && !orange)

I just finished cutting up some apples and mandarin oranges (plus mango + grapefruit).

So maybe I could come up with a more complex problem :-)

Box 3 is the only box to pick from. Since it’s labeled a+o and we know it’s wrong,

whatever fruit is shown defines the box as all apples or all oranges.

So we know box 3.

Assume box 3 is the real apple box. With that we can look at box 2 which

based on the wrong label must be either all apples or apples+oranges.

But since box 3 is the apple box, box 2 is the apples+oranges box

which leaves box 1 as the oranges box.

So if box 3 shoes apple, we have box1=orange, box2=apples+orange, box3=apple

If box 3 shows orange, we have box1=apple+orange, box2=apples, box3=orange

Since all of the boxes are mislabeled they can’t be what they are labeled to be so that eliminates one of the options.

To solve:-

Pick one fruit from box 3 (Apples and Oranges)

If its an apple. Then that box must be the box containing just apples. Therefore, the box marked \oranges\ can’t be the box containing just apples, and it can’t be the box containing just oranges either so it must be the box containing apples and oranges. The remaining box is therefore the box containing just oranges.

If its an orange. Then that box must be the box containing just oranges. Therefore, the fox marked \apples\ can’t be the box containing just oranges, and it can’t be the box containing just apples either so it must the the box containing apples and oranges. The remaining box is therefore the box containing just apples.