How did she do that?

Question

The King wants to test his 2 very smart advisors Sadin and Lena. In the courtyard he sets up 4 closed wooden boxes on a table. All are the exact same size. In front of each box there is a name plate.

The King announces: “ Sadin and Lena, see these 4 sandalwood boxes? Each one has a fruit inside. One has a Pomogranate, one has an Orange, One has a Pineapple and one has an Apple. The names on the name plates in front of these boxes are not exactly right. Only one box has the right fruit name in front of it. All the others are mislabeled. Your job is to put all the boxes in the right places so they are properly labeled. You can open as many boxes as you want. BUT, one who opens the least number of boxes shall win! So what do you say?”

Sadin thought about it. He said, “ It is obvious that you have to open 2 boxes to accomplish that His Majesty”

Lena smiled and said,” I can easily beat that His Majesty”

Sadin was shocked.”Show me!”

Lena went to the table. She started rearranging the boxes without opening them. When she was finished rearranging the boxes, she smiled and said,” Check them yourself Sadin. I did not open a single box”

Sadin checked all the boxes. All were now correctly labeled.

What happened? Both had no prior knowledge of this test.

Explain: 1 Why did Sadin say he must open 2 boxes?

2 Why did Lena say she could do better?

3 Why was Lena able to do it without opening a single box?

Answer 1

1. Sadin says he needs to open 2 boxes.

Since the king anounced that exactly one fruit was in the right box, it means that the three other fruits are in a 3-cycle.

Sadin opens one box first (wlog, the one with an "apple" label).

if it contains an apple, there are only two options for the 3-cycle: pomegranate -> orange -> pineapple or pineapple -> orange -> pomegranate. Opening any of the three other boxes gives him all information needed.

Meanwhile,

if it contains another fruit (wlog, orange), then the only two possibilities are apple -> orange -> pomegranate or apple -> orange -> pineapple. Once again, opening any other box solves the problem and he can rearrange all the box in front of the correct names.

2. Lena says she could do better...

... because she could.

3. She even has 2 solutions:

First :

Because the boxes are identical, you can expect the one with an ananas to feel heavier than the one with a pomegranate, which itself weights more than the one with an orange, and the one with an apple is the lightest. Lena can lift each box, compare weights, and solve the problem.

Alternatively :

Wooden boxes let the smells out, and ripe fruits are odorant. Lena can easily tell an orange from an apple or an ananas by smelling them.

Of course, these two solutions can be combined for more confidence in the result.

She could even :

Shake the boxes to get clues from the sound the fruit inside makes. After all, since the wood prevents sight and taste seems unpractical, then touch, smell, and sound are the available senses to get information about the inside of the boxes !