# Puzzle about convict who chooses length of his imprisonment

This is the puzzle I deal with for few days right now and I can't figure it out. I translated it from other language if I can improve post please comment.

In the end of a trial judge sentenced a convict for some years of prison, but he let the convict to choose how many years he will spend there.

Judge placed 12 boxes in the circle. In clockwise order the numbers on the boxes were: 7, 10, 1, 3, 6, 11, 8, 4, 5, 9, 0, 2.

Judge said that in every box there is a certain number of coins. In clockwise order amount of coins in boxes were: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

Convict had to choose a box. The number of coins determined years of his/her imprisonment.

Convict asked judge how many boxes have number equal to number of coins inside. Judge said he won't say because then convict could determine empty box. Convict selected empty box.

How he found out which box was empty?

• There are 12 possibilities for how the two scales correspond up. Check all of them one by one. If the story makes sense there will be exactly one position that doesn't share a number of matches with any others. – Henning Makholm Apr 14 '14 at 23:08
• Also, I'm sure that crazy sentencing procedure is going to be overturned on appeal. So ultimately it doesn't matter. – Henning Makholm Apr 14 '14 at 23:10

Call a box "accurately labeled" if it has the same number of coins in it as its label. There are several ways in which none of the boxes could be accurately labeled, so at least one of them must be accurately labeled. The sets of boxes that could be accurately labeled are $\{0,6,8\},\{1\},\{2\},\{3,9\},\{4,5,10\},\{7\},\{11\}$. The only one of these sets with a unique cardinality is $\{3,9\}$, so those must be the accurately labeled ones. This means the box labeled 7 has no coins in it.

• Please could you explain what do you mean by unique cardinality? Why for instance $\{4,5,10\}$ is the wrong set? – Yoda May 31 '17 at 5:56
• Ah it's the only one set with two elements in it, got it. – Yoda May 31 '17 at 6:01
• So that means that box nr 7 is the right one? – Yoda May 31 '17 at 6:21

I found the answer by writing down the numbers

0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10

on a piece of paper and writing down the numbers

7 10 1 3 6 11 8 4 5 9 0 2

on another piece of paper and sliding one against the other and counting the number of matches.

5 positions have 0 matches.

3 positions have 1 match.

3 positions have 2 matches.

1 position has 3 matches. That is the one that saves the convict from prison.

• But how he know which one from this 3 matches is 0 coins? – Yoda Apr 15 '14 at 0:40
• The method in this solution is correct, but the result is wrong, please look at the awarded answer. – Yoda Jun 4 '17 at 14:07