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At the Korean Nation Police Agency (NPA) in Migeun-dong, Seoul, researchers discovered that 73% of the Senior Inspectors (경감, 警監) all came from the small village of Inducheon. At first, they thought it was nepotism. But the truth was far stranger. In this village, the elders set puzzles for the children that trained them to solve mysteries. These children became highly successful detectives.

The rules of this game are very simple. The child plays a detective who has to solve a murder mystery (with younger children they use a toy robbery). There is a group of people, some innocent, some guilty, and each makes a number of statements.

  1. Innocent people generally tell the truth. They might be mistaken in one of their statements. But if they lie twice, they are guilty.
  2. Guilty people cannot be trusted in general - they lie and tell the truth at will. But if they make a statement about another guilty person, they will always say that person is innocent.
  3. The smallest conspiracy that is consistent with the statements is the answer

The suspects are Armandina, Berenice, Laurice, Nguyet, Rosalind, Virgil.

In the following table, each line represents the statements made by one of the characters. For example: The I in the first line (Armandina) under the letter V represents a statement by Armandina that Virgil is innocent.

Statements     | A | B | L | N | R | V |
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Armandina      | I |   |   | I |   | I |
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Berenice       |   | I |   | I |   |   |
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Laurice        | I |   | I |   |   | G |
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Nguyet         | I | I |   | I | G |   |
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Rosalind       |   |   | G | G | I | G |
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Virgil         | I |   | G |   | I | I |
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Who are the murderers? Can you figure it out?

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Laurice and Nguyet are the murderers.

Reasoning:

Start by looking at Rosalind and Virgil. If Rosalind is guilty, then Virgil must be innocent, because Rosalind is accusing him. If Virgil is innocent, then he is wrong about Rosalind, therefore Laurice must be guilty. Laurice is accusing Rosalind, so she would not be able to be guilty. If Virgil is guilty but Rosalind is innocent, then Laurice must be innocent because Virgil is accusing him. If he is innocent, and Rosalind is innocent, then Nguyet must be guilty, because Rosalind is accusing both N and L. If Nguyet and Virgil are both guilty, then so must Armandina, because she says they're both innocent. That leaves 3 conspirators.

If Virgil and Rosalind are both innocent, then Rosalind is incorrect about Virgil, so L and N are both guilty. No one else says both of them are innocent, and no one thinks one is innocent while also believing someone else to be guilty. Thus, L and N is the smallest conspiracy I was able to find within the rules.

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;tldr:

The key is Rosalind. She cannot be guilty. Because she cannot be guilty, neither can Laurice. Moreover, because she is not guilty and is wrong about Laurice, both Nguyet and Virgil must be guilty. But if both of them are guilty, then Armandina lied about the two of them and must also be guilty. The smallest conspiracy is Armandina, Nguyet, and Virgil.

1st Premise:

Say R is guilty. Then L, N, and V are not guilty. Then L is wrong about V, so A is not guilty. But V is also wrong about L and about R. That is inconsistent with V being innocent. So R cannot be guilty. That means that there are at least 2 conspirators, and that at most one of L, N, and V are innocent.

1st Conclusion:

R is not guilty. Also, at most one of L, N, and V are not guilty.

2nd Premise:

Say L is guilty. Then V is not guilty. But that means R is wrong about both L and V, meaning R is guilty, which is inconsistent. So L is not guilty.

2nd Conclusion:

L is not guilty. Add that to the 1st Conclusion, and we surmise N and V must both be guilty.

So...:

If A is innocent, then he is mistaken about BOTH N and V. That won't work, so A is guilty.

What about...?

That takes care of A, L, N, R, and V. But what about B? If B is innocent, then she could have been wrong about N. N lied about B. So that is consistent. If B is guilty, then she lied about N and N lied about her. Also consistent. So we cannot rule B in or out of the conspiracy.

Bottom line (solution):

A, N and V are guilty. B might be guilty, might not. R and L are not guilty. That leads to 2 solutions consistent with the evidence: 1. A, B, N, V guilty 2. A, N, V guilty But we know that the solution with the smallest number of conspirators is correct. Therefore B is not guilty, and our conspirators are A, N, and V.

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  • $\begingroup$ Good effort. But L,N also works and is smaller. With 3 in the conspiracy, it is also possible that it would be ALN or BLN (as well as ANV as you found). $\endgroup$ – Dr Xorile Dec 26 '18 at 22:52
  • $\begingroup$ L, N does not work. As noted, R is not guilty. R said L, N, and V were guilty. R is wrong about, at most, one of these. L is not guilty. That means BOTH N and V must be guilty. $\endgroup$ – Matt Jan 28 at 12:28

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