logical deduction - Who Are The Guilty Parties? #2

You've barely got back from the island where your last case was, when you hear that you are needed again:

There is a group of people at an isolated villa. In a curious twist, the aristocrats have all been murdered. You have been sent to collect statements from each person.
- Each person in the group is either guilty or innocent.
- Each person is either friends or not friends with each other person. The friendship is mutual.
- Each guilty person is friends with every other guilty person.
- Each innocent person has witnessed a number of events, which allows them to deduce the innocence or guilt of a number of people in the group. This may or may not have been reciprocated.
- The guilty people know who the guilty and innocent parties are.
- Each person is either boastful or modest (to be explained below).

The statements that each person makes are as follows:
- Innocent modest people will simply make a list of truthful statements.
- Innocent boastful people will do the same, but will also make a statement about a random other person about whom they know nothing. If the person is their friend, they will say s/he is innocent. Otherwise they'll just say something (innocent or guilty at random).
- Guilty modest people will say that friends are innocent (regardless of whether they are); they will say that non-friends are guilty (which is inaccurate, because their non-friends are innocent by the above definitions).
- Guilty boastful people will do the same. But they will also make one truthful statement (either that one of their guilty friends is guilty or that one of their innocent non-friends is innocent).

This situation seems all too common these days...

On this particular day, you go out to the island and get to sit down with the main characters:
Elton, Roxann, Lamar, Laine, Diego, Marylee, Mika, Mendy, Tracey, and Ruth

This is what you're able to glean from each one:

Statements of Elton:
- I am innocent
- Mika is guilty
- Lamar is innocent
- Laine is innocent

Statements of Roxann:
- I am innocent
- Elton is innocent
- Laine is innocent

- Lamar is guilty
- Mika is guilty
- Tracey is innocent
- Diego is innocent

Statements of Lamar:
- I am innocent
- Laine is innocent
- Mika is innocent
- Roxann is guilty
- Tracey is guilty

Statements of Laine:
- I am innocent
- Mika is guilty
- Mendy is guilty
- Tracey is innocent

Statements of Diego:
- I am innocent
- Elton is guilty
- Ruth is innocent
- Lamar is guilty

- Roxann is innocent

Statements of Marylee:
- I am innocent
- Mika is innocent
- Elton is guilty
- Laine is innocent

Statements of Mika:
- I am innocent
- Marylee is innocent
- Roxann is guilty

- Diego is innocent
- Laine is innocent
- Tracey is guilty

Statements of Mendy:
- I am innocent
- Laine is guilty
- Roxann is innocent
- Diego is innocent
- Lamar is innocent
- Mika is guilty

Statements of Tracey:
- I am innocent
- Laine is innocent
- Mika is guilty
- Elton is innocent

Statements of Ruth:
- I am innocent
- Mika is innocent
- Elton is innocent
- Laine is innocent

It seems like a hopeless case, but then you remember occam's razor. The simplest solution is likely to be it, in this case, the solution with the fewest number of co-conspirators is likely to be the answer.

Who murdered the aristocrats?

Answer

Answer:

Lamar, Mika, Mendy, and Marylee murdered them.

Also, if we'd rather not use "Occam's Razor" as our evidence in court, we can comfortably arrest Lamar, Mika, and Mendy, while letting Roxann and Laine go (as these are consistent across all six possible solutions).

Logic:

Let's start with a statement of "facts":

Name - Innocent (Not Innocent)

Elton – Lamar, Laine (Mika)
Roxann – Elton, Laine, Tracey, Diego (Lamar, Mika)
Lamar – Laine, Mika (Roxann, Tracey)
Laine – Tracey, (Mika, Mendy)
Diego – Ruth, Roxann (Elton, Lamar)
Marylee – Mika, Laine (Elton)
Mika – Marylee, Diego, Laine (Roxann, Tracey)
Mendy – Roxann, Diego, Lamar (Laine, Mika)
Tracey – Elton, Laine (Mika)
Ruth – Mika, Elton, Laine

So let's try to pair them into two groups, where "innocent" means same group and "guilty" means other group.

Assumption: Roxann and Tracey are in opposite groups.

In this case, both Lamar and Mika are boastful, with this as their only boast (either they are claiming an innocent or a friend guilty, depending on whether they're innocent or guilty). As a result, Mika must be friends with Laine (if Mika is guilty) or Laine must be innocent (if Mika is innocent). Therefor, when Laine says Mika is guilty, that must be his boast (either they're both innocent, or they're both friends). Therefor, Mendy must be in the other group from Laine.

Group 1: Mendy

Group 2: Laine

If Mika was innocent, then Marylee, Diego, and Laine must be innocent (because Roxann or Tracey is a boast), which means Laine's claim that Mika is guilty is a boast, so Tracey is innocent, which means her claims about Mika are a boast, so Elton is innocent, which means his claim about Mika is also a boast, so Lamar is innocent. But then Diego is innocent and wrong about both Elton and Lamar. Thus, our original guess was wrong, and Mika is guilty.

Group 1: Mendy

Group 2: Laine

Guilty:

Mika

If Roxann was guilty, then Mika must be her boast, so Lamar is truly innocent. Tracey would also be innocent, since she's in the other group from Roxann. But then Lamar is boasting twice - he's wrong about both Mika and Tracey. This can't be, so Roxann is innocent.

Since Roxann is innocent, Tracey must be Guilty (and a boast), so Elton, Laine, and Diego are innocent and Lamar must be guilty (and Mendy, being in the other group from Laine, must be guilty). Diego is wrong about Elton, so that must be his boast, and Ruth is innocent. Marylee is wrong about both Mika and Elton, so Marylee must be guilty.

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Possible Solution 1:

Innocent:
Roxann, Elton, Laine, Diego, Ruth

Guilty:
Mika, Tracey, Lamar, Marylee, Mendy

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Alternative: Roxann and Tracey are in the same group

Group 1: Roxann, Tracey

Neither Mika or Lamar can be in their group, because both of them claim they're both guilty. For innocent or guilty, that would be two boasts. Since we only have two total groups, Mika and Lamar are together.

Group 1: Roxann, Tracey

Group 2: Mika, Lamar

Assumption: Laine and Mika are in the same group

This means Laine is with Mika and Lamar

Mendy says Laine and Mika are both guilty, so Mendy is not in their group (otherwise, that would be two boasts).

Group 1: Roxann, Tracey, Mendy

Group 2: Mika, Lamar, Laine

But now Laine says that Tracey is innocent and Mika is guilty. if Laine is innocent, then these are both boasts based on who shares a group. Thus, Laine must be guilty, and friends with Tracey (where Mika is still a boast).

Innocent:
Roxann, Tracey, Mendy

Guilty:
Mika, Lamar, Laine

But Roxann is innocent and said Laine was innocent - that's a boast, so Elton is innocent. But then Elton says Lamar and Laine are innocent, which are both boasts! This is inconsistent, so our assumption was false.

Conclusion: Laine and Mika are in different groups

Group 1: Roxann, Tracey, Laine

Group 2: Mika, Lamar

Assumption: Mika is innocent

This means we have:

Innocent:
Mika, Lamar

Guilty:
Roxann, Tracey, Laine

Mika is innocent and says that Laine is innocent, so that must be her only boast. As a result, Marylee and Diego are innocent.

Innocent:
Mika, Lamar, Marylee, Diego

Guilty:
Roxann, Tracey, Laine

But then Diego is innocent, but claims Roxann is innocent and Lamar is guilty - these are both boasts, which is inconsistent. Thus, our assumption was false.

Conclusion: Mika is guilty

Now we have:

Innocent:
Roxann, Tracey, Laine

Guilty:

Mika, Lamar

If Mendy were innocent, claiming Lamar innocent and Laine guilty would be two boasts, so Mendy is guilty.

Innocent:
Roxann, Tracey, Laine

Guilty:
Mika, Lamar, Mendy

At this point, we've used all of Laine's statements, so we can stop worrying about him.

Assumption: Diego and Elton are in the same group

In this case, Diego must be boasting when he says Elton is guilty, so Lamar can't be a boast, which means Diego (and Elton) are innocent. Also, Diego's claim about Ruth can't be a boast, so she's innocent.

Innocent:
Roxann, Tracey, Laine, Diego, Elton, Ruth

Guilty:
Mika, Lamar, Mendy

If Marylee were innocent, her claims about Mika and Elton would be boasts, so Marylee is guilty.

---

Possible Solution 2:

Innocent:
Roxann, Tracey, Laine, Diego, Elton, Ruth

Guilty:
Mika, Lamar, Mendy, Marylee

---

This means Ruth is boasting about Mika, Tracey is consistent, Mendy is friends with Roxann and Diego and boasting about Mika, Mika is friends with Deigo or Laine or both, Marylee might be friends with Laine, Diego is boasting about Elton, Lamar might be friends with Laine, Roxann is consistent, and Elton is boasting about Lamar. Everybody's statements line up, so this is a possible solution.

Alternative: Diego and Elton are in different groups

Innocent:
Roxann, Tracey, Laine

Guilty:
Mika, Lamar, Mendy

in this case, Roxann has no useful knowledge to add (we know she boasted, but not about whom). Tracey may or may not be boasting, but we don't gain any knowledge from her.

Assumption: Ruth is Innocent

In this case, Mika is a boast, so Elton is innocent, thus Diego is Guilty. Marylee can't be innocent because then Mika and Elton would be boasts, so Marylee is guilty. This gives:

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Possible Solution 3:

Innocent:
Roxann, Tracey, Laine, Ruth, Elton

Guilty:

Mika, Lamar, Mendy, Diego, Marylee

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Which is also consistent (though it does imply that Laine is a very friendly person!)

Alternative: Ruth is guilty

In this case, we have:

Innocent:
Roxann, Tracey, Laine

Guilty:
Mika, Lamar, Mendy, Ruth

[Diego and Elton in different groups]

At this point, for all remaining possibilities, there must be at least five guilty parties (Mika, Lamar, Mendy, Ruth, and Diego or Elton). But remember our last clue: "The solution with the fewest number of co-conspirators is likely to be the answer". And we already have a solution with only four guilty people! Thus, the solution with four guilty people must be the correct answer (assuming that Occam's razor works for detective work). Done!

For completeness:

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Possible Solution 4:

Innocent:
Roxann, Tracey, Laine, Elton

Guilty:
Mika, Lamar, Mendy, Ruth, Diego, Marylee

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Possible Solution 5:

Innocent:
Roxann, Tracey, Laine, Deigo, Marylee

Guilty:
Mika, Lamar, Mendy, Ruth, Elton

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Possible Solution 6:

Innocent:
Roxann, Tracey, Laine, Deigo

Guilty:
Mika, Lamar, Mendy, Ruth, Elton, Marylee

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