A Space Station 13 tricky situation

Question

You are the Captain of the Space Station 13, and everything went bad very quickly. Most of the crew died, and it took you not so long to figure out that some people are not who they're supposed to be (hence the bloody mess). Your task is to find out who is who in order to prevent the bad guys from reaching the escape shuttle.

There are 8 other crewmen with you: the Security Officer, the Medical Doctor, the Chief Engineer, the Cook, the Cargo Technician, the Botanist, the Janitor and the Clown (every station needs a Clown). You know that 4 of them are innocent, and that the 4 others are a Traitor, a Changeling, a Cultist and a Space Wizard. You also know for facts that:

• Everybody knows who is who (well... except you, the Captain, which is quite a shame).
• The only exception to the previous fact is that nobody knows who the Changeling is (except himself of course), and everybody believes the crewman the Changeling is impersonating is innocent.
• All innocent crewmen say the truth, or what they believe is the truth.
• The Cultist also says the truth (or what he believes is the truth), as part of a complicated ritual to summon N̻͔͓̳͇͖̝͠a̲͙̟͓̐ͧ̇̍̈ͥ̚͜r̲̠͇̲͈̐-͙̜͕̪͕͓͍̒̎̀S͈͕̅̾̕ͅi̙͋̎̒ͣ͗e̱͋̈́.
• The Traitor, Changeling and Space Wizard all lie (or what they believe is a lie).
• The Space Wizard will put a curse on you after you interrogated him, making you think the next crewman you interrogate said the opposite of what he actually said. For example, if you hear the crewman you interrogate just after the Space Wizard saying "The Janitor is a Cultist", what he actually said was "The Janitor is NOT a Cultist".

And this is what everyone said to you (in order of interrogation):

Security Officer: "The Medical Doctor is innocent."

Chief Engineer: "The Botanist is not innocent..."

Janitor: "The Clown is a Space Wizard!"

Cargo Technician: "The Chief Engineer is a Cultist!"

Medical Doctor: "The Cook is a Changeling!"

Clown: "The Janitor is a Traitor!"

Botanist: "The Security Officer is innocent."

Cook: "The Cargo Technician is a Traitor!"

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Remember, your goal is to assert the identity of everyone, not just sort bad guys and innocents. The Traitor must be shot in the head, the Changeling burnt to ashes, the Cultist spaced and the Space Wizard properly dismembered.

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_{If you liked this puzzle, try the rest of the story!}

Answer 1

Traitor: Janitor, Changeling: Cook, Cultist: Botanist, Space Wizard: Cargo Technician

The rest is innocent. Explanation:

First, observe that there are three people lying and five speaking the truth. Depending on where the Space Wizard is, two to four people's statements are false, the rest is true. For simplicity's sake, I assume all crew members are male.
The Medical Doctor's statement is false, so either (1) the Cargo Technician is the Space Wizard and the Medical Doctor is either innocent or the Cultist, or (2) the Cargo Technician is not the Space Wizard and the Medical Doctor is guilty but not the Cultist. We'll see that (2) leads to a contradiction.
In case (1) the Cook lies and must be the Changeling or the Traitor. If he's the Traitor, the Clown lies and must be the Changeling. Then the Janitor also lies, which is impossible since we've already identified three lying crew members.
So the Cook is the Changeling. Now, if the Clown lies, he must be the Traitor himself. Everybody else speaks the truth, including the Cargo Technician, who claims the Chief Engineer is the Cultist, but then the Botanist is not innocent and must be lying, which is a contradiction.
So the Clown speaks the truth, the Janitor is the Traitor and everybody else speaks the truth. According to the Chief Engineer, the Botanist is the last guilty member; he must be the Cultist. You can check that the other statements are now true as well.
Case (2) leads to a contradiction: the Medical Doctor lies, so he's guilty and the Security Officer lies as well. The Botanist's statement is false, so the third (and last) lying member is either the Botanist himself (a) or he lies because the Clown is the Space Wizard (b) making him the third lying member.
Now if the Cook speaks the truth, the Cargo Technician would be the fourth lying person which is impossible. So the Cook's statement is false because the Botanist is the Space Wizard, but then the Janitor is also lying, which leads to a contradiction.

It's a ... tricky situation ... to explain; I might have made some mistakes and used 'lying/speaking the truth' where I meant 'statement is false/true' or vice versa. Captain Glorfindel out.

Answer 2

(Credit to @Glorfindel who got the same answer first and has now added what's probably a more coherent explanation than mine. So vote for @Glorfindel! I'll leave this here in case it helps as an alternate explanation)

If

nobody knows who the changeling is

Then the Medical Doctor's statement must be

a lie

Which would mean that the

Security Officer and therefore the Botanist are also lying

But that fails because

We end up with too many bad guys since most of the other accusations must be true

So the other alternative is that

the Cargo Technician is the Space Wizard - the Doctor actually says "not a changeling".

That then means both the

Janitor and Cook

are lying, so they're the other two lying roles. If the

Clown

is telling the truth, then

The Janitor is the Traitor, and the Cook must be the changeling

I got stuck at this point and couldn't make the very last bit fit, thought I'd gone wrong, but then I check Glorfindel's answer and that gave the missing link that:

Everyone else is telling the truth, which means both The Botanist and the Chief Engineer. If the Botanist isn't innocent and is telling the truth, they must be the Cultist.

So in summary:

Cargo Technician is the Space Wizard. Janitor is the Traitor. Cook is the Changeling. Their three statements are false. The Botanist is the Cultist, and tells the truth along with everyone else (including the Medical Doctor, who that Captain mishears)

Answer 3

Warning: This was just a brute force thing I worked on here and there throughout the day, so I'm late in answering.

A good place to start in this determination is the person that makes a claim about the Changeling. The Changeling is believed to be speaking a true statement by all personnel. Our friend the Medical Doctor has claimed that the Cook is a Changeling.

All players understand the Changeling's response to be true. The Changeling lies, so the response must be false. Because only the investigator is afflicted by the Space Wizard's curse, we know that this disparity cannot exist amongst the crew. Hence, the statement "The Cook is a Changeling!" is not spoken by someone that speaks the truth.

Let's just plot out all possibilities for the Medical Doctor, then.

Here, let's assume that a lying MD won't risk claiming that an Innocent Cook or Cultist Cook is a Changeling. Similarly, a truthful MD wouldn't risk claiming that an Innocent Cook or Cultist Cook is not a Changeling.

 MD(T)
   |- Ck(SW)
        |- SO(Ch)
             |- B must be lying, but no liars left. Contradiction.
   |- Ck(Ch) + CT(SW)
        |- SO must be lying, but no liars left. Contradiction.
 MD(Ch)
   |- Ck(T)
        |- SO(SW)
             |- B must be lying, but no liars left. Contradiction.
   |- Ck(SW)
        |- SO(T)
             |- B must be lying, but no liars left. Contradiction.
 MD(SW)
   |- Ck(T)
        |- SO(Ch)
             |- B must be lying, but no liars left. Contradiction.
 MD(Cu) + CT(SW)   --- ".. not a Changeling!"
   |- Ck(T)
        |- SO(Ch)
             |- B must be lying, but no liars left. Contradiction.
   |- Ck(Ch)
        |- SO(T)
             |- B must be lying, but no liars left. Contradiction.
 MD(I) + CT(SW)    --- ".. not a Changeling!"
   |- Ck(T)
        |- SO(I)
             |- B(I)
                 |- CE(Ch)
                      |- J must be lying, but no liars left. Contradiction.
             |- B(Cu)
                 |- CE(I)
                      |- J(Ch) + Cl(I)
                          |- An I would not claim a liar was Ch. Contradiction.
                      |- J(I) + Cl(Ch)
                          |- An I would not claim a liar was Ch. Contradiction.
        |- SO(Cu)
             |- B(Ch)
                |- CE(I) + J(I) + Cl(I)
                     |- One of J or Cl must be lying. Contradiction.
   |- Ck(Ch)
        |- SO(Cu)
             |- B(T)
                 |- CE(I) + J(I) + Cl(I)
                     |- One of J or Cl must be lying. Contradiction.
        |- SO(I)
             |- B(I)
                 |- CE(T)
                      |- J must be lying, but no liars left. Contradiction.
             |- B(Cu)
                 |- CE(I)
                      |- J(T) + Cl(I)
                          |- No contradiction. Success!