# Logic puzzle - 3 suspects

An inspector knows that exactly one of 3 suspects committed a crime, and interviews them to find out which. Each person lies one time, and tells the truth the other time.

A says: I did not do it. B did it.

B says: I did not do it. I know that C did it.

C says: I did not do it. B does not know who it was.

Can the inspector figure out the culprit? If so, who is it?

• As VictorHenry points out, we need more information to solve this puzzle. Is it guaranteed that at least one of $A,B$ or $C$ committed the crime? May 11 '15 at 19:07
• It says, he(inspector) knows exactly that one of three persons is culprit and each person one time says true and one time lies
– lesa
May 11 '15 at 19:10
• The answer to the first question"Can the inspector figure out the culprit?" is "no" unless the inspector also knows that "Each person lies one time and tells the truth the other time." If he does, he should be able to solve the crime before interviewing B and C. May 13 '15 at 13:03

If we assume that there are no contradictions in the puzzle (i.e. there cannot be a situation where the "1 lie, 1 truth" rule is violated), then we don't even need to read what B and C say. Only A's statement matters.

If A's statements are Lie, Truth respectively, then his combined statement is that he did it (opposite of first statement) and B did it. This is clearly impossible, so it must be the case that A's statements are Truth, Lie respectively. This means A didn't do it, B didn't do it, so C must have done it. B and C's statements are irrelevant.

• You also has to show that the system is consistent. So both B's and C's statement has to contain one lie and one thruth if C did it. There is a catch in B's statements. May 12 '15 at 7:31
• Well, B is lying about knowing C did it. Doesn't mean he's not coincidentally guessing correctly. Similarly, since C is lying about having done it, he telling the truth about B not knowing (for sure) who it was. May 12 '15 at 9:03
• You can edit to remove the last line now since the assumption was added to the question criteria. May 12 '15 at 13:03
• there you go, upvote and you're at 5k rep :D Dec 3 '15 at 8:37

Each person makes one true statement and one false statement.

If A did it:
I was not is FALSE
B did it should be TRUE but is FALSE

If A committed the crime, then both of A's statements are false. This contradicts the rules of the riddle.

If A didn't do it:
I was not is TRUE
B did it should be FALSE.

At this time, we cannot confirm or deny this statement.

If B did it:
I was not is FALSE
I know that C did it. should be TRUE but is FALSE

Much like A, B cannot give two false statements. If B committed the crime, then B would know that B did it, thus making the second statement false.

If B didn't it:
I was not is TRUE
I know that C did it. should be FALSE

Again, we don't entirely know if B's second statement is false or not.

If C did it:
I was not is FALSE
B do not know who was it. is TRUE

Ok, so this is where things get interesting. If C is responsible, that we know that the second statement is true. That also means that the second case for B needs to be true as well. This means that B doesn't know who did it.

NOTE

This does not mean that it isn't C. It only means that B doesn't know C did it. He's just a lucky guesser, that's all.

This also means that A's second case should become true as well. A stated that B did it. We know, in this case, that is false. This means that 'A didn't do it' works out logically.

If C didn't do it:
I was not is TRUE
B do not know who was it. could be TRUE

Obviously C's claim of innocence is true. This means that C's statement that B is wrong must be FALSE. If that is the case, then B must be telling the truth on that line and lying on the first. We have already logically concluded that is contradictory and as such impossible.

Therefore, the only logically valid statement of the six above is C did it

• This is a bit confused. If A did it, then "B did it" isn't TRUE; and if B did it, then "I know that C did it" isn't TRUE either! May 11 '15 at 18:20
• That is why the statements are contradictory. Let me see if I can reword to make it more clear. May 11 '15 at 18:21
• OK, it makes more sense now. I still think my answer is a lot simpler and just as logically sound! May 11 '15 at 18:42

First assume A did it. Then A's statements "I did not" and "B did it" are both false. Contradiction, so A didn't do it.

Now assume B did it. Then A's statements "I did not" and "B did it" are both true. Contradiction, so B didn't do it.

The only possibility remaining is that C did it. Now A's 1st statement and B's 1st statement are true while A's first statement and C's first statement are false, so we can also deduce that B doesn't know who did it.

• The tricky bit here is that B doesn't know who did it. However, he can be totally guessing that C did it and be right. May 11 '15 at 18:40
• @tfitzger No, because B says "I know that C did it." Edit: OK, you're just saying that's what makes the puzzle tricky. I think both our answers are logically sound :-) May 11 '15 at 18:41

None of them did it! (The question doesn't specify that one of the three suspects actually did it.)

Here's how:

Assume that all three "I didn't do it" statements are true. That means we just have to reconcile the three following statements so that they are all false:

B did it.
B knows that C did it.
B doesn't know who did it.

Or in other words, show that the following three statements can be true:

B didn't do it.
B doesn't know that C did it.
B does know who did it.

Let's say Person D actually did it, and B knows it. Then:

B didn't do it. (TRUE!)
B doesn't know that C did it. (TRUE! He doesn't know that C did it, he knows that D did it!)
B does know who did it. (TRUE! He knows that D did it!)

Therefore, none of them did it!

• "An inspector knows that exactly one of 3 suspects committed a crime". I think that rules out your answer.
– Rene
May 13 '15 at 13:52
• Yeah, this was written before that update was made. May 13 '15 at 17:51

So if the system is consistent, following logic brings us a conclusion that

C did it.

How? that's simple

A says: (a) I did not do it. (b) B did it.
Assume that A(a) is TRUE and A(b) is FALSE

B says: (a)I did not do it. (b)I know that C did it.
being consistent with A(b), we know that that B(a) is TRUE, which means that B(b) is FALSE

C says: (a)I did not do it. (b)B does not know who it was.
being consistent with B(b), C(b) is TRUE since !B(b) and C(b) state same thing, which means that C(a) is FALSE.

so C did it.

The Culprit could be:

C

logic:

A says: I did not do it [TRUE]. B did it [FALSE].
B says: I did not do it [TRUE]. I know that C did it [FALSE](just a guess).
C says: I did not do it [FALSE](which means C's the culprit). B does not know who it was [TRUE](which means that B was just guessing).

The culprit is

C

Because,

suppose C's first statement is true, second must be false....... that leads B's first statement is true, second is false....... that leads first statement is true, second is false........ That gives no culprit.

But An inspector knows that exactly one of 3 suspects committed a crime(I won't say, 'none of them')

that clearly means C's first statement is false, and second is true.... that leads B's first statement is true, second is false...... that leads A's first statement is true, second is false. That is it.

As many have already explained that first statement is proof enough to spot the convict, i.e

A says:

I did not do it.    B did it.
TRUE(Its not A)     FALSE(Its not B) => Its C
FALSE (Its A)       TRUE( Its B)     => Contradiction. Hence, not possible.
Conclusion 1 => C did it.


Explaining other statements:

B says:

I did not do it.       I know that C did it.
TRUE (Conclusion 1)  FALSE
Conclusion 2=> B does not know that C did it


C says:

I did not do it.       B does not know who it was.
FALSE (Since C did it)  TRUE (Conclusion 2)


Hence, all the statements have one TRUE and FALSE statement and the convict is C.