Logic puzzle - 3 suspects

Question

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?

Answer 1

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.

Answer 2

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

Answer 3

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.

Answer 4

Here's another possible answer:

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!

Answer 5

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.

Answer 6

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).

Answer 7

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.

Answer 8

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.