Cop, thief and spy!

The only person who can say "I am a thief" is a spy, since cops never lie and thieves can never tell the truth. Thus, Person 2 is a spy. This means that Person 3 is either a thief or a cop. If Person 3 were a cop, then they would be lying by saying "I am a spy," which is not consistent with the behavior of a cop. Therefore, we deduce that Person 3 must be the thief. (This would make Person 1 the cop.)

As a check, we can test the validity of everyone's statements based on their role:

• Person 1 says "I am a cop" : This is true, so it is consistent with them being a cop.
• Person 2 says "I am a thief" : This is false, so it is consistent with them being a spy (since a spy can either tell the truth or lie).
• Person 3 says "They are both lying, and I am a spy" : Both parts of this statement are not true, so it is consistent with them being a thief.

Person 1: I am a cop........................this must be true. because , cops never tell lie and without person 1 ,nobody claiming themselves as a cop.so, no collapse in that

Person 2: I am a thief...................................................as thief always tells lie. so he will never concede himself as a thief.so, he must be the spy.

at last, there left only 1 person ---Person-3. so, he must be the thief.

Claiming to be a cop can be done by anyone, whether they are a truthful cop, a lying thief or a lying spy.

Claiming to be a thief is an action unique to a lying spy (special circumstances, allowable by the spies pass).

Claiming to be a spy can be done by a lying thief and a truthful spy.

Since we are told that there is one of each, then Person 2 is a spy by their uniqueness, Person 3 is a lying thief and Person 1 is a truthful cop.

It’s simple. The only person who can tell Statement 2 would be a spy (since its impossible for anyone else to say that without contradictions) and then now Statement 3 is a lie, thus person 3 is a thief.