$\textbf{Heist}$
The chief detective hurries down to the police station after hearing big news: there is a heist at Pi National Bank! The police has brought in seven known gang members seen leaving the crime scene. They belong to the nefarious True/False Gang, so named because each member is required to either always tell the truth or always lie. Although everyone is capable of engaging in wrongdoing, the chief also knows from his past cases that any crime committed by this gang always includes one truth teller. When the chief shows up, he asks the gang members the following questions:- Are you guilty?
- How many of the seven of you are guilty?
- How many of the seven of you tell the truth?
Here are their responses:
- Person $1$: Yes; $1$; $1$
- Person $2$: Yes; $3$; $3$
- Person $3$: No; $2$; $2$
- Person $4$: No; $4$; $1$
- Person $5$: No; $3$; $3$
- Person $6$: No; $3$; $3$
- Person $7$: Yes; $2$; $2$
After looking these answers over, the chief correctly arrests those responsible gang members.
Who out of these seven are arrested?
$\textbf{Answer}$
Person $2$, $3$ and $4$.
$\textbf{Analysis}$
Truth tellers must give the same answers to the $2^{nd}$ and the $3^{rd}$ questions. Hence, they must be among one of the following groups:
- Person $1$
- Person $2$, $5$, and $6$
- Person $3$ and $7$
- Person $4$
If the person $1$ is the only truth teller, then he is the only guilty person. However, this will contradict to person $3$ to $6$ who are supposed to lie.
Similarly, if person $4$ is the only truth teller, then there should be four guilty people. This means that these four people should answer No to the first question. This is not the case.
If person $3$ and $7$ are the only truth tellers, then there should be one liar who is guilty. However, three answer No while we expect only one will deny his involvement.
Hence, the truth tellers must be $2$, $5$, and $6$. And the criminals are $2$, $3$, and $4$.
$\textbf{Note}$
To solve this type of problems, it is always helpful to focus on extracting something certain out of given uncertainties. In this case, while we do not know who are truth tellers, realizing that "all the truth tellers must give the same answer to the second and third questions" is the key to solve this problem.