Question

The police arrested four criminals - P, Q, R and S. The criminals knew each other. They made the following statements:

P says " Q committed the crime."
Q says " S committed the crime."
R says " I did not do it."
S says " What Q said about me is false."

Assume only one of the arrested four committed the crime and only one of the statements made above is true. Who committed the crime?

• A. R
• B. P
• C. S
• D. Q

Answer 1

The correct option is A R

Suffix C as criminal, NC as not criminal.

Case I:

Criminals	P	Q	R	S
Assumption	T	F	F	F
Result	$Q_C$	$S_{NC}$	$R_C$	$S_C$

Q and R are criminal in the result is impossible because only one person committed the crime.

Case II:

Criminals	P	Q	R	S
Assumption	F	T	F	F
Result	$Q_{NC}$	$S_C$	$R_C$	$S_C$

S and R are criminal in the result is impossible because only one person committed the crime.

Case III:

Criminals	P	Q	R	S
Assumption	F	F	T	F
Result	$Q_{NC}$	$S_{NC}$	$R_{NC}$	$S_C$

$S_{NC}\text{and}{S}_C$ in the result which is contradiction. [S committed crime and same time not committed crime which is contradiction]

Case IV:

Criminals	P	Q	R	S
Assumption	F	F	F	T
Result	$Q_{NC}$	$S_{NC}$	$R_C$	$S_{NC}$

R is criminal in the result.
Hence, this case satisfies only one person committed the crime.

So, option (a) is correct.