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19 votes
19 votes

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?

  1. $P$
  2. $R$
  3. $S$
  4. $Q$
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6 Answers

Best answer
18 votes
18 votes
Option B. R
Assuming S is saying the truth.
P says Q committed the crime (FALSE) means Q has not
Q says S committed the crime(FALSE) means S has not
R says I did not (FALSE) means R has committed the crime.
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42 votes
42 votes

Consider Subscript C as criminal, NC as not criminal.

Case I: Consider P as True             

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 also $S_{NC}$ and $S_C $ in the result which is a contradiction. [S committed the crime and same time not committed crime which is contradiction]

 

Case II: Consider Q as True    

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: Consider R as True  

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

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

 

Case III: Consider S as True  

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

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

Therefore R Committed the crime which is option B in Question

6 votes
6 votes

S says: “What Q said about me is false”.

We can paraphrase this as

S says: I did not do it,

So, one of S and Q is telling the truth, the other is lying. And it is given that exactly one of P,Q,R,S is telling the truth.
=> Either S or Q is the only person telling the truth.

Suppose S is telling the truth => R is lying (so are others) => R did it.

Suppose Q is telling the truth => R is lying => R did it.

 

Option B

4 votes
4 votes
R committed the crime.

If we take S's statement as true and the remaining 3 statements as false. If what S says is true, then :

i. Q said that S committed crime which turns out to be false.

ii. R said that he didn't do it (if false) then it means R committed the crime.

iii. And the last statement said by P also becomes false.

In considering any of the other statements as true the remaining of the three will contradict with each other.
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