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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. $Q$

edited | 3.9k views
+2

this question is just picked from    (48:41)    #nptel..........

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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.
by (459 points)
selected
+1

you are saying that

Assuming S is saying the truth.

means S says what Q says about me is false

and Q is saying that S did not committed  crime (after negation )

now come to the point        Q is saying  S did not committed crime but S is saying what Q says about me is false means S committed the crime (contradiction)

so i think answer is P

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Q says : S committed the crime !  ===> it is false then S didn't committed the crime.

S says : What Q said about me is false ==> we are treating this as true

then where is contradiction ? @Gurdeep Saini

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@Shaik Masthan

The contradiction comes from R statement that says R didn't committed the crime as it is false both R and S will be culprit but the question says only one is culprit.

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I wonder why not the answer is P

When we assume R is saying truth. Then

P says “Q committed the crime.” (false) Q is free

Q says “S committed the crime.”(false) S is free

R says “ I did not do it.” (true) R is free

S says “What Q said about me is false”.(false) S is free and no contradiction

Which leaves P who is the committed the crime according to question one of them committed the crime and only one.
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S says “What Q said about me is false”.(false) S is free and no contradiction

if what S says is false, then what Q said is true, ==> Q said S committed the crime.

Then R and S both are criminals ==> contradiction , right ?

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

by Junior (527 points)
+1
well written and nicely explained
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thanks
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Very nice method.. One observation:

q says s did it and

S says what q said about me is false this means that q is saying i didn't do it.

Also r is saying i didn't do it.

If both of them are false then both r,s are criminals which violates the statement that only one person committed the crime.

Therefore one of 3 or 4 must be true.

Now we can apply your method for above 2 cases...
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thank you so much very nice explanation
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.
by (187 points)
edited
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Thanks for editing. :)
Option B-R

⇒ I true: 'Q' is Criminal
II, III, IV are false.
II is false: 'S' is not criminal.
IV is also false: 'S' committed crime.
There is conflict among I, IV now as P, S both should be criminals.
⇒ II true: 'S' is criminal.
I, III, IV are false.
IV is false. So, 'S' is not criminal.
III is false. So. 'R' is criminal.
There is conflict among II, III as S, R both should be criminal.
III is true: Criminal ≠ R.
I, II, IV are false.
I is false. So, 'Q' is also not criminal.
II is false. So, 'S' is not criminal.
IV is false. So, 'S' is not criminal.
Q, R, S are not criminals. So, 'P' is criminal.
IV is true: I, II, III are false.
II is false. So, 'S' is not criminal.
IV is true. So, 'S' is criminal.
I is false. So, 'Q' is not criminal.
III is false. So, 'R' is criminal.
There is conflict, as R, S both should be criminals.
by (117 points)
edited
+1 vote

Option A :  Suppose P committed crime. In that case

P says “Q committed the crime.” {incorrect as P committed crime}

Q says “S committed the crime.” {incorrect as P committed crime}   -------line1

R says “ I did not do it.”               { correct as P committed crime}

S says “What Q said about me is false”. {we already seen at line1 that  whatever Q  said is incorrect, hence this statement is correct}

Hence as can be seen there are two correct statement. Hence A is not correct option.

Option B :  Suppose R committed crime. In that case

P says “Q committed the crime.” {incorrect as R committed crime.}

Q says “S committed the crime.”{incorrect as R committed crime.} ---------line2

R says “ I did not do it.”              {incorrect as R committed crime.}

S says “What Q said about me is false”. {we already seen at line2 that  whatever Q  said is incorrect, hence this statement is correct }

Hence as can be seen there is only correct statement. Hence B is not correct option.

Option C :  Suppose S committed crime. In that case

P says “Q committed the crime.”{ incorrect as S committed crime}

Q says “S committed the crime.”  {correct as S committed crime}  ------line3

R says “ I did not do it.”                 {correct as S committed crime}

S says “What Q said about me is false”. {as can be seen on line3 whatever Q said is actually true, hence this statement is false/incorrect}

Hence as can be seen there are two correct statement. Hence C is not correct option.

Option C :  Suppose Q committed crime. In that case

P says “Q committed the crime.” {correct as Q committed crime}

Q says “S committed the crime.” {incorrect as Q committed crime} ------line4

R says “ I did not do it.”                {correct as Q committed crime}

S says “What Q said about me is false”.{correct as seen at line 4 whatever Q said is incorrect , hence this statement is correct}

Hence as can be seen there are two correct statement. Hence D is not correct option.

by Loyal (5.3k points)
0

@Bikram sir this method seems correct to me but i can't understand where it is going wrong. Can you look into this??

+2
Given that only one person committed the crime and only one given statement is correct.

When P is correct and Q, R, S are false: it implies, Q committed crime, S did not commit crime, R committed crime, S committed crime. This situation is impossible.

When Q is correct and P, R, S are false: it implies, Q did not commit crime, S committed crime, R committed crime, S committed crime. This situation is impossible.

When R is correct and P, Q, S are false: it implies, Q did not commit crime, S did not commit crime, R did not commit crime, S committed crime.

This situation is impossible. When S is correct and P, Q, R are false: it implies, Q did not commit crime, S did not commit crime, R committed crime, S did not commit crime.

This situation is possible means R has committed crime.

So, option (B) is correct ......

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

by Loyal (6.9k points)