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Math Help - investigation of a crime

  1. #1
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    investigation of a crime

    A, B, and C are being investigated . One of them committed a crime. Here are their claims made during the investigation.
    A: 1-I did not do this; 2-B did not do this.
    B: 1-A did not do this; 2-C did that.
    C: 1-I did not do this; 2-A did this.

    It was established that one of them lied twice, another one told the truth both times and the third one once lied and once told the truth. Who committed the crime?
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  2. #2
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    Hello, spred!

    A, B, and C are being investigated . One of them committed a crime.
    Here are their claims made during the investigation.

    . . \begin{array}{c}A \\ \hline \text{I did not do this} \\ \text{B did not do this} \end{array}  \qquad\quad \begin{array}{c}B \\ \hline \text{A did not do this} \\ \text{C did it} \end{array} . . . \begin{array}{c}C \\ \hline \text{I did not do this} \\ \text{A did it}\end{array}

    It was established that one of them lied twice, another one told the truth both times
    and the third one lied once and told the truth once.

    Who committed the crime?

    Here are their statements, written in symbols:

    . . \begin{array}{cc}A\!: & A\text{ is innocent and }B\text{ is innocent} \\ \\[-4mm]<br />
B\!: & A\text{ is innocent and }C\text{ is guilty}\\ \\[-4mm]<br />
C\!: & C\text{ is innocent and }A\text{ is guilty}\end{array}


    Who made two true statements?


    Was it A?
    If so, he said: A is innocent and B is innocent.
    . . Hence, C is guilty.
    But B said: A is innocent and C is guilty.
    . . So he too made two true statements.


    Was it B?
    If so, he said: A is innocent and C is guilty.
    . . Hence, C is guilty.
    But A said: A is innocent and B is innocent.
    . . So he too made two true statements.


    It must have been C.
    He said: C is innocent and A is guilty.
    . . Hence, \boxed{A\text{ is guilty.}}


    The rest of the statements check out:

    B said: A is innocent and C is guilty.
    . . B made two false statements.

    A said: A is innocent and B is innocent.
    . . A made one false statement and one true statement.

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