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Math Help - Logic Puzzle

  1. #1
    Newbie
    Joined
    Jan 2010
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    Logic Puzzle

    hey

    a puzzle in the style of Raymond Smullyan

    Problem:

    There is an Island on which there are three different people: One Group always says the truth, the other always lies and the third group switches, they say the truth then a lie, truth and so on.

    A scientist comes to the island and meets with three people A,B,C (he knows that every group is represented). He asks one question:"To which group do you belong?"

    As first answer (A1): "C always tells the truth"
    As second answer (B1): "B switches"
    B1: "A always lies"
    B2: "C switches"
    C1: "A switches"
    C2: "I always tell the truth"
    Figure out which person belongs to which group?

    modification:
    A1: I always tell the truth
    A2: B always lies
    B1: I switch
    B2: C always lies
    C1: I tell the truth
    C2: A always lies
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  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
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    Thanks
    539
    Hello, hiddy!

    Please give the original wording of the problem.
    As given, the problem does not make sense.


    There is an Island on which there are three different people:
    One group always says the truth, the other always lies,
    and the third group switches. (They tell the truth then lie, and so on.)

    A scientist comes to the island and meets with three people A,B,C.
    He knows that every group is represented.
    He asks one question:"To which group do you belong?"

    If he asks one question, why are they giving two answers?

    Since he asks "To which group do you belong?"
    why are they giving information about the others?



    As first answer (A1): "C always tells the truth"
    As second answer (A2): "B switches"
    B1: "A always lies"
    B2: "C switches"
    C1: "A switches"
    C2: "I always tell the truth"

    Figure out which person belongs to which group,

    Let the three groups be: . \begin{Bmatrix}T &=& \text{Truth}  \\ L &=& \text{Liar} \\ S &=& \text{Switch} \end{Bmatrix}


    Here are their statements:

    . . \begin{array}{|cc|cc|cc|}<br />
A_1\!: & C = T &  B_1\!: & A = L &  C_1\!: & A = S \\<br />
A_2\!: & B=S &  B_2\!: & C = S &  C_2\!: & C = T \end{array}


    Suppose \,A_1 is false: .  C \ne T
    . . .Then \,C_2 is false: . C \ne T

    Since \,A and \,C both lied,
    . . then \,B is the Truthteller.

    Then both of \,B's statements are true: . A = L,\;C = S
    . . That is, A is the Liar, \,C is the Switcher.

    Since \,C is the Switcher, then \,C_1 must be true: . A = S
    . . That is, A is the Switcher.

    We have a contradiction.
    . . Hence, \,A_1 is true: . C = T

    Then \,C is the Truthteller: . A = S

    . . Hence, \,A is the Switcher.

    . . Therefore, \,B is the Liar.

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