# Logic Puzzle

• Oct 11th 2010, 09:48 AM
hiddy
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
• Oct 12th 2010, 06:39 PM
Soroban
Hello, hiddy!

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

Quote:

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?"

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: .$\displaystyle \begin{Bmatrix}T &=& \text{Truth} \\ L &=& \text{Liar} \\ S &=& \text{Switch} \end{Bmatrix}$

Here are their statements:

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

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

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

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

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

. . Hence, $\displaystyle \,A_1$ is true: .$\displaystyle C = T$
Then $\displaystyle \,C$ is the Truthteller: .$\displaystyle A = S$
. . Hence, $\displaystyle \,A$ is the Switcher.
. . Therefore, $\displaystyle \,B$ is the Liar.