Logic Puzzle

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?"

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.