A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013 On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and $C$ says that both $A$ and $B$ are lying.
Then find out - who is lying and who is telling the truth.
So I have tried by supposition - For example let B be telling the truth- $$
\begin{array}{c|lcr}
n & \text{A} & \text{B} & \text{C} \\
\hline
 & T & T& T\\
 & \ & T &  \
\
\end{array}
$$
What this means is if we assume $B$ to be true it implies that $C$ is true  and that implies that $A$ is true and B is true . So there is a possiblity that $B$ is speaking the truth . I am thinking in this way...But am not going further about the Liars.
 A: If $B$ lies, then $C$ is telling truth and hence $A$ lies which means $B$ is telling the truth and hence contradiction. 
So if $B$ is telling the truth, then $C$ lies about  $A$ and $B$. But also $A$ lies because $B$ is telling the truth. Hence: $B$ is telling the truth and $C$ and $A$ lie.
Note that if $C$ lies then either $B$ is telling the truth or $A$ is telling the truth.
A: A more formal and direct approach is to write $\;T(x)\;$ for "$\;x\;$ tells the truth", and recognize that "$\;x\;$ says $\;\phi\;$" implies that $\;T(x) \equiv \phi\;$: either $\;x\;$ is a truth-teller and $\;\phi\;$ is true, or $\;x\;$ is a liar and $\;\phi\;$ is false.
Using this, what you are given implies
\begin{align}
(1) \;\;\; & T(A) \equiv \lnot T(B) \\
(2) \;\;\;& T(B) \equiv \lnot T(C) \\
(3) \;\;\;& T(C) \equiv \lnot T(A) \land \lnot T(B) \\
\end{align}
So starting with the most complex equation $\;(3)\;$, we can simply calculate
\begin{align}
& T(C) \equiv \lnot T(A) \land \lnot T(B) \\
\equiv & \;\;\;\;\;\text{"using (1); double negation"} \\
& T(C) \equiv T(B) \land \lnot T(B) \\
\equiv & \;\;\;\;\;\text{"contradiction; simplify $\;\phi \equiv \text{false}\;$ to $\;\lnot\phi\;$"} \\
& \lnot T(C) \\
\end{align}
So now you can draw your conclusion about $\;C\;$, and then the rest follows.
A: Starting from the bottom. C can only either be a truth-teller or a liar.
If C is a truth-teller, then his claim is true that both A and B are liars. But A would be telling the truth by claiming that B is a liar, so A can't be a liar. So C can only be a liar and it is not true that both A and B are liars. At least one is a truth-teller. But which one? 
(Actually, only one of them is a truth-teller. We've seen that it can't be that both A and B are liars, as C claimed, and it also can’t be that both A and B truth-tellers, because A’s claim that B is lying would be true, and B can’t lie as a truth-teller.)
If C is a liar, which we’ve established from the opposite case, what are the other two, by necessity?
B would be a truth-teller, since his statement that C is lying is a truth.
Now that B is a truth-teller, it follows by necessity that A is lying about B being a liar, so A is a liar.
C can only be a liar, and from that, A is a liar, and B is a truth-teller.
A: I figured out an answer in gym class. The person who is telling the riddle is lying about how C said that A is lying. I have tried and there is almost no other way it will work. A and C are telling the truth, and B is lying. Boom. 
