The liar, the switcher and the truth-teller; who's who?

Question

A, B and C are three people. One of them is a liar who always tells lies, another is a saint who always tells the truth, and the third is a switcher who sometimes tells the truth and sometimes lies. They make the following statements: A: I am the liar. B: A is the liar. C: I am not the liar. Who is the liar among A, B and C? Briefly explain your answer.

P, Q and R are three more people, one a liar, one a saint, and the third a contrarian who tells a lie if he is the first to speak or if the preceding speaker told the truth, but otherwise tells the truth. They make the following statements: P: Q is the liar. Q: R is the liar. R: P is the liar. Who is the liar among P, Q and R? Briefly explain your answer.

X, Y and Z are three more people, one a liar, one a switcher and one a contrarian. They make the following statements: X: Y is the liar. Y: Z is the liar. Z: X is the liar. X: Y is the liar. Y: X is the liar. Who is the liar among X, Y and Z? Briefly explain your answer.

Solution

A cannot be the liar, since then he would have told the truth, and A cannot be the saint because then he would have lied. Therefore A must be the switcher.

B is therefore lying, so is the liar. C is telling the truth and is the saint.

We can show this reasoning in a table. The top half of the table shows the six possible combinations of T (truth-telling saint), L (liar) and S (switcher). The bottom half shows which combinations become impossible at each step in the conversation.

A is B is C is	T L S	T S L	L T S	L S T	S T L	S L T
A says A=L	x	x	x	x	$\checkmark$	$\checkmark$
B says A=L	–	–	–	–	x	$\checkmark$
C says C $\neq$ L	–	–	–	–	–	$\checkmark$

Again, we draw up a table, this time using C to mean contrarian.

P is Q is R is	T L C	T C L	L T C	L C T	C T L	C L T
P says Q=L	$\checkmark$	x	$\checkmark$	$\checkmark$	$\checkmark$	x
Q says R=L	$\checkmark$	–	x	x	$\checkmark$	–
R says P=L	x	–	–	–	$\checkmark$	–

The only possible combination is PQR=CTL. Let’s check that this is consistent with the statements.

P: Q is the liar. (FALSE. Consistent with P=contrarian, so lies if speaks first.) Q: R is the liar. (TRUE. Consistent with Q=saint.) R: P is the liar. (FALSE. Consistent with R=liar.)

So R is the liar.

X is Y is Z is	L S C	L C S	S L C	S C L	C S L	C L S
X says Y=L	$\checkmark$	$\checkmark$	$\checkmark$ (truth)	$\checkmark$ (lie)	$\checkmark$	x
Y says Z=L	$\checkmark$ (lie)	x	$\checkmark$	$\checkmark$	$\checkmark$ (truth)	–
Z says X=L	$\checkmark$	–	x	$\checkmark$	$\checkmark$	–
X says Y=L	$\checkmark$	–	–	$\checkmark$ (lie)	x	–
Y says X=L	$\checkmark$	–	–	x	–	–

Let’s check LSC is consistent.

X: Y is the liar. (FALSE, consistent with X=liar). Y: Z is the liar. (FALSE, consistent with Y=switcher). Z: X is the liar. (TRUE, consistent with Z=contrarian, previous statement was false). X: Y is the liar. (FALSE, consistent with X=liar). Y: X is the liar. (TRUE, consistent with Y=switcher).

So X is the liar.