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.

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 | – | – | – | – | – | \(\checkmark\) |

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.

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, 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.

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.