Review question

# The liar, the switcher and the truth-teller; who's who? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8279

## Solution

1. 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 – – – – – $\checkmark$
1. 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.

1. 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.