Truth teller question.

Question

In a village there are two types of people. Type F who always lie and Type T people who always say the truth. X says "according to Y, I always lie". Assume both X and Y belong to same village. Which one of them is possible ?

1)Both X and Y belong to type T

2)Both X and Y belong to type F

I am able to figure out that both cannot belong to type T. But I am not able to figure out the 2nd statement.

Answer 1

For these 'truth-tellers and liars' (often presented as 'Knights and Knaves') puzzles, you can often use some simple algebra.

In general, you use a propositional letter $A$ for some person $A$ being a truth-teller (knights), then you know that they are a truth-teller if and only if what they are saying is true.

Thus, for example, if $A$ says "$B$ is a liar", then what $A$ says can be symbolized as $\neg B$, and that is true if and only if $A$ is a truth-teller, so you would get $A \leftrightarrow \neg B$

For this particular problem, you have $X$ saying something about what $Y$ would say ... which really just means that you first symbolize the latter part. That is, the "$Y$ would say that I ($X$) am a liar" becomes $Y \leftrightarrow \neg X$, and since $X$ is saying all that, you get:

$$X \leftrightarrow (Y \leftrightarrow \neg X)$$

which can be simplified as follows:

$$X \leftrightarrow (Y \leftrightarrow \neg X) \Leftrightarrow \text{ (Commutativity of } \leftrightarrow \text{)}$$

$$X \leftrightarrow (\neg X \leftrightarrow Y) \Leftrightarrow \text{ (Associativity of } \leftrightarrow \text{)}$$

$$(X \leftrightarrow \neg X) \leftrightarrow Y \Leftrightarrow $$

$$\bot \leftrightarrow Y \Leftrightarrow $$

$$\neg Y $$

So, we know that $Y$ is telling lies, while $X$ can be of any type. Hence, 2 is the only viable option out of 1 and 2

P.s. Algebra rules!

Answer 2

Both of them lie means Y cannot say that X always lies. There is no contradiction. Thus, the answer is 2).

Answer 3

Substitute "say the truth" with +1 and "lie" with -1.

If Y says that X "says the truth", it means that Y*X=1, if Y says that X "lies" it means that Y*X = -1.

Furthermore, X says that according to Y, X "says the truth", means that X*Y*X=1, while X says that according to Y, X "lies", means that X*Y*X=-1

Now, if X=1 and Y=1, then X*Y*X=1

If X=-1, Y=-1, then X*Y*X=-1

So, the second option is valid, not the first one.

(Another possibility is Y is type F, X is type T, but not Y type T and X type F)