Who broke the vase ? logic question.

Question

One of Kane, Dave, Ron or Rose broke a vase. The following is what each of them had to tell about the person who broke the vase:

1. Kane: Dave broke it.
2. Dave: Kane lied.
3. Ron: Kane broke it.
4. Rose: I did not break it.

(a) If only one of these statements is true who broke the vase?

(b) If only two of these statements are true who broke the vase? Justify your answer.

I tried to answer this by taking ,

Let p = Kane broke the vase , q = Dave broke the vase , r = Ron broke the vase and s = Rose broke the vase.

So the statements Dave broke it = q , Kane lied = -q , Kane broke it = p and I did not break it (Rose) = s

Both q and q cannot be either true or false simultaneously.

Then how to proceed ? I'm stuck here

Answer 1

This problem can be more easily visualized/solved by making a table:

Assuming the person in the left column broke the vase, we can conclude which statements are true or false. It is important to notice that this arrangement only works if it is assumed that every other character knows who broke the vase.

Both Parts A and B are now Trivial:

A) Rose must have broken the vase, as that is the only way in which only one person (Dave) has told the truth.

B) There are two solutions:

-1) Dave broke the vase, Kane and Rose are telling the truth.

-2) Ron broke the vase, Dave and Rose are telling the truth.

Answer 2

The key to both parts of this problem is noticing that if Rose is lying then he broke the vase, and if Dave is not lying then Kane is, and vice-versa.

a : So suppose Rose is not lying. Then, the rest are, which means that Dave is lying, so that Kane can't be lying. This is a contradiction. Hence Rose broke the vase if only one person is saying the truth. That person saying the truth must then be Dave.

part b : Suppose that Rose is not lying. Then one of the rest are also not lying. Suppose Kane is not. Then, Dave broke the vase. The rest seems to make sense : Dave says Kane is lying, which is not true, and Ron says that Kane broke the vase, which isn't true either. So Dave must have broken the vase.

However, there's a bit of problem. Suppose both Rose and Dave are saying the truth. This means that Rose did not break it, and Kane is lying that Dave broke it, and Ron is lying that Kane broke it. Hence, Ron must have broken it, and this is consistent as well.

Therefore, there seem to be two answers to part b, namely Dave and Ron.

Answer 3

Assuming all four know who broke it, let's use symbols: $+$ (telling truth), $-$ (telling lie), $\times$ (broke), $o$ (did not break). Let's consider $a)$ case by case: $$1) \ Kane \ + \ (Dave \ \times) \Rightarrow Rose \ o \ (Rose \ +) \ ? \ (2 \ +)$$ $$2) \ Dave \ + \ (Kane \ - \ ) \Rightarrow K \ - \ (Dave \ o \ ) \Rightarrow Ron \ - \ (K \ o\ ) \Rightarrow Rose \ - \ (Rose \ \times) \ !$$ $$3) \ Ron \ + \ (Kane \ - \ ) \Rightarrow Kane \ - \ (Dave \ o \ ) \Rightarrow Dave \ + \ (Kane \ - \ ) \ ? \ (2 \ +)$$ $$4) \ Rose \ + \ (Rose \ o \ ) \Rightarrow Ron \ - \ (Kane \ o \ ) \Rightarrow Kane \ - \ (Dave \ o \ ) \Rightarrow Dave \ + \ (Kane \ o \ ) \ ? \ (2 \ +)$$

Answer 4

Using symbols probably makes this harder rather than easier. Just try some things. If Kane broke it, then Kane is lying, but the others are telling the truth, so you have three true statements. Well, that's not what you are looking for. So, try something else.

Answer 5

Changing the names of protagonists, it becomes obvious who broke the vase in situation (b).

$\begin{array}{l} \text{Clovis :} & \text{The soldier broke it}\\ \text{Soldier :} & \text{Clovis lied}\\ \text{Gregoire :} & \text{Clovis broke it}\\ \text{St Remi :} & \text{I did not broke it}\\ \end{array}$

:-)