Maneki Neko (Lucky Cat) Triple Trouble

This is the age old liar and truthteller problem-with a twist.

For the sake of simplicity, I will mark the Lucky Cats as ${ M }_{ 1 }$ , ${ M }_{ 2 }$ , and ${ M }_{ 3 }$ .

${ M }_{ 1 }$ says he isn't cursed,

${ M }_{ 2 }$ says she isn't and that ${ M }_{ 1 }$ is cursed,

and ${ M }_{ 3 }$ 's speech is muffled.

You ask the other two what he said, ${ M }_{ 1 }$ responds that ${ M }_{ 3 }$ said that he is cursed,

and ${ M }_{ 2 }$ responds that ${ M }_{ 3 }$ said that he is not cursed and ${ M }_{ 2 }$ is.

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Now that we've summarized the data, we can get to solving the problem. First, let's assume that ${ M }_{ 1 }$ is the truthteller.

The first two situations check out just fine with ${ M }_{ 1 }$ being the truthteller, but the third situation-where he responds with what ${ M }_{ 3 }$ said according to him-gets a little tricky.

If ${ M }_{ 1 }$ is the truthteller, ${ M }_{ 2 }$ and ${ M }_{ 3 }$ must be lying. In that case, when ${ M }_{ 1 }$ says that ${ M }_{ 3 }$ said that he( ${ M }_{ 3 }$ ) is cursed, it must be true. If ${ M }_{ 3 }$ is lying, though, he couldn't say that he is cursed and have it be true. Therefore, ${ M }_{ 1 }$ being the truthteller doesn't check out.

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This time, let's assume that ${ M }_{ 2 }$ is the truthteller. The first two situations check out just fine for her, and if what ${ M }_{ 1 }$ claimed that ${ M }_{ 3 }$ said is false, that checks out too. The final situation checks out for her as well.

${ M }_{ 2 }$ is the truthteller, making ${ M }_{ 1 }$ and ${ M }_{ 3 }$ the cursed Lucky Cats. Just in case you want it, here's a situation describing how ${ M }_{ 3 }$ wouldn't be the truthteller:

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If ${ M }_{ 3 }$ was the truthteller(making ${ M }_{ 1 }$ and ${ M }_{ 2 }$ liars), the first situation would check out just fine for him, but in the second situation, ${ M }_{ 2 }$ tells a partial truth, and since the cursed Lucky Cats can do nothing but lie, ${ M }_{ 3 }$ being the truthteller wouldn't make sense.

To solve this problem we just need this:

"The first responds, 'He said yes.' The second responds, 'He said no, and accused me of being cursed.'"

Since they tell the opposite of each other, one of them must be lying, while the other one must be telling the truth, i.e. the Third Cat is a liar.

Now, no cat could have said "Yes" to the question whether they are cursed or not. So, the first Cat lied saying "He [the third cat] said yes."

With this, we get our liars, The first and the third.

To Solve the problem you need to analyze details carefully. We labeled 1st, 2nd and 3rd cats as A,B and C respectively.

1. A said "no" while B said "no and A is lying"

Based on these answers we assume that both of them are the liars, but it turns out that their statements would be unacceptable and would contradict one another since the truth would be:

A = Yes, I am the liar B = Yes, I am the liar but A isn't

On these analysis, We can conclude that EITHER ONE OF THEM IS CURSED AND THE OTHER IS NOT. We also found out that C IS SURELY A LIAR

Now we need determine which one is lying between A and B

1. C's speech is muffled. A claimed that C said "Yes" B claimed that C said "No and B is cursed"

On those statements, We turn it into the opposite since we are certain that C IS LYING

If A is telling the truth C's statement would be like this if C didn't LIE: "No, I'm not the liar" If B is telling the truth C's statement would be like this if C didn't LIE: "Yes, But B is not Cursed"

No we have the answer

first and third cant be lying at the same time,so one of them is the truth teller.the third will never say that he is cursed,so first is lying and second is the only truthteller

Let us consider the first two answers from MN1 and MN2. We note that:

• If MN1 is cursed and a liar, MN2 is not cursed and a truth teller.
• If MN1 is not cursed, then MN2 is.

Therefore either MN1 or MN2 is cursed.

• This implies that MN3 is cursed, and would have answered "No", which you didn't hear.
• But MN1 said MN3 said "Yes". This is because MN1 is a compulsive liar.
• The statement of MN2 was true.

Therefore, the two MN cursed are $\boxed{First}$ and $\boxed{Third}$ .