We're all mad here! IV

Logic Level 3

Everyone in Wonderland is either sane or mad. A sane person believes a statement if and only if it is true, while a mad person believes a statement if and only if it is false.

The King of Hearts, Queen of Hearts, and Alice all know whether each of the three is sane or mad (although they might not believe it).

The King says, "I believe that Alice believes I'm sane."

Alice says, "I believe that the Queen believes the King is mad."

The Queen says, "I believe that Alice is mad!"

Each person told the truth. Which of the three are mad?

You can see more problems like this in my Wonderland Puzzles set .

None All three Queen of Hearts The King and Queen of Hearts Alice King of Hearts Alice and the King of Hearts Alice and the Queen of Hearts

2 solutions

Niranjan Kumar
Aug 30, 2015

By the 1st statement, which was made by the King, we can come to a conclusion that if King is mad then Alice is also a mad and vice versa.

REASON: Assume King to be a mad, then Alice believes a False statement, thereby making her a mad. Assume King to be sane, now Alice believes a True statement making her sane too.

Now in the 2nd statement assuming 'King is mad' as True, Alice becomes a sane but that is contradictory to our conclusion from 1st statement.

So considering 'King is mad' as False Queen becomes a mad (since she is mad she believes 'Alice is mad' which is a false) and Alice becomes sane and so does the King.

This is my own solution so I'm sorry if I'm wrong. Please bear with the bad English.

The first part is actually not quite right - the king's statement tells us that Alice is sane, not that she and the King.are the same.

Reason: If the King is mad, then his belief that Alice believes he's sane is false. Then Alice does not believe he is sane. Since that's correct, Alice is sane.

If the King is sane, then his belief is correct, so Alice does believe he's sane. Since that's correct, Alice is sane.

Maggie Miller - 5 years, 9 months ago

I made that conclusion because, in the question its mentioned that only the statements can be true or false, not the beliefs. Please correct me if I'm wrong.

Niranjan Kumar - 5 years, 9 months ago

The three statements made are true, but the statements are a claim about a belief. If truly believed something ridiculous like "The sky is red," then even though "The sky is red" is a false statement, the statement "I believe the sky is red" is true.

Suppose the King is mad. We are told that the King believes Alice believes he is sane. This statement is true - that is, the King truly does believe that Alice believes he's sane. But the King is mad, so he believes false statements. Therefore, Alice does not believe he's sane. Then Alice believes a correct statement, so she is sane.

Maggie Miller - 5 years, 9 months ago

@Maggie Miller – So you're trying to say that, there is difference between making a false statement and believing a false statement, right?

If I say 'The sky is red' then it is false. But, 'I believe that the sky is red' is true. Am I right? Have I got your point?

Niranjan Kumar - 5 years, 9 months ago

@Niranjan Kumar – Yes, exactly - the syntax is important! It's the difference between stating a fact vs. stating an opinion.

Maggie Miller - 5 years, 9 months ago

@Maggie Miller – Yup thanks for the explanation!

Niranjan Kumar - 5 years, 9 months ago

If the Queen believes Alice is mad, then Alice's belief is false which would mean Alice is mad??

Alina Khan - 5 years, 8 months ago

The Queen believes that Alice is mad, but since the Queen is mad, the Queen's beliefs are all false. Therefore, the Queen is wrong and Alice is actually sane. :)

Alice's belted is trye: since the King is sane, the Queen believes he is mad.

Maggie Miller - 5 years, 8 months ago

but but but but.... when I compared the individual argument of Alice, before I think about the king, she also looks kinda insane:

Alice think that the Queen think that the king is mad, but the Queen think that Alice is Mad and not something about the king. If Alice was sane, she would say: "I believe that the Queen believe that I'm mad". We end up with Alice and the King being mad.

And Queen goes out being sane.

I'm new in logic, so it has a good chance that I'm missing something. I even made a graphic! Could you answer me where I'm getting wrong?

Marco Antonio - 5 years, 6 months ago

They are not making comments regarding the statements made by the others, but on the reality or known truths about each other. If it will make you feel better, assume that they spoke to you as a detective individually (one on one & one by one) in a soundproofed interrogation room.

Saya Suka - 7 months, 3 weeks ago

The Queen's statement shows that either Alice or herself is mad (she rightly accused a mad Alice or a sane Alice is wrongly accused by a madwoman). Whichever the case is, the premise of a mad king stated by Alice is false (like a product of a plus & a minus equal a minus). To be exact, a SQ will believe the truth about the K, but MA will distort it into a false statement (if MA says SQ says MK, then the reality is SQ says SK, so King is sane). Even if it's the opposite with a MQ & SA, the king is still sane (SA says MQ says MK, then the reality is MQ does say MK, but wrong in her belief as any other mad people would, so King is sane). So the first established fact of a SK is reached, and whatever he believes can be trusted by us. SK says A says SK, so A says SK is the reality as well as a right belief as we already know. So we have a SA & a MQ. The Queen is MAD.

Saya Suka - 7 months, 3 weeks ago

Saya Suka
Oct 24, 2020

The Queen's statement shows that either Alice or herself is mad (she rightly accused a mad Alice or a sane Alice is wrongly accused by a madwoman). Whichever the case is, the premise of a mad king stated by Alice is false (like a product of a plus & a minus equal a minus). To be exact, a SQ will believe the truth about the K, but MA will distort it into a false statement (if MA says SQ believes MK, then the reality is SQ believes SK, so King is sane). Even if it's the opposite with a MQ & SA, the king is still sane (SA says MQ believes MK, then the reality is MQ does believe MK, but wrong in her belief as any other mad people would, so King is sane). So the first established fact of a SK is reached, and whatever he believes can be trusted by us. SK says A says SK, so A says SK is the reality as well as a right belief as we already know. So we have a SA & a MQ. The Queen is MAD.

S = Sane, M = Mad, Q = Queen, K = King & A = Alice.

We're all mad here! IV

You can see more problems like this in my Wonderland Puzzles set .

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