We're all mad here! III

Logic Level 3

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

One day, the Kings of Spades, Diamonds, and Clubs were playing croquet. The Kings all knew whether each of the three were mad or sane (although they may or may not have each believed the facts).

The King of Spades announced, "I believe that the King of Diamonds believes at least one of us are mad!"

The King of Clubs responded, "Well, I believe that I believe that I am sane."

Exactly which of the three Kings are mad?

Also try this problem and this problem .
Spades Diamonds Diamonds and Clubs Spades and Clubs They are all sane. They are all mad. Spades and Diamonds Clubs

Maggie Miller by Maggie Miller , 27, USA

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1 solution

Maggie Miller
Jul 17, 2015

Since the King of Spades believes that the King of Diamonds believes that one of the Kings is mad:

  • If the King of Diamonds is mad, he doesn't believe that at least one of the Kings is mad. Therefore, the King of Spades is also mad.
  • If the King of Spades is sane, then the King of Diamonds must be sane as well. Then the King of Clubs is mad.

Whether or not the King of Clubs is mad, he believes that he is sane. Then since he believes that he believes he is sane, he must be sane.

Therefore, the Kings of Spades and Diamonds are mad while the King of Clubs is sane.

4 Helpful 0 Interesting 0 Brilliant 2 Confused

nice problem

Mohamed Hessien - 5 years, 10 months ago

If Club is mad. He believes he is sane is false. And since he is mad he can believe that he is sane (which is false). So Club can be mad as well. My answer to who is mad is Club. But at the same time your answer too is correct if Club is sane. .... Not a fully conclusive question. Either Club could be mad or both S and D could be mad.

amit beesen - 5 years, 8 months ago

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No, Clubs cannot be mad. He believes he believes he is sane.

Suppose Clubs is mad. Then he is not sane, so he believes he is sane. Therefore, "Clubs believes Clubs is sane" is true, so Clubs does not believe it. That is, Clubs does NOT believe that he believes he is sane. :)

Maggie Miller - 5 years, 8 months ago

I found the wording kind of confusing. I thought the king of spades was only referring to himself and the king of clubs when he said "at least one of us".

Tristan Goodman - 1 year, 3 months ago

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