We're all mad here! II

Logic Level 2

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 Mad Hatter, the March Hare, and the Dormouse had tea. The three were good friends, and all knew whether each of the three were mad or sane (although who knows if they really believed it).

The Mad Hatter said, "I believe that the March Hare and Dormouse are both mad."

The March Hare replied, "I believe that the Dormouse believes the Mad Hatter is sane."

The Dormouse finished, "I believe that the Mad Hatter is sane."

Exactly which of the three are mad?

If you're stuck, try this problem .
Mad Hatter March Hare and Dormouse Mad Hatter and March Hare March Hare All three are sane. All three are mad. Mad Hatter and Dormouse Dormouse

Maggie Miller by Maggie Miller , 27, USA

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3 solutions

Elizabeth Wang
Sep 10, 2015

Since March Hare believes that dormouse believes that Mad Hatter is sane, and Dormouse affirms it, then march Hare is sane. Then Mad Hatter cannot be sane, as he believes March Hare is mad, however, March Hare is not. Therefore, Dormouse is mad, since he believes that Mad Hatter is sane, but he is not.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Abin Das
Jul 22, 2015

Assume Mad hatter is sane.Then both March and Dormouse are mad. If it is so then Dormouse has to believe that Mad hatter is mad.But it is not so.

Hence Mad hatter is mad.Then atleast one of the two others are mad.Dormouse should be mad as he believes Mad hatter to be sane.If both are mad,Dormouse should believe that Mad hatter is sane and hence March should believe that Dormouse believes Mad hatter is mad.Hence March should be sane to believe what is as given.

Hence Mad hatter and Dormouse are mad.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Maggie Miller
Jul 17, 2015

Since the Mad Hatter believes that the March Hare and Dormouse are both mad:

  • If the Mad Hatter is sane, then both the March Hare and Dormouse are mad.
  • If the Mad Hatter is mad, then at least one of the March Hare and Dormouse are sane.

Since the March Hare believes that the Dormouse believes that the Mad Hatter is sane:

-If the Mad Hatter is mad, then the Dormouse believes that the Mad Hatter is sane if and only if the Dormouse is mad. Then the March Hare is sane if and only if the Dormouse is mad.

Therefore, we've narrowed the situation to the following possibilities: 1. The Mad Hatter is sane and both the March Hare and Dormouse are mad. 2. The Mad Hatter is mad and exactly one of the March Hare and Dormouse are mad.

Therefore, exactly one of the three is sane.

Since the Dormouse believes that the Mad Hatter is sane, either both the Dormouse and Mad Hatter are sane or both the dormouse and Mad Hatter are mad. They cannot both be sane since only one of the three is sane, so they are both mad. Then the March Hare is sane.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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