Which Alice?

Solutions to the Puzzles
AND NOW WHAT ABOUT THE QUEEN OF HEARTS? What we
proved in the last puzzle would apply to the King and Queen of
Hearts just as well as the King and Queen of Clubs: It is not possible
that the King of Hearts believes that the Queen of Hearts believes
that the King of Hearts believes she is mad. Since the Queen of
Hearts does believe that the King believes this, then she is mad.
As for the King, it is not possible from these data to determine what
he is.
THE DODO, THE LORY, AND THE EAGLET Since the Lory
believes that the Dodo is mad, then the Lory and the Dodo are of
opposite types (if the Lory is sane, then the Dodo really is mad; if
the Lory is mad, then the Dodo isn't really mad but sane). Since the
Eaglet believes that the Dodo is sane, he is opposite to the Lory (who
believes the Dodo is mad), hence he is like the Dodo. (Alternatively,
one could prove this by reasoning that if the Eaglet is sane, then the
Dodo really is sane, and if the Eaglet is mad, then the Dodo is not
really sane but mad.) Therefore the Eaglet and the Dodo are alike,
and the Lory is opposite to them both. Since the Lory is opposite to
the Eaglet, then the Lory must believe that the Eaglet is mad.
Therefore the Dodo's belief is correct, so the Dodo is sane. So, the
Dodo and the Eaglet are both sane and the Lory is mad.
THE KNAVE OF HEARTS I will prove that if Seven is mad,
then Six must be sane — and therefore that the Knave was right in
believing that Six, Seven are not both mad.
Well, suppose Seven is mad. Then Seven's belief about Five is
wrong, so Five is sane. Then Five's belief is correct, so One and
Four are either both mad or both sane. Now, it is not possible that
One and Four are both mad. (Because if Four is mad, his belief is
wrong, which makes Three and Two both mad, but Three's being
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