Which Alice?

ALICE IN PUZZLE-LAND
the Gryphon) must also be sane. So the solution is that the Lobster
is mad, and the Gryphon and Mock Turtle are both sane.
AND NOW, WHAT ABOUT THESE TWO? The Queen (of
Spades) believes that the King believes that she is mad. If she is
sane, then the King really does believe she is mad, which means the
King must be mad. If she is mad, the King doesn't really believe she
is mad, but if he were sane, he would. So in this case, again the King
is mad. Therefore in either case, the King must be mad. As for the
Queen, she could be either.
THE KING AND QUEEN OF CLUBS It is impossible that the
King believes the Queen believes the King believes the Queen is
mad, for suppose the King did believe this. Suppose the King is
sane. Then the Queen really does believe that the King believes she
is mad, but as we saw in the last puzzle, this means the King is mad.
So if the King is sane, he is mad—therefore the King can't be sane,
he is mad. Therefore his belief is false, hence the Queen doesn't
really believe that the King believes she is mad. Now, the Queen is
either sane or mad. If she is sane, her belief is correct, hence it
is true that the King doesn't believe she is mad, so he believes she is
sane. Then the King is right, and we have the impossibility of the
mad King believing something true. On the other hand, if the
Queen is mad, then her belief is wrong, so the King does believe she
is mad, which again makes the King sane, which he isn't. So we get a
contradiction in either case.
This proves that it simply is not possible that the King believes the
Queen believes the King believes she is mad. So, had the Duchess
told this to Alice, then she would have had to be mad! But, of
course, she didn't tell this to Alice; all she said was "What would you
say if I told you—"
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