Which Alice?

ALICE IN PUZZLE-LAND
mad means that One is sane rather than mad. Hence if Four is mad,
One must be sane, so One and Four can't both be mad.) Therefore
One and Four are both sane. Since Four is sane, then Three and
Two are not both mad—at least one of them is sane. However,
Three can't be sane, because he believes One is mad. Therefore it
must be Two who is sane. Therefore One and Two are both sane.
This means that Six's belief is correct, so Six must be sane.
We have therefore shown that if Seven is mad, then Six must be
sane. Therefore it is not possible that Seven and Six are both mad.
Since the Knave believes that they are not both mad, then the
Knave must be sane.
THE GRYPHON'S EVALUATION In Problem 15, we proved
that the Cook is sane. So if the Duchess's story were correct, the
Cook would be sane. But then the Duchess tells Alice that the Cook
believes the Duchess to be mad. This would mean that the Duchess
must be mad (because the Cook, who is sane, believes she is).
Therefore, if the Duchess's entire story were true, she would have
to be mad, which would mean that her story is not true. So if the
story were true, we would have a contradiction. Therefore her story
is not true.
Incidentally, the above argument is not intended to prove that
the Duchess is mad; there is no reason to believe that she is. All it
showed was that if her story were true, she would have to be mad,
hence her story is not true. This means only that the Duchess is not
correct in all her beliefs—not that she is incorrect in all her beliefs!
Chapter 4
HOW MANY? However many tarts the Dormouse has, call that
number one portion. So the Dormouse has one portion of tarts and
150