Which Alice?

Solutions to the Puzzles
carrying a black card; so he can't be Tweedledum—again he must
be Tweedledee. So in either case, the speaker is Tweedledee.
ROUND SIX If the first one were holding a red card, we
would get the following contradiction: Suppose the first one were
red. Then his statement is true; hence his brother is Tweedledee, so
he is Tweedledum. Thus he is Tweedledum carrying a red card.
This makes the second one's statement true. But then, how could
the first one, who is truthful, lie and say that his brother is
Tweedledee holding a black card? So it is impossible that the first
one is carrying red; he must be carrying a black card.
Since the first one is not red, then the second one's statement
cannot be true; so the second one is also carrying a black card. If the
second one were Tweedledee, then he would be Tweedledee
carrying a black card, which would make the first one's statement
true. But the first one's statement is false (because the first one is
carrying a black card); so the second one can't be Tweedledee. This
proves that the first one must be Tweedledee.
ROUND ONE (ORANGE AND PURPLE) The speaker couldn't
have been Tweedledum carrying an orange card, or he would have
told the truth and said, "My card is orange." The speaker couldn't
have been Tweedledum carrying a purple card, or he would have
lied and said, "My card is orange." Therefore, the speaker was not
Tweedledum; so it was Tweedledee (who either carried a purple
card and told the truth, or an orange card and lied).
ROUND TWO (ORANGE AND PURPLE) A useful principle,
which will be used in this problem and some of the later ones, is
this: If the two cards are of the same color, then one of them is lying
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