Which Alice?

ALICE IN PUZZLE-LAND
and the other is telling the truth (because if they are both orange,
Tweedledum is telling the truth and Tweedledee is lying; if they are
both purple, then Tweedledee is telling the truth and Tweedledum
is lying). On the other hand, if the cards are of different colors,
then the brothers are either both lying or both telling the truth.
Now, let us consider the present problem. Since both brothers
claim to be Tweedledum, then clearly one is lying and one is telling
the truth. Therefore, both cards must be of the same color.
Suppose they were both purple. Then the first one's second
statement is false; hence his first statement is false; hence he is
Tweedledee, which would mean that Tweedledee with a purple
card is lying, which is not possible. Therefore, both cards are
orange. Hence the first one's second statement is true, which means
that his first statement is also true, so he is Tweedledum. So, the
first one is Tweedledum, the second is Tweedledee, and both are
carrying orange cards.
ROUND THREE (ORANGE AND PURPLE) Looking at the first
two statements, we can tell they are obviously either both true or
both false. Therefore, their cards are of different colors (see the
principle discussed at the beginning of the last solution). So the first
one lied when he said that the cards are of the same color. So the
first one also lied when he said he was Tweedledee. So he is
Tweedledum.
ROUND FOUR (ORANGE AND PURPLE) Since the two make
contradictory claims, then one is lying and one is telling the truth.
Therefore, their cards are of the same color (same principle!). If the
cards are both purple, then the first one is telling the truth; so he
must be Tweedledee (because his card is purple and he is telling the
truth). If both cards are orange, then the first one is lying, hence
again must be Tweedledee (because his card is orange and he is
lying). So in either case, the first one is Tweedledee.
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