Which Alice?

Solutions to the Puzzles
for example, he was told that A told the truth and that B and C both
lied. Then whoever A accused was guilty (since A told the truth), so
he might have accused B (in which case B was guilty) and B and C
both lied and accused A (or maybe B accused C, and C accused A).
On the other hand, it could be that A accused C and that B and C
both accused A, in which case C was guilty. So if A was the only one
who told the truth, then either B or C could be guilty. Similarly, if B
was the only one who told the truth, then either A or C could be
guilty, and if C was the only one who told the truth, then either A or
B could be guilty. So if the White King had been told either that A
was the only one who told the truth, or that B was, or that C was, he
could never have known who was guilty. So he was not told any of
these three things.
Could he have been told that all three told the truth? No, this is
impossible, because the guilty one certainly lied (since he accused
one of the others, and both the others were innocent).
This leaves only the case that exactly one lied. Well, if exactly one
lied, then the one who lied must be the guilty one, because if an
innocent one lied, that would make two lies—his and the guilty
one's. So, therefore, the White King was told one of three things:
Case 1: A lied, B told the truth, C told the truth.
Case 2: A told the truth, B lied, C told the truth.
Case 3: A told the truth, B told the truth, C lied.
We now see how the White King knew who was guilty, but how
can we know, since we don't know which of the three cases the King
was told? Well, Humpty Dumpty either asked the White Knight
whether any two consecutive statements were false, or he asked
whether any two consecutive statements were true. The first
question would have been a pointless one (since there is only one
false statement), and had he asked it, the answer would have been
no, and this wouldn't have enabled Humpty Dumpty to know which
of the three cases held. So Humpty Dumpty asked whether any two
consecutive statements were true. Had he been told "Yes," he would
have ruled out Case 2, but couldn't have known who was guilty. But
Humpty Dumpty did know, so he must have gotten No for an
answer and then realized that Case 2 must be the one; so B is guilty.
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