MHB Which Fairies are Telling the Truth?

[Ackbach]

Here is this week's POTW:

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Merlin and Arthur are magically transported to a land of pink and green fairies. Merlin explained to Arthur, "Real pink fairies always tell the truth, and real green fairies always lie. However, it is within my power to change a pink fairy to a green fairy, or a green fairy to a pink fairy, or both. A pink fairy disguised as a green fairy still always tells the truth, while a green fairy disguised as a pink fairy still always lies. Now, here are three fairies. Each will make a statement. Provide a classification of each fairy and answer the question, 'Does A appear to be a green fairy or a pink fairy?'"

A: B is a pink fairy.
B: If A is a pink fairy, then C is a green fairy.
C: B is a green fairy or A is what she appears to be.

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[Ackbach]

No one answered this week's question, which I must now attribute to the book Merlin Book of Logic Puzzles, by Margaret C. Edmiston. Here is her solution:

Spoiler

A and B are pink; C is green; A appears to be green. We show that A and B must be the same kind of fairy (in essence, that is, not necessarily in appearance). Suppose they are two different types. First, suppose A is pink and B is green. This cannot be the case, because if A were a pink fairy, she would not make the false statement that B is pink. Now, suppose A is green and B is pink. This cannot be the case either, because if A were a green fairy, she would not make the true statement that B is pink. So A and B are either both pink fairies of both green fairies. Suppose they are both green. Then A is green, so the "if" part of B's statement ("A is a pink fairy") is false. This means that B's statement is true, a contradiction of the assumption that B is a green fairy. So both A and B are pink fairies. Therefore since A is a pink fairy and B has made a true "if-then" statement, it follows that C is a green fairy.

Then, remembering what it means for an "or" statement to be false, we know that both parts of C's statement are false (since C is green). Hence, "A is what she appears to be" is a false statement. So A must appear to be green.