# “Knights and Knaves” puzzle to solve using equivalences.

Context for the puzzle:

You are on an island inhabited by knights and knaves. Knights always tell the truth, knaves always lie. You meet 5 natives:

• Carl says "I am the same type as Peter".

• Zippy says "Carl is a Knave or Jack is a Knight".

• Jack says "I am a knight and Bill is a Knave".

• Peter says "Bill is a Knight".

• Bill says "I am a Knight and so is Carl"

From this, I have formulated the following equivalences:

1. C ≡ P
2. Z ≡ ¬C ∨ J
3. J ≡ J ∧ ¬B
4. P ≡ B
5. B ≡ B ∧ C

I am trying to deduce which of the people are knights and which are knaves, however, after doing some substitution and manipulation using logical laws, I get to a point where I cannot make any more simplification and hence get stuck.

Could someone please point me in the right direction towards solving this problem?

Thank you.

It's not $C ≡ P$ - that would be true in the case of both $C$ and $P$ being knaves. You want $C ≡ (C ≡ P)$, which simplifies to simply $P$.