Knights, Knaves, and Normals Puzzle - A: "B is a normal", B: "C is a normal", C: "B is a normal"

Question

You are on an island inhabited by three types of people: knights (always make true statements), knaves (always make false statements) and normals (sometimes make true statements and sometimes make false statements).

You come across 3 people Adam, Ben and Carl. You know that one is a knight, one is a knave and one is a normal. They say the following:

• Adam: "Ben is a normal."
• Ben: "Carl is a normal."
• Carl: "Ben is a normal."

Determine which person is what or whether you do not have enough information.

My Attempt

Adam:

• Adam is a knight => Ben is a normal
• Adam is a knave => Ben is a knight or Ben is a knave
• Adam is a normal
  • If Adam is true, Ben is a normal
  • If Adam is false, Ben is a knight or Ben is a knave

Ben:

• Ben is a knight => Carl is a normal
• Ben is a knave => Carl is a knight or Carl is a knave
• Ben is a normal
  • If Ben is true, Carl is a normal
  • If Ben is false, Carl is a knight or Carl is a knave

Carl:

• Carl is a knight => Ben is a normal
• Carl is a knave => Ben is a knight or Ben is a knave
• Carl is a normal
  • If Carl is true, Ben is a normal
  • If Carl is false, Ben is a knight or Ben is a knave

1. Adam is a knight => Ben is a normal => Carl is a knave (can't be normal since Ben is one and can't be a knight since A is one) => Ben is a knight or Ben is a knave which contradicts the fact that Ben is a normal.

2. Adam is a knave => Ben is a knight (can't be a knave) => Carl is a normal => Ben is a knight (can't be a normal or a knave).

3. Adam is a normal => Ben is a knight or knave (can't be normal) => Carl is normal (not possible, so eliminated) or Carl is a knight (can't be knave) => Ben is a normal (not possible since Adam is a normal).

Therefore, Adam is a knave, Ben is a knight and Carl is a normal

My Question Does my solution seem correct and make sense?

Thanks in advance.

Answer 1

Your solution is correct and also seems valid, though I didn't read it that closely.

I can't resist posting a shortcut analysis.

Assume Ben normal.
Then one of Adam or Carl must be a knight, and the other a knave.
This yields a contradiction, because you can not ever have a knight and knave make the same statement.

Therefore, Ben is not normal.
Therefore, at least one of Adam or Carl must be a knave, since they have both lied about Ben.

Therefore, Ben is also not a knave, and therefore Ben must be a knight.

At this point, everything falls into place.

Answer 2

Since I am very lazy, I like methods that don't involve much "thinking" ;-) (I mean thinking through options, assumptions, and trees). So I did it this way:

First, notation: A is a knight is AT, A is a normal is AN, A is a knave is AF.

Now, construct pretty automatically the following XORs (where pq is p ^ q):

AN + ATBN + AFBT = 1

BN + BTCN + BFCT = 1

CN + CTBN + CFBT = 1

Multiply both sides of the equations, noting that any terms with AXAY, BXBY, or CXCY (X ≠ Y) are 0.

You are left with AFBTCN = 1. That is Adam is a knave, Ben is a knight, and Carl is normal.