# Knights and Knaves

A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet four inhabitants: Bozo, Marge, Bart and Zed.

• Bozo says," Bart and Zed are both knights".
• Marge tells you that both Bart is a knight and Zed is a knave
• Bart tells you," Neither Marge nor Zed are knaves".
• Zed says that neither Bozo nor Marge are knaves.

Can you determine who is a knight and who is a knave?

I am having extreme difficulty with this can anyone help me? I assume is starts like this.

So

$Bo\equiv(Ba\land Ze)$

$Ma≡(Ba\land \lnot Ze)$

$Ba\equiv(Ma\lor Ze)$

$Ze≡(Bo\lor Ma)$

Where

$Bo$= Bozo is a knight

$Ma$= Marge is a knight

$Ba$= Bart is a knight

$Ze$= Zed is a knight

• possible duplicate of Knights and knaves: Who are B and C? (task 26 from "What Is the Name of This Book?") – DonAntonio Dec 4 '12 at 3:12
• @Don: Could you please describe the isomorphism you see between this problem and the duplicate you suggested? Superficially they look quite different. – joriki Dec 4 '12 at 3:20
• @amWhy: The same question to you; your suggested duplicate also looks quite different superficially. – joriki Dec 4 '12 at 3:22
• To all, I mistakenly voted to close, prematurely. The "possible duplicate" that was generated as a comment was not accurate. I have deleted that comment. Apologies. @joriki - You needn't be so argumentative, joriki. – amWhy Dec 4 '12 at 3:24
• @amWhy: I'm sorry if I came across as argumentative; that wasn't my intention; I was only asking for an explanation and expressing a different view. Please explain why my comment struck you as argumentative to help me avoid that in the future. – joriki Dec 4 '12 at 3:33

First, your symbolic translations of Bart’s and Zed’s statements are incorrect. Bart actually said $$\text{Ma}\land\text{Ze}\;,$$ and Zed said $$\text{Bo}\land\text{Ma}\;.$$

A quick way to solve it is to suppose that Bart is a knight. Then he’s telling the truth, so Marge and Zed are also knights. But that’s impossible, because Marge said that Zed is a knave: if she’s a knight, she’s telling the truth, and Zed isn’t knight. Thus, Bart cannot be a knight and must therefore be a knave. Can you finish it from there?

Bozo says, "Bart and Zed are both knights."

Marge tells you that both Bart is a knight and Zed is a knave.

Bart tells you, "Neither Marge nor Zed are knaves."

Zed says that neither Bozo nor Marge are knaves.

Marge and Bart contradict each other. If Marge is telling the truth, Zed lies, and Bart tells the truth. Bart, however, Zed and Marge tell the truth. Thus, Marge and Bart are Knaves. Therefore, Zed is a knave (Because Marge said he was a knight). Bozo says Bart and Zed are knights, which has been established false. Thus all are knaves.

The way I typically solve these problems is to look for contradictions.

If Bozo is a knight, that means that Bart and Zed are knights, and if Zed is a knight, then Bozo and Marge are knights. If Marge is a knight, then Zed is a knave, which contradicts what we have, so Bozo is a knave.

If Marge is a knight, then Bart is a knight and Zed is a knave. This means that (from Bart) that Zed is not a knave, so Marge is a knave.

If Bart is a knight then that means Marge is a knight, which we already established to be false, so Bart is a knave.

If Zed is a knight, then that means Bozo is a knight, which we already establish to be false, so Zed is a knave.

You have chosen a good formalization here: with the corrections from Brian M. Scott's answer, you are given that \begin{align} (0) \;\;\; & Bo \equiv Ba \land Ze \\ (1) \;\;\; & Ma \equiv Ba \land \lnot Ze \\ (2) \;\;\; & Ba \equiv Ma \land Ze \\ (3) \;\;\; & Ze \equiv Bo \land Ma \\ \end{align} Now, looking at the shape of these formulae, we note that $\;Bo\;$'s $(0)$ and $\;Ma\;$'s $(1)$ have a similar structure where we may expect to get a contradiction, and these $\;Bo, Ma\;$ are used symmetrically in $(3)$. Therefore we calculate \begin{align} & Ze \\ \equiv & \;\;\;\;\;\text{"by (3)"} \\ & Bo \land Ma \\ \equiv & \;\;\;\;\;\text{"by (0); by (1)"} \\ & Ba \land Ze \;\land\; Ba \land \lnot Ze \\ \equiv & \;\;\;\;\;\text{"logic: contradiction; simplify"} \\ & \text{false} \\ \end{align} Using this in $(0)$ and $(2)$ immediately leads to $\;Bo \equiv \text{false}\;$ and $\;Ba \equiv \text{false}\;$, respectively. And plugging that last conclusion into $(1)$ gives us $\;Ma \equiv \text{false}\;$.

Therefore all are knaves.

Let M denote Marge is a knight and ~M denote Marge is a knave (not a knight). Similarly for the others.

Here are the base statements, where comma denotes and:

Bo => Ba , Z

Ba => M, Z

M => Ba, ~Z

Z => Bo, M

Notice that if one of them is a knave, for example ~M => ~Ba OR ~Z (that is at least one of these is true).

M => Ba, ~Z => ~Ba, ~Bo (since they each claim Z, but Marge claims ~Z). But this contradicts Ba. Thus ~M. Also, ~Z (Z claims M). This makes them all knaves, since Bo and Ba both claim Z.