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## Question

In the Land of Knights and Knaves [LKK], Knights told the truth, and Knaves always lied. When the rare averitas flu virus infected some of the inhabitants, their behavior was reversed: sick Knights began to lie, and infected Knaves had to tell the truth!

Your job is to find out the ones who have been infected so they can be given the costly AFV vaccine. Just as you've finished with the previous group (or even if you're not finished... too bad, they're here anyway ), another group of five LKK citizens barge in. Here's what they have to say:

Vlad: Zedekiah is a knight as much as I am.

Wally: Vlad is a liar! He's a knave. I'm a knave too.

Xavier: If there are an odd # of knights in this group, then I need a shot, otherwise, I'm a knave.

Yitzhak: I'm a knave.

Zedekiah: Don't trust Yitzhak, he's lying! Trust me, I'm a knight!

Well, at least you know their names. Who needs a shot, and who doesn't?

Special thanks to bonanova for the puzzle format and introductory text.

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Wally:"I am a Knave" ---> Wally is sick.

Wally: "Vlad is a Knave" ---> Wally and Vlad are both K's or both N's

Wally: "Vlad is a liar" ---> Wally and Vlad are a liar and truthteller, in some order.

Yitzhak: "I'm a Knave" ---> Y is sick.

Zedekiah: "I'm a Knight" ---> Z is well.

Zedekiah: "Yitzhak lies" ---> Y and Z are liar and truthteller in some order.

Taken together, Y and Z are both K's or both N's.

Vlad: "Z is K as much as I am" ---> V, W, Y, Z are 4 K's, 4 N's, or 2K's and 2N's in some order.

Vlad and Zed are well; Wally and Yit are sick.

That leaves Xavier.

Xavier: If number of Knights is odd, I need a shot, otherwise, I'm a Knave.

Well since the other four have an even number of Knights,

"If Knights are odd" translates to "If I am a Knight," and "otherwise" translates to "If I am a Knave."

If I'm a Knight, then I need a shot [meaning sick and therefore lying].

otherwise

If I'm a Knave, then I'm a Knave. [seems that part could have been left out: x AND tautology <==> x.

So it's If I'm a Knight then I'm lying.

Well, the statement I'm lying is untenable, so we must negate the premise.

That says X is a Knave. No info on need for a shot

So Wally and Yitzhak need shots. And maybe Xavier.

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OK, here's my shot:

Vlad's statement is equivalent to "If I am a knight, Zed is a knight". Vlad can only be lying if he is a knight (and Zed a knave). So Wally's assertions that Vlad is a lying knave cannot be true.

Wally is lying.

Wally's statement gives us:

Wally is a sick knight.

Zed is a knight.

Zed's statement gives us:

Zed is honest & therefore well.

Yitz is lying.

Yitz's statement gives us:

Yitz is a sick knight.

That leaves Xavier. As I understand it, his statement evaluates to:

Knights...X sick....X knave...Statement

ODD.......TRUE......N/A.......TRUE

ODD.......FALSE.....N/A.......FALSE

EVEN......N/A.......TRUE......TRUE

EVEN......N/A.......FALSE.....FALSE

We already have four knights, so an odd number of knights means Xavier is a knight, and so must be lying if he is sick. That contradicts his statement, so Xavier is a knave. This means his statement is true:

"If there are an odd # of knights in this group [false], then I need a shot [true], otherwise, I'm a knave [true]."

Xavier is a sick knave.

So, the ones who need shots are Wally, Xavier, and Yitzhak.

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