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Truth-Tellers and Liars

Time to play detective! Every puzzle in this collection contains a set of statements, but it's up to you to figure out which of those statements are true and which of them are false.

Truth Tellers and Liars: Level 3 Challenges

         

You are offered two cupcakes. One is poisoned and the other is safe to eat.

You happen to be in a village full of knights (who always tell the truth) and knaves (who always lie), but you can't tell which is which by their appearance.

You ask one of them, "Which cupcake is safe to eat?" To this he makes the following two statements,

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"If I were a knave, I'd say the one on the right."  
"But I'd say the one on the left, if I were a knight."

Which cupcake is safe to eat?

For every \(n \geq 1\), there is an \(n^\text{th}\) Pessimist sentence saying that not all later Pessimist sentences are true. For example, the \(5^\text{th}\) Pessimist sentence says:

"For at least one \(m > 5\), the \(m^\text{th}\) Pessimist sentence is false."

Where \(t\) and \(f\) are the number of true and false Pessimist sentences, what is \(t+f\)?

You happen to be in a village full of knights (who always tell the truth) and knaves (who always lie), but you can't tell which is which.

You approach two people, Dwight and Dave, one of whom is a knight and the other is a knave. One of them makes the following two statements:

"If I am a knight and he is a knave, then I am Dwight and he is Dave.

But If I am a knave and he is a knight, then I am Dave and he is Dwight."

Hmm... knight, knave, Dwight, Dave...

Is the speaker a knight or a knave? Dwight or Dave?

\(\)
Clarification: As with all knights and knave problems, treat this question as a formal logic question.

If it's Tuesday, then this sentence is true.

Suppose it's Wednesday; is the sentence above true?

You find yourself in a village that has knights (who always tell the truth) and knaves (who always lie), and come across an individual who says,

"If I am a knight, then I always lie."

Is this person a knight or a knave?

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