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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.

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

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

An islander on Truth & Untruth is either a Truth-tellers who *always* tell the truth or a Lying-liars who *always* lie.

You come across three islanders and ask them, "How many Truth-tellers are there among the three of you?"

- The one lying down responds first, but she talks softly and you don't understand what she says.
- The one seated beside her says, "My friend just said that there is one truth-teller among us."
- But then the standing one counters, "Don't believe that, he's lying!"

What can you determine about the seated and standing inhabitants?

I met two men who lived there and asked the taller man "Are both of you Knights?".

He replied with a "Yes" or "No", but from his answer, I could not figure out what type of person each man was.

I then asked the shorter man "Is the taller man a Knight?".

He replied with a "Yes" or "No", and after that I knew which type of person each man was.

Were the men Knights or Knaves?

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 are in a room with 4 doors and a guard. The guard says, "Welcome to the end of your quest! Behind one of these four doors lies the treasure you have been searching for. However, behind the other 3 doors lies certain death. Each door has a sign on it, but you are not sure which signs are truthful or which are lies. However, you are certain that the door that contains the treasure is truthful. Choose your door wisely and remember, "if you pick wrong, death will be brought upon you."

Here are what the four signs read:

- Door 1: Exactly one of doors 3 and 4 is truthful.
- Door 2: Both odd-numbered doors are untruthful.
- Door 3: Neither of the odd-numbered doors contains the treasure.
- Door 4: One of the even-numbered doors contains the treasure.

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