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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. See more

Level 3

         

On the island of Truth & Untruth, people are very consistent. All islanders are either Truth-tellers who always tell the truth or 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?

You find yourself on the island of knights and knaves, where every inhabitant is of one of two types: a knight who always tell the truth, or a knave who always lie.

You come across two inhabitants, Artemis and Hera. Hera says: "If I am a knight, so is Artemis."

What type is Artemis and what type is Hera?

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.

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 \(t+f\)?

An island has only two types of people: Knights (who always speak the truth) and Knaves (who always lie).

I met two men who lived there and asked the taller man if they were both Knights. He replied, but from his answer, I could not figure out what type of person each man was, so I asked the shorter man if the taller man was a Knight. He replied, and after that I knew which type of person each man was.

Were the men Knights or Knaves?

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