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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

         

\[\begin{array}{|l|} \hline \small{\text{ (1) Both sentences in this box are false. }} \\ \small{\text{ (2) Superman exists. }} \\ \hline \end{array} \]

If these statements are both logical, then does Superman exist?

If this statement is true, then the other statement is also true.
If this statement is false, then the other statement is also false.

How many of the statements above is/are true?

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 "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?

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