Tricky Doors

Logic Level 3

A prisoner is trapped in a room with two doors. He knows that there is a dangerous monster behind one door and the other leads him to freedom.

On the doors there are inscriptions, which read as follows:

Left door: "Exactly one of these inscriptions is true."
Right door: "The monster is behind this door."

Can the prisoner determine which door has a monster behind it?

Warning (Hint): Do not make any implicit assumptions

Yes, the left door Yes, the right door No, there is not enough information

1 solution

Agnishom Chattopadhyay
Mar 18, 2018

One might indeed incorrectly reason as follows: Suppose the inscription on the left door is correct, in that case the inscription on the right door is wrong, which means the monster is behind the left door. On the other hand, if the inscription on the left door is false, the inscription on the left door has to be false as well, meaning that the monster is behind the left door, again.

Doesn't that show that the monster must be behind the left door? Well, not quiet.

We are missing just one point. The prisoner cannot guarantee with certainty that the messages on the doors have got something to do with what is actually behind the door. The problem never mentions that the inscriptions in fact are related to what is behind the doors. It is indeed physically possible to hide the monster behind the right door, and still write down these inscriptions on the doors

Inspired by this blog post , consider the following scenario:

Suppose you have two boxes. You know one of them has some treasure inside it, the other one does not. But you have two labels with you, "Exactly one of these labels is correct." and "The treasure is inside this box". If you arbitrarily paste these labels, you could go through the same deduction. But does that mean just pasting labels on the boxes let you figure out which box had the treasure?

I love this problem! Even though I got it wrong, I learned something from it and that's what really matters.

The "incorrect" reasoning (incorrectly) assumes that the inscriptions on the door are either true or false, which is not necessarily the case. For example, $x:= x \text{ is false}$ is neither true nor false.

If the inscriptions on the first and second doors are $a$ and $b$ respectively, then $a:=(a\wedge \neg b)\vee (\neg a \wedge b)$ and $b:= \text{ The monster is behind this door.}$ . It is true that $b$ is true or false, but the same can't be said about $a$ . If $b$ is false, then $a$ can be either true or false. But if $b$ is true, then $a$ is neither.

Mursalin Habib - 3 years, 2 months ago

Tricky Doors

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