Two doors, each either trapped or safe, have signs whose truth depends on certain circumstances. Each sign reads “Both doors are safe”.

puzzlerecreational-mathematicssoft-question

A friend of mine sent me the following puzzle:

There are two doors, and behind them either a trap, or a safe passage
to somewhere, and on the doors, it is written something about whether
the doors are safe or not. It is known that if the first door opens
up to a safe place then the what is written on that first doors is
true, but it is opens up to a trap, what is written on that that first
door is false. Moreover, if the second door opens up a safe place, what is
written on that second door is false, and if this is a trap, what is
written on that second door is true.

Now, you look at the doors, and see that on the first door, it is
written that

Both doors are safe

and on the second door, it is written that

Both doors are safe

The question asks what are there behind the doors, i.e are they safe to pass or not.

I've tried making a list according to whether the given statements are correct or not, but the conditions listed in the question are not always (if the statement on the door is true) then (something), so I couldn't figure out how to approach such a problem and prove the results that I get somehow, so;

How to solve this puzzle, and prove that the results that we get are indeed correct ?

Best Answer

Here's Raymond Smullyan's solution to the Fourth Trial in the "Ladies or Tigers?" chapter of his 1982 book of logic puzzles, The Lady or the Tiger. Interpret Smullyan's "Room I" and "Room II" as OP's first and second doors, respectively; moreover, a room with a lady represents a safe door, and a tiger corresponds to a trapped door.

Since the signs say the same thing, they are both true or both false. Suppose they are true; then both rooms contain ladies. This would mean in particular that Room II contains a lady. But we have been told that if Room II contains a lady, the sign is false. This is a contradiction, so the signs are not both true; they are both false. Therefore, Room I contains a tiger and Room II contains a lady.

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