# Monty's Labyrinth

(A fun combination of two classic puzzles, hope seasoned puzzlers enjoy.)

Having escaped through almost all of a labyrinth, you are met by three magical doors. Two of them lead to certain and immediate death, and one leads to freedom. They are magical because which door leads to which changes every time a new person approaches.

However, there are two guards that guard these doors. One always lies and one always tells the truth. Due to the magical nature of the doors, they do not know which door leads to which outcome.

As you approach the three doors, the guards instruct you to pick a door. After you do this, the guards are given a flash of magical insight, and one of the doors (excluding potentially the one you picked) that leads to certain death is revealed to them. The guards then allow you to ask a single question, and pick one of the other doors if you wish.

With this one question, what is the best possible outcome you can achieve?

Edit - As some have identified, this is exploiting the same techniques as the Monty Hall problem (hence the name), the difference lies in the fact you do not know which is the dud door, and cannot simply ask the guards.

• Possible duplicate of The Monty Hall problem Dec 8 '18 at 0:48
• @Arpeyji See my comment to the other answer. Dec 8 '18 at 1:01
• Just to narrow it down a bit: is it always the same bad door that is revealed to both of them? Also, do the guards know this? And come to that, do they know that the other guard is revealed the fact? And once more, do you need to direct your question to only one of them, or can you expect an answer from both of them?
– Bass
Dec 8 '18 at 10:10
• @Bass Yes, it is always a bad door that is revealed. The guards do know this. They know that the other one is told. You should direct your question to only one of them. Dec 8 '18 at 10:23

## 3 Answers

The answer:

I would ask the guards: "What would the other guard say if I asked which door other than the door I have chosen, leads to death?" whichever door they point to, it is the other door that leads to death. So I would change my door to their answer. Giving me a chance of 2/3 every time as if I had chosen death in my first guess, I will certainly find the door that leads to freedom in my second guess.

• Great job! I'll wait for a little while to accept this to encourage some other answers. Dec 8 '18 at 15:09
• Can you look at my last edit? I think I found a hole in the question. Dec 8 '18 at 15:21
• Oh, you can ask a question, you cannot get them to point to anything. Dec 8 '18 at 15:22
• Well, still, it would be impossible for them to answer? Dec 8 '18 at 15:23
• They only know one door leads to death. They do not know which is the right door. Dec 8 '18 at 15:25

I think the question that should be asked is: "If I asked your friend which door was revealed to lead to certain death, which door would he not answer with?" Note that this gives us exactly what was revealed to the guards, namely one of the doors which leads to certain death. No matter what happens, it is always optimal to switch. If you originally picked the right door, then you will die. Otherwise, the door that was revealed could've been yours or another, with equal probabilities. The probabilities of living, then, are 1/2 and 1 respectively. So, our answer is 3/4*2/3=1/2.
It is impossible to do better, since the guards have the maximum amount of information, and this question allows you to extract all of it.

• Ah I should clarify that the bad door revealed to them is never the one you picked, but this answer is very close. Dec 8 '18 at 10:33

I was wrong. So let's evaluate and actually read both the article and the question...

And then try this:

Question: "Assuming my chosen door is bad, what door would you choose?"
The door I choose we'll call door A. If door A is good and B is known: T(ruther) points to C and L(iar) points to B.
If A is good and C is known: T points to B and L points to C.
If door A is known, thus bad: T points to random door and L points to random door?
If door A is bad, and say B is known: T points to C and L points to B
Most of the time they point to opposite doors, so probably could get a better question.

Hmm

Question: "Assuming my chosen door is good, what door would you choose?"
A is good: T points to A, L points to B or C
A is known: T points to B or C, L points to A.
Thus this is a bad question.

I don't know, but I'm not currently deleting my answer because the comment on it can be useful to others.

Let's try this:

"If I didn't want the known door, what door would the other guard point to?".
I think the issue with this question is that the liar could still point to either the known door or the other unknown door.

Final try for now:

" Only answer if the door I picked is not the known door: of the remaining doors, what would the other guard say is the known door?"
So, if the door I picked is the known door, would the liar still answer?
My goal: if I can just get the honest guy to deterministically answer without the liar messaging it up, then I don't care what they have to say in the end.
What it gets me: essentially the same or exact opposite (?) Of this variant of the original problem:.
"The host knows what lies behind the doors, and (before the player's choice) chooses at random which goat to reveal. He offers the option to switch only when the player's choice happens to differ from his.", Which results in a 50% chance!

• rot13: Juvyr gur znguf oruvaq vg vf gur fnzr nf Zbagl Unyy, gur gevpx yvrf va gur dhrfgvba lbh unir gb nfx gur thneqf gb svther bhg juvpu qbbe lbh fubhyq fjvgpu gb. Dec 8 '18 at 1:01