# Classic 2 Guards But With 3 Doors Instead

So the rules are almost the same :

You get to ask just one question to whichever guard you want to (just 1 guard; asking both guards is not allowed)

There are 2 guards, one tells the truth and one lies (You don't know which is which)

This time, there are 3 doors. You know exactly that there is 1 door that leads to life, 1 door that leads to death, and 1 door that either leads to life / death. Only the guards know which is which (They also know where the random door - the one that either leads to life / death - leads to)

Now, here is the question :

Is it still possible to go to the door of life?

Bonus :

Is it possible to determine which door is which with just 1 question?

• Are we allowed to ask both guards our question or only one or the other? Sep 8, 2018 at 4:53
• You can only ask one guard. Thanks for pointing that out @AndersGustafson :D Sep 12, 2018 at 13:03

If I asked the other guard, which door would he say would have death behind it?

The reasoning is:

If you asked the truth telling guard, he would tell you that the lying guard would point you to any life door, whether there was one or two. He would then point to a life door. If you asked the lying guard, he would tell you that the truth guard would also point to a life door. He would also point to a life door.

• Hmm didn't think of this :D Will add a bonus question for u lol Sep 8, 2018 at 3:01
• Haha sounds good @KevinL! I’ll give the bonus some more thought. Sep 8, 2018 at 3:04
• OK then, hope you get it right :D Sep 8, 2018 at 3:04
• Suppose door #1 is life. If the lying guard were ask to point to a death door, thye truth telling guard would either respond by pointing to door #2, or he would respond by pointing to door #3. In the event that the truth-telling guard would have pointed to door #2, a lying guard could (quite falsely) report that the truth-telling guard would have pointed to #3. Likewise if the truth-telling guard would have pointed at #3, the lying guard could (again falsely) say he would have pointed at #2. So I don't think your strategy works. Sep 28, 2018 at 20:23

Bonus Question:

You could number the door 1, 2 and 3, and tell the guard, "I am thinking of a number, either 0 or 1. If I asked you whether the sum of my number and the number of the door to life is greater than two, would you say yes?

Yes = door 3
No = door 1
Don't know = door 2

Reasoning:

The "would you say yes" is to circumvent the lying, because the truthy person would say yes, and we would say " yes, I would say yes", and the falsy person would say no (because he lies) and so he says "yes I would" (because he lies). Now, the next part: if the door to life is 1, he would say "no it's not" because 1 + 0 / 1 is never greater than 2, if it is 2, then it could be greater than two if I pick 1, so he'll say "I don't know". If it is 3, then it will always he higher than 3 so he will say yes.

• I was thinking the exact same thing, hahah! As they say, "great minds think alike" :D Sep 8, 2018 at 10:28
• Well, I just knew that the only way was to have him say either: Yes, No, or Maybe Sep 8, 2018 at 10:40
• Hmm... I think I might have a different answer to this puzzle heheh :P Sep 8, 2018 at 10:45
• Oh yes! That was where it was from! I remembered that idea, but I forgot where! Sep 8, 2018 at 10:59
• But the guards know where the random leads to which makes this strategy not work :D Sep 8, 2018 at 11:29

I have an answer for the bonus question:

Yes, it is possible. You could ask one guard, "which door or doors can possibly lead to death?"

Reasoning:

If you ask the truth-telling guard, they'll point to both the "death" door and the "random" door, so you simply go through the remaining one. If you ask the lie-telling guard, they'll simply point to the "life" door (since you gave them the chance to point to any number of doors, and the "life" door is the only one that cannot possibly lead to death), and you go through that one. In summary, if the guard points to a single door, go through it; if the guard points to two doors, go through the one that isn't pointed at.

• However, they know whether the random door leads to life or death, and could direct you accordingly, defeating this particular strategy.
– hat
Sep 8, 2018 at 8:01
• Just as jesse said, this strategy would not work in this case Sep 8, 2018 at 11:28

If the lying guard is guaranteed to be "maximally" untruthful, and the doors are labeled 1, 2, and 3, one could ask a guard to identify every number between 1 and 6, inclusive, which either identifies a door that is safe, or is three higher than a random door. If the guard identifies one doors in the range 4-6, the guard is telling the truth, and any doors the guard identifies in the range 1-3 are safe (the number 4-6 would indicate which door was random). If the guard identifies two doors in the range 4-6, the door not identified in the range 4-6 is the random door, and doors not identified in the range 1-3 are safe.

Personally I don't like such presumptions, however, since the above would really be asking not one question, but six, and asking the guard to lie about the answers individually. I think a much better rule would allow a lying guard, asked one question, to give any response that would answer the question without being entirely truthful. Under that formulation, if door #1 was random and safe, #2 was safe, and #3 was death, the truthful guard would answer #1, #2, #4; the lying guard would be required to either specify at least one of #3, #5, and #6, or fail to specify at least one of #1, #2, #4, but could respond with any one of 63 possible combinations of numbers (out of 64 total). Under such rules, however, it would be necessary to ask more questions or have more information.

• So what you mean is that it's not possible to do it with only 1 question or...? Sep 29, 2018 at 2:33
• @KevinL: It would not be possible with what I would regard as "one question", but may be possible with what the author would consider to be one question./ Sep 29, 2018 at 3:14
• Yeah, I just wanted to see someone prove that it's impossible :D Sep 29, 2018 at 3:24
• @Kevin: Even if one knew that the person being asked the question was a liar, the best a known-liar's answer can do is eliminate one possibility, given a model where the liar's rule is "If there is exactly one correct answer, and at least one allowable answer that doesn't match it, choose an answer from among the alternatives that doesn't match; else choose an answer arbitrarily from among the offered alternatives". If one were allowed to formulate an unanswerable question, and treat involuntary non-responsiveness as a form of "answer", one could get a third state that way, but then... Sep 29, 2018 at 19:20
• ...a liar should be entitled to answer any question by saying nothing (since that would be a form of response). Sep 29, 2018 at 19:20