# How can I formulate the two guards puzzle to disallow references to the other guard?

An old logic puzzle goes as follows:

There is a gate with two doors, each leading to a different city – Tannenbaum or Belvedere. You wish to get to Tannenbaum, but you do not know which of the two gates leads there. Two guards stand at the gates. One always tells the truth, and one always lies, but you also do not know which is which. Can you ask one of the two guards just one question to determine which gate leads to Tannenbaum?

The traditional answer to this question is to ask one of the guards what the other guard would say if you asked him which gate led to Belvedere – that is:

If I were to ask the other guard which gate led to Belvedere, which gate would he point to?

However, there is another, not-often-referenced solution that also works – that is, to ask the guard what he would say if you asked him which gate leads to Tannenbaum:

If I were to ask you which gate led to Tannenbaum, which gate would you point to?

This is different from simply asking "Which gate leads to Tannenbaum?" as you are wrapping the question in a layer of hypotheticality – the liar would point you to Belvedere if you had asked him which gate led to Tannenbaum, so he must lie about that and point you to Tannenbaum anyway.

Is there a natural way to reformulate this puzzle so that questions involving the other guard are disallowed, thereby forcing the reader to consider this alternate solution?

You could have just one guard, which is either a truthteller or a liar, but you don't know which.

• Or you could keep the two guards, and each one is either a truthteller or a liar, but you don't know which. I think this might be just the thing OP was looking for! You can now ask guard $A$ what guard $B$ would say, but it doesn't help; you have to ask guard $A$ what guard $A$ would say. – MJD Mar 26 '14 at 18:40
• I think this works the best. – Joe Z. May 17 '14 at 4:03

My current formulation adds the qualifier that the guards always tell the truth or always lie, but this is just a natural condition of theirs - they don't know that they always tell the truth and always lie – they just kind of do.

Originally I'd simply said "the guards don't know anything about the other guards; only you know that one of them always tells the truth and one lies", but somebody defeated this by introducing another hypothesis:

What would a guard who answered questions the opposite way of how you do say if I were to ask him which gate led to Belvedere?

• In this case there is a difference between actual truth and observed truth. The guards when asked that should always think that the other guard tells the truth. – ruler501 Mar 22 '14 at 2:50
• I'm not sure I get you. The truth-teller would think the other guard always lies, wouldn't he? – Joe Z. Mar 22 '14 at 2:56
• You said that you are the only one that knew that one lies and one tells the truth. So shouldn't they both have to assume some truth value about the other? – ruler501 Mar 22 '14 at 2:57
• Wait, are you talking about the formulation above the line or below the line? I thought you were talking about the one below the line. – Joe Z. Mar 22 '14 at 3:11
• If it's the one above the line, then the guard would answer "I don't know". – Joe Z. Mar 22 '14 at 3:32

I'm trying here to cover all options:

To discourage refers to the other guard one needs guards to be independent. There are several ways to do it:

1. Just remove the other guard from the puzzle, leaving only one there.

• Note, that in this and any other case the use of other guard is still possible. One just need to ask about imaginary guard, which has opposite type to the guard you are asking. (That's why I wrote "discourage", not "forbid")
2. Allow guards both to be of different kind and to be of the same kind. For example, say that they belong to a tribe, which consists from truth-tellers and liars, but you do not know how they are.

3. Say that guards do not know each other. For example, because they are from different tribes.

4. Introduce one more guard of known kind. In this case for any 2 guards you would know whether they are both of the same kind or of different kind.

• Note, that in this case one still can ask what would answer the second if he asked what would answer the first.
5. One can combine 2. and 4. introducing many unknown guards, but it would be so much different from 2.

You could say something along the lines of "the guards refuse to acknowledge the existence of one another." This has the disadvantage of tipping the reader off to the solution you wish to avoid, but it would work.

• You're right that I'd prefer one that didn't tip it off so overtly. – Joe Z. Mar 22 '14 at 15:00

You want to get to Tannenbaum. Your version uses Tannenbaum in the question and forces both gaurds to point you in the right direction. If you use Tannenbaum in the traditional question they both point you in the wrong direction. Make 2 conditions.

1) The question must be about Tannenbaum.

2) The 'player' must take the path that is pointed to.

Then only your version works and the traditional question becomes non-functioning. BTW , thanx for this question , I never thought about it that way.:)

• You have a typo. "Make 2 cconditions." should read "Make 2 conditions." – Brian J. Fink Mar 22 '14 at 4:04
• Ahhh , thanx , buttons sticking ... – neofoxmulder Mar 22 '14 at 4:19
• I don't think the puzzle works if requirement 1 is in place. The traditional questions, and the question OP wants as the correct answer, are about the answers one of the guards would give to hypothetical question. – Taemyr Nov 11 '14 at 8:39

You ask one question and get one bit of information. You want that bit to indicate the route to Tannenbaum, so you can't learn anything else. In particular, you can't learn whether the guard that answers tells the truth or lies. There are a number of ways to do that, all causing the lying to happen an odd or even number of times so you get the bit you want. The reference to the other guard is one way to make sure you get one true answer and one false answer. Your last question is in the same vein-you get two answers from the one guard instead of one from each. In each case you know the result of the sequence, so I don't see the difference.

• The difference is that the "odd number of times" solution is by far the more well-known one, and I want to disallow that to get people to think about it again. – Joe Z. Mar 22 '14 at 14:59

If you put a wall between the two guards so that they don't know the other guard exists, you can state the problem like this:

There is a gate with two doors, each leading to a different city – Tannenbaum or Belvedere. You wish to get to Tannenbaum, but you do not know which of the two gates leads there. Two guards stand at the gates with a wall between them that prevents them from knowing the other exists. One always tells the truth, and one always lies, but you do not know which is which. Can you ask one of the two guards just one question to determine which gate leads to Tannenbaum?

This should probably be reworded before using. Helps prevent people from finding it with google.