Oh what a lie!

Posted by Thomas J. Brown at 1324 hrs

Remember in the 1986 movie Labyrinth, when Jennifer Connelly's character gets to those talking doors where one always tells the truth and the other always lies? For some reason I was thinking about that just now and trying to remember how to solve the problem.

I don't think the film ever tells you how, so a few years ago a friend and I sat down and solved it. So in case you don't know the answer, here's a quick rundown of the problem and its solution:

In the film, Sarah (played by the ever-hot Jennifer Connelly), comes to two talking doors. They tell her that one of them always tells the truth and the other always lies (then have a humorous argument over which is the truth tell and which is the liar ("No, I always tell the truth and you always lie." "Oh what a lie!")). Sarah has to choose which door to go through by figuring out which door is telling the truth. She doesn't figure it out.

Why is it so hard? Here are some obvious questions and the answers that would be given:

"Are you the liar?"
Truth-teller (T): No.
Liar (L): No.

"Are you the truth-teller?"
T: Yes.
L: Yes.

So obviously you need to ask one door about the other, right?

"Is he the liar?"
T: Yes.
L: Yes.

"Is he the truth-teller?"
T: No.
L: No.

Now take it one step further...

"If I ask him if you're the liar, what will he say?"
T: Yes.
L: No.

"If I ask him if you're the truth-teller, what will he say?"
T: No.
L: Yes.

These two questions work because the truth-teller can honestly tell you what the liar would say, but the liar has to lie about what the truth-teller would say. So in the first example (which is the one I prefer), if you ask the liar if the truth-teller is the liar, he will lie to you and say yes. The truth-teller can tell you that, so you know he's telling the truth, simply because he said yes. If you ask the truth-teller is liar is the liar, he will say yes, but the liar cannot tell you the correct answer, so he must say no. Doing so reveals him as the liar.

Ergo, it is possible to solve the problem by asking a single question, which is sometimes a rule that is applied to the problem to make it seem harder than it actually is.

Anyway, there you go. Remember that in case you're ever on a reality show and they stick you in a similar situation.