Points of Convergence


Liar / Truth Teller and the Wonders of Inductive Reasoning
January 29, 2010, 7:10 am
Filed under: Uncategorized

Maybe it’s prompted by my return to grad school, or maybe by a surplus of free time, but I thought I’d write a little piece on the basic and very essential act of Thinking. Actually, I had a dream last night that I already wrote this, so I guess it could be called “Dreaming of Thinking.” Or, as someone that took the wrong train today for several stops on the way to work, it would be better to call it “Dreaming Instead of Thinking.” But then, that has absolutely nothing to do with the lecture at hand. Clearly, not much thought was put into this opening paragraph.

Let’s get into it—I now present you with the classic Liar / Truth Teller Riddle:

“A man finds himself standing in front of two doors: one which leads to heaven and the other which leads to hell. In front of the doors are two guards, one which always lies and the other which always tells the truth. The man does not know which door is which, nor which guard is which. He must ask one question of one of the guards to determine which door will lead him to heaven…”

You may have seen this one before, and you may know the answer. I don’t care about that. Pretend you don’t. Pretend you’re on a reality TV show with no prior knowledge of this riddle, and maybe substitute “Heaven” and “Hell” with the Price is Right showcase containing the new car, and the kind of lame showcase with the weird ‘70s furniture, respectively.

As you can see, the stakes are high. But let’s cut to commercial and discuss this a bit before doing anything rash. How do we go about selecting an appropriate question? How do we know how to evaluate the answer we receive?

Of course, just as you may not need a flashlight to unlock your front door when it’s pitch black, you may not need a strategy at all to come up with the correct answer—“fumbling in the dark “ can be quite effective, and the correct answers often come out of the blue with seemingly no antecedent.

However, for us mere mortals, we can choose to use our magical gift of reason to light the way for us. I’ll describe a simulated thought process:

The first step is to truly UNDERSTAND the problem. This means stripping away the extraneous information. To evaluate each statement, ask yourself whether you can substitute the given subject for another without changing the problem. Start simple: A MAN finds himself STANDING in front of TWO doors. Suppose we want to be politically correct; could we change this to a woman? Of course we can. Maybe she isn’t standing. Could she be hopping on one foot? Sure, it makes no difference. Now, what if we changed it to THREE doors? Uh oh…then the contents of the third door is undefined in the description. Better leave it as two doors.

After completing this excision, we have the statements that matter:

(i) Two Doors; one with the desired outcome, one with the undesired outcome.
(ii) One Guard Per Door; one always lies, one always tells the truth. The identity of either is unknown to you.
(iii) One Question Available to Ask Either Guard to Obtain Desired Outcome.

Now, we have some information to work with. We see immediately that asking questions like, “Is it the door on the left?” will not increase our odds beyond that of pure chance; there’s no correlation between left and right and lies and truth stated. In fact, there isn’t much to go on…but that’s a good thing.

Let’s examine each statement for any outstanding features. When we do this, we notice that statement (ii) is the only one of note—it gives the CONDITIONS of the problem (the other two simply state the “rules of the game”). Therefore, it is not unreasonable to make an assumption that, if there is, in fact, a solution to this riddle, it should exploit this condition. Let’s ask some more questions—specifically, about truth and lies…

If we ask a question that only involves the truth, such as, “Is the Truth Teller guarding the correct door?” we still fail to increase our odds beyond chance; If the Truth Teller is, indeed, guarding the desired door, he will say yes; if not, he will say no. But, if the liar is asked, he will say the opposite. Of course, we have no way of knowing who we asked, so the answer would be meaningless.

This serves to illustrate the fact that you can be on the right PATH to a solution and STILL be no closer to the answer; it’s often an all-or-nothing proposition. However, this doesn’t mean that our assumption about statement (ii) is incorrect.

After trying out analogous questions involving the Liar only, it becomes apparent that the impediment to our quest to win that sweet 1979 LeBaron lies in the fact that each guard will give opposite answers. If we ask X to the Truth Teller, we receive X. If we ask X to the Liar, we get O. This is a major bummer.

Now, imagine what the solution, if there is one, would hypothetically contain—what would overcome this barrier? Obviously, there would need to be a question that would receive the same answer from either guard. That way, the answer wouldn’t be due to chance.

So, now we have two pieces of information about the solution: It must exploit the fact that one guard always lies and one always tells the truth. Also, it must force the same answer from either guard. We can extend this last one by recognizing that such a question would, out of necessity, be interpreted differently by each guard; this means it must be a general question that does not refer to the Liar or Truthteller by NAME.

So now we just need to find a way to throw a monkey wrench in the Truth/Lie machine. A question that somehow exploits the properties of the Liar and Truthteller, without explicitly referring to them, AND forces the same answer from both.

This last bit may be sufficient to spark the answer, but if not, there is one more interesting property about Lies and Truth: If we represent both as binary logic, a lie about a lie will equal the truth…whereas the truth about the truth will STILL be the truth…

Similarly, If you ask the Liar to tell you what HE WOULD SAY if you WERE TO ASK him which door to choose, you’ll negate his lie and get the CORRECT door. If you ask the Truth Teller to tell you what he would say, you’ll also get the CORRECT door. Both answers are the same; drive that new car off the set and go get your pets spayed or neutered.

The other solution, of course, would be to “bounce” the answers off of one another by asking, “Which door would the OTHER guard tell me to choose?” In this case, both will tell you to choose the incorrect door, so you just choose the opposite one. Both questions, when you pick them apart, serve the same function and satisfy the necessary conditions of our hypothetical solution.

Pretty amazing stuff, this ability we have as thinking humans! Don’t feel bad if it took you awhile to get the answer—it took me a long time, and I felt very dumb as a result. It wasn’t until I actually sat down and went through all the steps I described that I figured both solutions out (I don’t think there are any other solutions that aren’t a variation on the one I gave, but I could be wrong).

Now, in our everyday lives, we encounter much more difficult situations with many more variables, and many, many more outcomes. But, when we pick these situations apart, we are often surprised at how much of our confusion stemmed from a simple misreading of a problem, or our inability to visualize what a solution might LOOK like. Thinking is hard! But, if you can find a way understand the problem AND understand what the solution might entail, all the stuff in the middle usually finds a way to work itself out.
As an alternative, you can always just bid $1 and wait for the other contestants to mess up. –SCB

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4 Comments so far
Leave a comment

Someone must have pointed out that you are wrong about this:

Similarly, If you ask the Liar to tell you what HE WOULD SAY if you WERE TO ASK him which door to choose, you’ll negate his lie and get the CORRECT door. If you ask the Truth Teller to tell you what he would say, you’ll also get the CORRECT door. Both answers are the same;

You can only negate the liar’s answer if you know he is the liar. Since you don’t know which is the liar you will get two answers UNLESS you ask them each what the other would say.

Jack

Comment by jackdev

Hi Jack,
The liar will negate himself automatically, because it is a hypothetical question. Think about it: If you ask a liar to tell you what he would answer if you were to ask (instead of directly asking him), he will give the opposite of the answer he would normally give…so, he will be giving the correct answer in this case, which is the same answer that the truthteller would give when asked the same question. It’s essentially the same mechanism as having them tell you what the other one would say- only in this case, you are asking them to answer a hypothetical about themselves from an outside point of view.

Comment by scottbradlee

Actually the whole point is you just have to ask the guard 1 type of question to test water before asking another one question to determine if he speak the truth or not and it can be any question, not necessary it had to concern the door.

Just ask one of the guard : “Am I an man or a woman/gay”, and another guard “Is this the correct door to heaven”(It does not violate asking 1 question to each guard) it is very easy to see who is telling the truth and who is lying.

Comment by Steve

I thought of a different solution and wondered if anybody else had come up with it (or if there’s some reason it wouldn’t work):
point to a door and ask one of the guards “Can you tell me that this door leads to heaven?” If it does, the truth-teller will say “yes,” and the liar will also say “yes” because he actually can’t – that would be telling the truth. If it does not, the truth teller will say “no” because he can’t lie, and the liar will also say “no” because he can (he would, in fact, be obligated to). Although I suppose this might just be a variation on the first solution…

Comment by Foxface




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