A Familiar Problem

As usual, I’m spending my afternoon watching Yu-Gi-Oh, and watching two early episodes that I’ve never seen before involving two duelists named Para and Dox. Accoridng to the dynamic duo, one of them tells only the truth and one of them tells only lies. Joey and Yami got one question to try to determine information they needed, in this case the door being guarded by the dynamic duo of the three in the room. Joey handled the situation beautifully and got the answer they needed. Yami immediately disregarded the information, telling Joey that they couldn’t trust either one. I’m fine with the fact Yami protested, but his argument would have held so much more weight if he had pointed out that there were three doors in the room and if Dox had pointed to a different door than Para.

In short, the problem was misconstructed.

This problem is a very familiar one. It’s shown up in more places than I can count. My first encounter with it was in a Childcraft book when I was in elementary school. I was the brave space explorer Laura on the planet Blarg inhabited by the known truth-tellers, the Zorbats, and the known liars, the Gazgles. She had to ask them which city she was in, because somehow there was a member of the opposite population in the city.

The theory behind the problem is simple: you ask one of half a dozen questions that will result in both entities telling you the same answer in such a way that a useful meaning exists in the answer. Some of the better examples have been: Am I in the land of ?, Take me to your village., Point to the door you are guarding. What’s truly amazing is that for how prolific this particular mind bender is that so many writers completely mess it up or bring in irrelevant information that only confuses the situation for literal minded folks like our dear animated friend Yami.

Another favortie botch of this problem is from the movie Labyrinth. Granted, Sarah’s not the brightest light bulb, but this writing botch took some talent. She’s confronted with two doors with playing card guards (who are possibly only a hair smarter than our lovely heroine). They explain the rules before engaging in an argument over which guard is the liar. After what passes for deep thought on her part, Sarah finally presents her question to one of the guards:Would he [points to other guard] tell me that this door leads to the center of the labyrinth?After a moment of debate he responds yes, and she promptly engages in weak logic and selects the wrong door.

Did her question have a chance of actually being useful? Let’s label the doors and their respective guards A and B. She asks Guard A if Guard B would say Door A went to the center of the labyrinth. Guard A says Guard B would say that Door A went to the center of the labyrinth. Pretend Guard A is the one who tells the truth. Guard B would indeed say that his own door went to the center of the labyrinth. Do we know that Door A is the correct door? Well, in this scenario, Guard B would have to be lying and Door B would be the correct door. Let’s reverse things and pretend Guard A is lying. This means Guard B would say that Door A was not the correct door. Because we know at this point Guard B has to be telling the truth, we know that Door B has to be the correct door.

Either way, Door B is the correct answer, despite the overly complex question. But our determined heroine completely mishandles her logic, either because she doesn’t remember the exact question she asked or because of that light bulb problem. In her misguided logic, she picks the wrong door.

I suppose I’m just simply amused at how easily this rather simple and well-known problem is presented incorrectly.

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