You and your team have crash-landed on an ancient planet. Can you appease the three alien overlords who rule it and get your team safely home? Created by logician Raymond Smullyan, and popularized by his colleague George Boolos, this riddle has been called the hardest logic puzzle ever. Alex Gendler shows how to solve it.
Transcript:
Created by logician Raymond Smullyanand popularized by his colleague George Boolos, this riddle has been called the hardest logic puzzle ever.
You and your team have crash-landed on an ancient planet. The only way off is to appease its three alien overlords, Tee, Eff, and Arr, by giving them the correct artifacts. Unfortunately, you don’t know who is who. From an inscription, you learn to ask three yes or no questions, and each addressed to anyone lord.
Tee’s answers are always true, Eff’s are always false, and Arr’s answer is random each time. But there’s a problem. You’ve deciphered the language enough to ask any question, but you don’t know which of the two words ‘ozo and ‘ulu’ means yes and which means no. How can you still figure out which alien is which?
At first, this puzzle seems not just hard but downright impossible. What good is asking a question if you can neither understand the answer nor know if it’s true? But it can be done. The key is to formulate our questions so that any solution yields useful information carefully.
First of all, we can get around to not knowing what ‘ozo’ and ‘ulu’ mean by including the words themselves in the questions; secondly, if we load each question with a hypothetical condition, whether an alien is lying or not won’t matter. To see how that could work, imagine our question is whether two plus two is four. Instead of posing it directly, we say, “If I asked you whether two plus two is four,would you answer ‘ozo’?
“If ‘ozo’ means yes and the overlord is Tee, it truthfully replies, “ozo.” But what if we ask Eff? Well, it would answer “ulu,” or no to the embedded question, so it lies and replies ‘ozo’ instead. And if ‘ozo’ actually means no, then the answer to our embedded question is ‘ulu,’ and both Tee and Eff still reply ‘ozo,’ each for their reasons.
If you’re confused about why this works, the reason involves logical structure. A double-positive and a double negative both result in a positive. We can be sure that asking either Tee or Eff a question this way will yield ‘ozo’ if the hypothetical question is true and ‘ulu’ if it’s false regardless of what each word means.
Unfortunately, this doesn’t help us with Arr. But don’t worry, we can use our first question to identify one alien lord that isn’t Arr. Then we can use the second to find out whether it’s Tee or Eff. And once we know that, we can ask it to identify one of the others. So let’s begin.
Ask the alien in the middle, “If I asked you whether the overlord on my left is Arr, would you answer ‘ozo’?”If the reply is ‘ozo,’ there are two possibilities. You could already be talking to Arr, in which case the answer is meaningless. But otherwise, you’re talking to either Tee or Eff, and as we know, getting ‘ozo’ from either one means your hypothetical question was correct, and the left overlord is indeed Arr. Either way, you can be sure the alien on the right is not Arr.
Similarly, if the answer is ‘ulu,’ you know the alien on the left can’t be Arr. Now go to the overlord you’ve determined isn’t Arr and ask, “If I asked ‘are you Eff?’ would you answer ‘ozo’?”Since you don’t have to worry about the random possibility, either answer will establish its identity. Now that you know whether its answers are true or false, ask the same alien whether the center overlord is Arr. The process of elimination will identify the remaining one. The satisfied overlords help you repair your ship, and you prepare for takeoff. Allowed one final question, you ask Tee if it’s a long way to Earth, and he answers, “ozo.”Too bad you still don’t know what that means.
