Created by logician Raymond Smullyan and popularized by his colleague George Boolos, this riddle is often referred to as the hardest logic puzzle ever. Imagine you and your team have crash-landed on an ancient planet. The only way to escape is by appeasing its three alien overlords: Tee, Eff, and Arr, by presenting them with the correct artifacts. However, there’s a catch—you don’t know which alien is which.
From an inscription, you learn that you can ask three yes or no questions, each directed to any one of the alien lords. Tee’s answers are always true, Eff’s are always false, and Arr’s answers are random each time. Complicating matters further, you have deciphered the language enough to ask questions, but you don’t know which of the two words ‘ozo’ and ‘ulu’ means yes and which means no. How can you still determine which alien is which?
At first glance, this puzzle seems not just difficult, but utterly impossible. How can you ask a question if you can’t understand the answer or know if it’s true? However, the solution lies in carefully crafting questions so that any answer provides useful information.
Firstly, you can bypass the uncertainty of what ‘ozo’ and ‘ulu’ mean by incorporating the words themselves into the questions. Secondly, by embedding a hypothetical condition into each question, whether an alien is lying or not becomes irrelevant.
Consider a question about whether two plus two equals four. Instead of asking directly, you could 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.” If you ask Eff, it would answer “ulu,” or no, to the embedded question, so it lies and replies ‘ozo’ instead. If ‘ozo’ actually means no, then the answer to the embedded question is ‘ulu,’ and both Tee and Eff still reply ‘ozo,’ each for their own reasons.
This logical structure works because a double positive and a double negative both result in a positive. Thus, asking either Tee or Eff a question in this manner will yield ‘ozo’ if the hypothetical question is true and ‘ulu’ if it’s false, regardless of what each word actually means. Unfortunately, this doesn’t help with Arr, but don’t worry; there’s a strategy to identify one alien lord that definitely isn’t Arr.
Start by asking 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 be talking to Arr, making the answer meaningless, or you’re speaking 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,’ then the alien on the left can’t be Arr.
Next, go to the overlord you’ve determined isn’t Arr and ask, “If I asked ‘are you Eff?’ would you answer ‘ozo’?” Without the random possibility, either answer will establish its identity. Once you know whether its answers are true or false, ask the same alien whether the center overlord is Arr. Through process of elimination, you can identify the remaining one.
With the overlords satisfied, they assist you in repairing 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.” Unfortunately, you still don’t know what that means.
Gather in small groups and assign roles: one student each as Tee, Eff, and Arr, and the rest as the questioners. Use the rules from the article to ask questions and determine who is who. This will help you understand the logic behind the puzzle and practice formulating questions.
Complete a worksheet that includes various scenarios and questions similar to the ones in the article. This will reinforce your understanding of how to craft questions that yield useful information regardless of the answer’s truthfulness.
Design a logic puzzle inspired by the hardest logic puzzle. Include a scenario, characters with specific truth-telling rules, and a set of questions. Exchange puzzles with classmates and try to solve each other’s creations.
Use a logic puzzle app or online game that involves similar principles. These games often provide immediate feedback and can help you practice and refine your logical thinking skills in a fun and engaging way.
Participate in a class discussion where you analyze the puzzle’s solution step-by-step. Discuss alternative strategies and what makes this puzzle particularly challenging. This will deepen your understanding and allow you to see different perspectives.
Logic – The study of reasoning and the principles of valid inference. – Logic helps us understand how to form valid arguments and make sound decisions.
Puzzle – A game or problem that tests a person’s ingenuity or knowledge. – Solving a puzzle requires critical thinking and the ability to see patterns.
Questions – Sentences that are used to ask for information or to clarify something. – Asking questions is an important part of critical thinking and helps us gain deeper understanding.
Aliens – Hypothetical beings from other planets or galaxies. – In science fiction, aliens often pose interesting logical dilemmas about communication and understanding.
True – A statement that is in accordance with fact or reality. – The statement “All squares are rectangles” is true based on the definitions of the shapes.
False – A statement that is not in accordance with fact or reality. – The claim that “The Earth is flat” is false because it contradicts scientific evidence.
Random – Without a definite aim, purpose, or reason; unpredictable. – When conducting an experiment, it is important to select a random sample to avoid bias.
Identify – To recognize or establish what something is. – In logic, we must identify the premises and conclusions of an argument to evaluate its validity.
Answer – A response to a question or problem. – Providing a clear answer is essential in critical thinking to demonstrate understanding.
Information – Data that is processed or organized to provide meaning. – Gathering accurate information is crucial for making informed decisions in any logical analysis.
