The Alien Overlords Riddle for MIT

Hello fellow thinkers. I enjoy challenging myself so, as most of you know, I sometimes give myself riddles that consist of logic or math that are solvable by intuition. Let's talk about one of my favorite 'riddles' that took me two hours to solve.


While applying for an on-campus course for MIT I was given a prompt:


Please describe and discuss your favorite mathematical or scientific problem or challenge. This can be anything from a cool problem you had for homework, to a project you have undertaken, to a research direction you find exciting.

The first thing that came to my mind was the mind-boggling Alien Overlords Riddle and I remembered the surge of dopamine that me and my ADHD mind felt after solving it. But even after solving it, the challenge didn't end. I had to explain my solution to the MIT admissions committee in under 330 words.


My main idea to limit my word count was to assume the reader already has some knowledge of logical concepts that are required to answer this riddle. But of course this may not be the case so I included that information in the Appendix and referenced it from my main body of text. Typically, Appendices do not contribute to word count. Here's my response... (Side note, I probably should not be writing this right now since I have GCSEs in 9 days)


INTRODUCTION

This is one of my favorite riddles as it incorporates Boolean logic into a riddle. While visiting another planet, we understand and speak their language but we do not know the meanings of “ulu” or “ozo”. They are either yes or no. We can only ask 3 yes or no questions to identify which of the three aliens is which. Here’s the twist:


MEET THE ALIENS

  1. Arr always answers randomly

  2. Tee always tells the truth

  3. Eff always lies


IDENTIFYING THE ALIENS / SOLUTION

Firstly, we need to find Arr to eliminate him from our questions as he provides useless answers. Let's ask the middle alien “If I asked you if the alien to your right is Arr, would you reply ‘ozo’?” If the reply is ‘ozo’, we are either talking to Arr and so the answer is meaningless, or we are talking to Tee or Eff and so they will either way reply with ‘ozo’ if true and ‘ulu’ if false (For more details on the logic of that statement and the logic associated with this solution see LOGICAL CONCEPTS).


Depending on the answer we will know whether Arr is on the (Left or Middle) or (Right or Middle). We can then use this answer to go to the one we know is not Arr and ask “If I asked you are you ‘Eff’, would you reply ‘ozo’?” If the reply is Ozo then that is Eff otherwise it is Tee since we know it isn’t Arr (For more information on why ozo always means true in this statement, see ALL SCENARIOS). Now ask the same alien “If I were to ask you if the alien in the middle is Arr, would you reply ‘ozo’?” and if the answer is ‘ozo’ then you know the alien in the middle is Arr and find the final alien through the process of elimination. The same applies if the answer was ‘ulu’.


Now we have identified each alien and we can now head back to earth, “Is it a long way back to earth” I ask. And they all reply ‘Ulu’ except for one. Too bad we still don’t know what that means.


APPENDIX


LOGICAL CONCEPTS

Facts that we can infer:

  • Knowing 2 of the aliens, we can use the process of elimination to find the third

  • If we format our questions: “If {Condition}, would you answer ozo?”, this will get us truthful answers from both Tee and Eff.

  • Eff goes through two stages before answering, resolving the first statement to true or false (the condition), using it to resolve the would you answer statement, then answering the question inverted to the answer of stage 2 since he always lies.

  • This whole problem’s solution uses the concept of how a positive positive = positive and a negative negative = negative

  • In terms of logic gates, the XOR gate embedded in the phrasing of the questions means that either way, when Tee or Eff are answering, we can get a truthful answer out of both without knowing the meaning of their answers.

ALL SCENARIOS


ASSUMING OZO MEANS YES

We ask Tee and Eff “If 2*2=4, would you reply ‘ozo’” If you are talking to Tee then 2*2=4 is true and so would you reply ‘ozo’ would also be true therefore the reply would be ‘ozo’? If you are talking to Eff, 2*2=4 is true, would you reply “ozo” would be false since he always lies so he would answer ulu but since he always lies he will therefore invert his answer again making the final answer true (ozo).


ASSUMING ULU MEANS YES

Let’s assume ulu=yes. We ask Tee and Eff “If 2*2=4, would you reply ‘ozo’” If we are talking to Tee then 2*2=4 is true and so would you reply ‘ozo’ would be false therefore the reply would be ‘ozo’. If you are talking to Eff, 2*2=4 is true, would you reply “ozo” would be true since he always lies but since he always lies he will therefore invert his answer again making the final answer true (ozo).


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