OpenLab #3: The MIU puzzle

We are going to play a game with strings of symbols.  This game was invented by a man named Douglas Hofstadter and found in his book Gödel, Escher, Bach. Here are the rules:

Suppose there are the symbols ‘M’, ‘I’, and ‘U’, which can be combined to produce strings of symbols called “words”, like MUI or MIUUU. The MIU game asks one to start with the word MI and transform it using the following rules, to obtain some goal word (which is given to you).  The rules state:

  1. You must always begin with the word MI.
  2. You may add a U to the end of any string ending in I. For example: MI to MIU, or MUUII to MUUIIU.
  3. You may double any string after the M (that is, change Mx, to Mxx, where ‘x’ represents any string of symbols). For example: MIU to MIUIU
  4. You may replace any III with a U. For example: MUIIIU to MUUU
  5. You may remove any UU. For example: MUUU to MU

WARM UP.  In each example, start with the axiomatic word MI and show, step-by-step, how to obtain the goal word (in each step, state which of the rules you used). These are just for practice (you do NOT need to submit your answers).

Example 1: Goal word MIU
Example 2: Goal word MIIU
Example 3: Goal word MIIUIIU
Example 4: Goal word MUUII
Example 5: Goal word MUUIIUIIU

Example: Goal word MUI
Step 1: MI  (we always start with this word)
Step 2: MI to MII (rule 3)
Step 3: MII to MIIII (rule 3)
Step 4: MIIII to MUI (rule 4)

Assignment (due Thursday, 9/15): Your assignment has three parts.

PART 1.  First, create an MIU puzzle — that is, make up a goal word, and post it in the comments. Your goal word should be between 8 and 16 letters long.  Try to make it tricky to reach, requiring at least four steps to reach (but the more the better!).  See if you can find a clever use of the rules!

PART 2.  The second part of your assignment is to solve someone else’s puzzle.   Type your solution step-by-step, indicating which rule you used at each step.  Leave your comment as a response to their puzzle.  Only one solution per puzzle!

PART 3.  The third part of your assignment is to write a short paragraph (at least 3 sentences) responding to the following prompt.  Be sure to respond to each part:

Writing Prompt, MIU puzzle:  Did you enjoy this assignment? Why or why not?  Describe a connection between this assignment and our work in the class.  (If you don’t believe there is a connection, try to imagine why we are doing this).  Leave your response in the comments.


  1. Armando Cosme

    Here’s mine, I told ya’ll I’d try to make it difficult 🙂

    • Ismail Akram

      Your word has to begin with “MI” and you haven’t stated a goal word to reach.

      • Armando Cosme

        That is the goal, If you begin with MI and then apply some rules, you should come to that word

    • Jonas Reitz

      Hi Armando,
      I love that you decided to go big — but the instructions say “Your goal word should be between 8 and 16 letters long”… could you give us another example? (maybe keep this one up as a challenge to your classmates).

  2. Tyniqua

    Here’s my goal word for my MIU puzzle: MUUUUUIU

    • Violanda

      Hello Taniqua, if I didn’t make any stupid mistake,I think that this is the way to solve you puzzle.
      MI (Rule 1)
      MII (Rule 3)
      MIIII (Rule 3)
      MIIIIIIII (Rule 3)
      MUUUUUI (Rule 4)
      MUUUUUIU (Rule 2)

    • Violanda

      Hello Taniqua, if I didn’t make any stupid mistake,I think that this is the way to solve your puzzle
      MI (Rule 1)
      MII (Rule 3)
      MIIII (Rule 3)
      MIIIIIIII (Rule 3)
      MUUUUUI (Rule 4)
      MUUUUUIU (Rule 2)

  3. Tyniqua

    Yes I enjoyed this assignment solving the puzzles, although I’m having trouble writing down Armando’s too many letters to even attempt to solve it, so I’m waiting on others. I do find a connection between the work we are doing in class and these puzzles in reference to the symbolism. Meaning, we use symbols to represent special numbers or sets in class and in the puzzle we take an arrangement of specific letters (symbols) and translate them into a word using the rules and limits set for us.

    • Armando Cosme

      Haha!!! I knew it’d be difficult. A little tip, I double alot!

    • Jonas Reitz

      Great connections – this is definitely one of the reasons we are looking at this puzzle! (nice puzzle, too).

  4. Ismail Akram

    Here’s mine:
    PART 1:
    Start with “MIUUIUUIUUIUU’
    End with “MIIUIIUIIUI

    Hint: Rules 3 and 4 will come in handy. Can be done in 4 steps.

    PART 3:
    As for the assignment, I found it quirky. This tasks asks for comfort with the rearrangement of variables within set rules. This should sound familiar as this is how we manipulate algebraic expressions. To connect this with our class, we need to become very confident and comfortable in manipulating expressions to achieve a desired outcome (or proof), within the bounds of the rules of course.

    • Tyniqua

      Hi Ismail,
      I thought we were suppose to start out with MI, but going by your start I followed the rules and removed all of the UU’s and was left with MIIII, which I ended up with the word MUIU.

    • Gary

      Hi Ismail,
      Is this the same way you solved your puzzle.
      MIUUIUUIUUIUU (start with)
      MIIUIIUIIUI (step 4)

    • Jonas Reitz

      Hi Ismail,

      Tyniqua is correct, solutions always have to start with the word MI. Could you gives us another example that fits the assignment?

      However, I do like your puzzle a lot – it presents a interesting new twist on the rules of the MIU game, in which you give not just a goal word, but a starting word and a goal word. Very cool!

      -Prof. Reitz

      • Ismail Akram

        Sure, I’d modify the question into
        Start: MI

        Gary is correct with the first one, funny enough I got my answer by applying R3 and R4 in a different order.

  5. Armando Cosme

    Part I: Oops, I didn’t realize it was supposed to be 8-16 letters!!
    Here’s a New Goal Word: MIUIUIIU

    Part III: I liked this alot, it’s very fun to create a problem, but it’s intresting to see if there is more than one way to solve one. I know this has something to do with logic. What “logically” can fit in which position. I still have yet to put the pieces together to what this topic is equivalent to in our class though.

    • Violanda

      MI (Rule 1)
      MII (Rule 3)
      MIIII (Rule 3 )
      MIIIIIIII (Rule 3)
      MIUIUIIUU (Rule 4)
      MIUIUII (Rule 5)
      MIUIUIIU (Rule 2)

  6. Violanda

    Here is my puzzle.
    The goal word is :MUIIUIIUIIUIIU

    • Armando Cosme

      Part 2: Solution to Violanda’s Goal
      1) MI (Rule 1)
      2) MII (Rule 3)
      3) MIIII (Rule 3)
      4) MIIIIIIII (Rule 3)
      5) MUIIU (Rule 4)
      6) MUIIUU (Rule 2)
      7) MUII (Rule 5)
      8) MUIIUII (Rule 3)
      9) MUIIUIIUIIUII (Rule 3)
      10) MUIIUIIUIIUIIU (Rule 2)

      • Violanda

        You made a mistake at “6)”,you can not add a letter U after U,only after letter I. This is what Rule 2 says.

        • Armando Cosme

          One More Time!!!!
          1) MI (R3)
          2) MII (R3)
          3) MIIII (R3)
          4) MIIIIIIII (R3)
          5) MIIIIIIIIU (R2)
          6) MUIIUU (R4)
          7) MUII (R5)
          8) MUIIUIIUIIUII (R3)
          9) MUIIUIIUIIUIIU (R2)

          • Jonas Reitz

            Nice fix!

  7. Gary

    Here is my puzzle.
    Start with MI

    • Tyniqua

      Goal word too long Gary, needs to be between 8 and 16 letters.

    • Hanan

      Hello Gary, while I was solving your puzzle I realized that you went over 16 letters by mistake. I still wanted to solve your puzzle because it looked challenging.

      Before solving your puzzle I checked to see if you duplicated your answer at all. I found that IUUUIU was repeated. So I had to find MIUUUIU

      MI (Rule 1)
      MII (Rule 3)
      MIIII (Rule 3)
      MIIIIU (Rule 2)
      MIIIIUIIIIU (Rule 3)
      MIUUIIIIU (Rule 4)
      MIUUUIU (Rule 4)
      MIUUUIUIUUUIU (Rule 3)

  8. Hanan Abuhashish

    Here is my goal word: MUIUIUUIUIU

    • Sonam

      Hi Hanan, here is my solution for your puzzle.

      1. MI (Rule 1)
      2.MII (Rule 3)
      3.MIIII (Rule 3)
      4.Mllllllll (Rule 3)
      5.MUIUI (Rule 4)
      6.MUIUIU (Rule 2)
      7.MUIUIUUIUIU (Rule 3)

      • Hanan

        Hi Sonam, you absolutely right .

  9. Hanan

    Yes, I enjoyed this assignment. I enjoyed this assignment because it was challenging and it helped me practice trial and error. It showed me that math has more to do with logic than just simple numbers and answers. I did feel like there was a connection between this assignment and the work in class. Both have to to with logic and reasoning.

  10. Sonam

    Hello all, here is my goal words: MULLUULLU

    • Sonam


      • Jpitt

        I think this is the solution to your puzzle:
        1. MI ( Rule 1)
        2. MII ( Rule 3)
        3. MIIII (Rule 3)
        4. MIIIIIIII (Rule 3)
        5. MIIIIIIIIIIIIIIII (Rule 3)
        6. MUIIUUIIU ( Rule 4)

  11. Jpitt

    Hi all sorry I’m late. My goal word is- MUUIIIU

    • Jpitt

      Forgot it needs 8 words. Goal word: MUUIIIUUU

      • Leonardo Perez

        Hey there, correct me if I’m wrong but I believe this is the solution to your goal word:

        1. MI (Rule 1)
        2. MII (Rule 3)
        3. MIIII (Rule 3)
        4. MIIIIIIII (Rule 3)
        5. MIIIIIIIIIIIIIIII (Rule 3)
        6. MUIIIIIIIIIIIII (Rule 4)
        7. MUUIIIIIIIIII (Rule 4)
        8. MUUIIIUIIII (Rule 4)
        9. MUUIIIUUI (Rule 4)
        10. MUUIIIUUIII (Rule 3)
        11. MUUIIIUUU (Rule 4)

        I believe you only used rules 3 and 4. As I said again I might be wrong or you might came up using other rules.

        • Jonas Reitz

          Hi Leonardo – with Rule 3, you always have to double *everything* after the M (not just some of the letters) – so in step 10, using Rule 3 would give:
          Can you come up with an alternative solution?
          -Prof. Reitz

      • Jpitt

        Sorry I was the one who messed up. Gave a unsolvable one by mistake cause I was in a rush lol. New goal word : MUIIUIIIU
        Hint: start with MI and go to MIIIIIIII with rule 4 then go from there.

  12. Jpitt

    Part 3: I enjoyed the assignment because it was fun making and solving the puzzles. What I realized is that everyone loves to use rule 3, where you just double everything, since it’s the rule that really stretches the problems making them seem unsolvable. I think these puzzles relate to sets where we create sets but there are certain rules that apply to each set like set-builder notation. We might be given incomplete sets with rules we might have to find ourselves and have to complete those sets with those rules.

  13. Leonardo Perez

    Hello everyone! Sorry that I’m late but here’s my goal word of the day, I see that mostly everyone came up with 8 letter word solutions haha, so here’s mine:


    Part three:
    This is an interesting assignment! Its like basically solving a puzzle and have to think here for a while on what rule applies next in order to reach to the final. I have seen that a lot has used the 3rd rule because it has to deal with doubling any string in order to extend and reach our goal word. From what we have learned in class we are using the term of “sets”. Since the purpose here is to make our goals words between a set of 8-16 letters, the third rule, is most likely to assist us to get our best answers. However to make things more interesting, this is where the other rules kicks in for us to play along as we get to reduce some letters from our long words that might contain more than a set of 16 letters. I also see that there’s a limit of how many letters we can use so it will become a challenge.

    • Kevin Truong

      1) MI ( RULE 1)
      2) MII (RULE 3)
      3) MIIII (RULE 3)
      4) MIIIIIIII (RULE 3)
      7) MIIIIUUUUU (RULE 4 x4)
      8) MIIIIUUU (RULE 5)

      So much llllllllllllllll I’m blind!!!

      • Jose De Leon

        III gives you U so you can reduce the numbers of Is.

  14. Kevin Truong

    OHHH man I’m really late for this one. But here is my goal word:


    Part 3:
    This assignment was pretty fun and it woke me up (thanks, now i can’t sleep…). Answering a classmate’s puzzle had me to go find a paper and pencil to do this because it was difficult to do the whole thing in my head. I believe this assignment has a connection to the work we do in class which it requires lots of brain power to find the solution. The class is called Logic and Proofs for a reason, these problems requires lots of logic and reasoning to figure out the answer.

    • Ismail Akram

      I guess I’ll have a crack at this.

      MII (R3)
      MII II (R3)
      MIIII IIII (R3)
      MI U II U I U U (R4)
      Goal: MIUIIUIUU

      • Jonas Reitz

        Ismail – I like the way you use spacing to help make each step easier to read. Nice innovation!
        – Prof. Reitz

  15. Jose De Leon

    MIU(Rule 2)
    MIIII(Rule 3)
    MUI(Rule 4)
    MUIIIII(Rule 3)

  16. Jose De Leon

    Puzzle made me feel to challenge my thinking on how to figure out rules and follow as well to comply with them.Initially, MI rule was difficult to put in place using UU or IU to double IU. So, adding III gave me a U.However, I enjoyed to be engaged monitoring possible answers.

  17. Violanda

    I really enjoyed this assingment as everybody else.One reason that I enjoyed it was that It didn’t have anything to do with writing essays because I really am a bad writer.However I believe that this puzzle have a lot to do with logic.So,since we are studying Proofs and Logic there is a connection between the puzzle and our class.

  18. Gary

    Start: MI

  19. Gary

    I enjoyed this puzzle because this was like a maze because once you start you can get stuck but eventually you’ll it out an finish. I think the MIU puzzle connects with logic because we started this and started learning about logic on the same day. I think this challenge is to help us see that there will be challenging questions in the future but if we keep trying we’ll be able to solve them like with the puzzles people post up.

  20. Sonam

    I really enjoyed to working on this assingment. MIU puzzle is a great game to practise logical thinking, and logical thinking skills is crucial when we try to proof somethings. Also this assingments gave us a great opportunity to play game with other calssmates, we were able to solve the problem together and understands the strengths and weaknesses of each other.

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