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 “axiomatic” word MI and transform it using the following four rules, to obtain some “goal” word. The rules state that you may:

- Add a U to the end of any string ending in I. For example: MI to MIU, or MUUII to MUUIIU.
- Double any string after the M (that is, change Mx, to Mxx, where ‘x’ represents any string of symbols). For example: MIU to MIUIU
- Replace any III with a U. For example: MUIIIU to MUUU
- 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

Here is a sample solution to Example 2:

MI to MII (rule 1)

MII to MIIU (rule 2)

**Assignment (due Tuesday, 10/29):** 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. Try to create a goal word that balances the following two requirements:

- The goal word should not be too long – definitely not more than 10 letters (but the shorter the better).
- The goal word should be 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 toimaginewhy we are doing this). Leave your response in the comments.

Goal word MUIU

I did enjoy this assignment even though it was a bit challenging at first. the only connection I think between this assignment and the work we’ve done in class is that basically we proven that something exist . In the case of the puzzle we proving that we can make such a word using a combination of the rules given. from reading a bit more on this topic I can also tell that some things are hard to prove or cant be proven.

Solution:

1. , by rule 2.

2. , by rule 2, again.

3. , by rule 3.

4. , by rule 1.

Very clear solution – thanks!

Thanks for giving the first response – nice puzzle!

The Goal Word is: MUIIU

I did enjoy this assignment because it involved some deep thinking in trying to figure out how to work this puzzle. It was fun because I was basically playing a game where I first had to learn the rules then I had to play along. Finally it was fun because once you create the puzzle, one of our classmates then has to rack their brains trying to figure out what steps to take to solve it. This assignment was similar to doing proofs and contradictions in my opinion. I feel that way because in writing proofs, one must know the rules and concepts in order to apply the correct steps to solving a proof. Similar to that, in creating the puzzle we had to go step by step remembering the rules and breaking down the puzzle to correctly identify what we needed to do next.

The Goal Word is: MIUUIIUUIU

I, really, enjoyed doing this puzzle. It was not only fun, but also, a bit challenging. I think that the purpose of doing it, was in proving step by step, according to rules that, for example: MI = MIU (rule 1). I think, the question was stated in a way if P then Q: if MI then MIU or conversely, if MIU then MI.

Solving Albina’s Goal Word:

1. MI = MII, by rule 2

2. MII = MIIII, by rule 2

3. MIIII = MIIIIU, by rule 1

4. MIIIIU = MIUU, by rule 3

5. MIUU = MIUUIUU, by rule 2

6. MIUUIUU = MIUUI, by rule 4

7. MIUUI = MIUUIIUUI, by rule 2

8. MIUUIIUUI = MIUUIIUUIU, by rule 1

Sweet!

Wow – tricky puzzle!

Solution to Ricky’s goal word: MUIIU

MUUUII-MUII-MUIIU

rule 4 – rule 1

Hi Albina — you end in the right place, but solutions always must start with “MI” Can you fix it by going from MI to your starting place of MUUUII?

The Goal Word is: MUUIU

Hint: is like Ricky’s word, but goes a bit further. (Not sure if we can hint, but I think I made mine too many steps!)

I think this was an interesting assignment, and does connect to what we have been learning in class. In order to create our own puzzles and solve others’, we have to think about different possible steps to make a final “proof”. It reminds me of thinking whether to use contradiction, contrapositive, direct, etc, and also reminds me of thinking of how to solve the middle blanks of a proof.

I’m taking a try here:

MI -> MII (RULE 2)

MII -> MIIII (RULE 2)

MIII -> MUI (RULE 3)

MUI -> MUIU (RULE 1)

MUIU -> MUIUUIU (RULE 2)

MUIUUIU -> MUIIU (RULE 4)

MUIIU -> MUIIUUIIU (RULE 2)

MUIIUUIIU -> MUIIIIU (RULE 4)

MUIIIIU -> MUUIU (RULE 3)

Wow – I did NOT expect that short goal word to require so many steps

The Goal word is M.

I thought this assignment was very interesting. It made me think a lot and long. I enjoyed making my puzzle and it ends up being only an M which was weird because I didn’t know it would become one letter but that’s cool! The rules were pretty simple which helped a lot. I think I would recommend this puzzle to my friend because she loves puzzles. Overall this was fun!

Great puzzle – so simple! Not sure how easy it will be to solve, though…

Hi Sonya, here is my guess:

MI to MII (rule 2)

MII to MIII (rule 2)

MIII to MU (rule 3)

MU to MUU (rule 2)

MUU to M (rule 4)

Hi Saloua,

This is a nice solution, BUT when using Rule 2, you have to double *everything* after the M — so your second line should read:

MII to MIIII (rule 2)

Unfortunately, this messes up the stuff below – sorry! Can you fix it?

-Prof. Reitz

Solution to Ricky’s puzzle :

MIIIII–> MUII (Rule 3)

MUII –> MUIIU (Rule 1)

You end in the right place, but your solution MUST always start with “MI”. Can you fix it by going from MI to your starting place of MIIIII?

Solution to Loudia’s puzzle :

MUUUI —> MUI (rule 4)

MUI —-> MUIU (rule 1)

(same as above – but I’m looking forward to seeing a revised version!)

Solution to Patty’s puzzle:

MIIIUI –> MUUI (rule 3)

MUUI –> MUUIU (rule 1)

Goal Word: MUUIIUUII

I thoroughly enjoyed the assignment. It was nice to play around with the rules and create conjectures as to what could and could not be done. I think that we’re doing this because it is practice for using a system of rules to go from some initial step to a finished form, much like a proof.

The Solution

MI -> MII (Rule 2)

MII -> MIIII (Rule 2)

MIIII -> MIIIIIIII (Rule 2)

MIIIIIIII -> MUUII (Rule 3)

MUUII -> MUUIIUUII (Rule 2)

Nice one!

Goal: MUIUIIUIU

That was a something else, I really did enjoy this though.

solution for Ricky:

1. MI to MII r.2

2. MII to MIIII r.2

3. MIIII to MIIIIIIII r.2

I then split the I’s M(III) (II) (III) not sure if this is how we are able to use them.

4. M(III) (II) (III) to get MUIIU r.3

solution for Adam’s

1. MI to MII r.2

2. MII to MIIII r.2

3. MIIII to MIIIIIIII r.2

4. M(III)(III)II to MUUII r.3

5. MUUII to MUUIIUUII r.3

I think this let us think about a solution and in the ways we can be able to find the solution while using given rules to the problem. Like how we did when solving proofs. It also shows us that by following given rules we can find an answer, which also goes back to proof solving. And, well, if there are different ways to go about in solving a problem, then this puzzle shows us that. Well I guess that goes back to the different ways we can solve a proof, that is direct, contrapositive and/or contradiction.

this was fun.

Good solutions!

Goal Word is MUIUIIUU

thank you professor for this assignment, it was fun. I think this puzzle is relevant to what we have done about proof. following the rules in different order to end up with a result.

The Goal word is MIIU

When I first started the assignment, I though you just add the letters together and try to reach your goal word. That made it challenging for me and I became frustrated. As I began to understand the assignment, I used the rules to help me create a goal word. Then I told myself this assignment is not that bad and was quite interesting. I think the puzzle connects with logic as certain rules give you a certain result and you have to alter it to say what it is you are looking for.

answer to puzzle

MI-MII rule 2

MII-MIU rule 3

I thought my previous goal word would have been longer, my mistake.

Goal word: MIUUI

That’s funny – I also thought your first goal word would have been longer, and I was surprised when I saw Loudia’s solution!

Goal Word: MUIUIIUIU

This assignement was very very puzzling for me but it was interesting. I really do hate puzzles. It was very challeneging for me which kept me interested and motivated to keep at it.

MI- MII rule 2

MII-MIIII rule 2

MIIII-MIIIIIIII rule 2

MIIIIIIII- MIIIIIIIIU rule 1

MIIIIIIIIU- MUIUIU rule 3

MUIUIU-MUIUIUUIUIU rule 2

MUIUIUUIUIU- MUIUIIUIU RULE 4

MUIUIIUIU

this took me like 7 tries until I finally figured it out.

Sweet! Great puzzle and great solution 🙂

Do I post the puzzle i created or just my goal word?

Post your own goal word, and then post a solution to someone else’s puzzle.

My Goal word: MUUIU

This assignment was a little hard because I had to figure out how each step worked this puzzle. This puzzle was also fun because I like a challenge like Sudoku. Also it was nice to create my own puzzle for someone to figure out. This assignment was similar to doing proofs. The reason why I think this way is because when writing proofs, you need to know the order to apply the correct steps to solving a proof.

Goal Word: MIIIUUI

I enjoyed this assignment. I found it fun to try to create a puzzle which works. The assignment is similar to direct proofs when trying to solve the puzzle. I would start at MI and use the definitions to lead me into the goal word similar to direct proofs where you start with p and use definitions until you end with q.

solution for Jean’s

MI – MII #2

MII – MIIII #2

MIIII – MIIIIIIII #2

MIIIIIIII – MUUII #3

MUUII – MIIU #1

Looks good! (I think in the last step you use both #4 and #1 ?)

Goal Word: MIIUIIU

I enjoyed this assignment because I like solving puzzles. But I’m admittedly bad at them. I think connects to our work in class because we need to use several steps to obtain our answers. It’s kind of like using facts to solve proofs. At one point, I tried to work backwards when attempting to solve a few of the puzzles here, which is similar to proving by contrapositive. Also, these puzzles aren’t as easy as some of them look. I kept running into dead ends, and then I’d have to backtrack to where I’d guess that I did something wrong.

-Darnell James.

MUIUIIUUI

This exercise was a bit tricky for me, i had to have the rules written down in front of me while working on this exercise. I believe this exercise is similar to doing proofs, because like proofs we must know the rules and follow those rules. There are certain steps in which we must take to get to the goal work, which is very similar to the way we proved things in class.

Solution to Renauthas Goal Word.

Goal Word: MUIUIIUUI

Solution:

MI

MII (rule 2)

MIIII (Rule 2)

MIIIIIIII (Rule 2)

MIIIIIIIIIIIIIIII (Rule 2)

MUIIIIIIIIIIIII (Rule 3)

MUIUIIIIIIIII (Rule 3)

MUIUIIUIIII (Rule 3)

MUIUIIUUI (Rule 3)

Goal Word: MUIIUUIIUI

I did enjoy this assignment because I enjoy solving puzzles in general. The reason why I think this relates to the stuff we are doing in class is because like proofs, we are always substituting stuff in our proofs given that we have some rules to work with that restrict us from doing other sorts of nonsense.