This assignment is a continuation of OpenLab #3, on the MIU puzzle. Ā Your assignment this time is a little different. Ā Consider the following statementĀ about the MIU puzzle:

Statement.Ā Anygoal word (any word with first letter M, followed by a combination of U’s and I’s) can be obtained from the starting word MI using the rules of the MIU puzzle.

**Assignment (Due FridayĀ 9/25 – Because of the nature of this project, please do not post your answer beforeĀ Wednesday 9/23)**. Do you think the statementĀ above is true or false? Ā I would like you to think about it and see if you can convince yourself of the answer. Ā Try out different things! Ā If one idea doesn’t go anywhere, see if you can come up with a different approach. Ā Ā **This is an assignment about metacognition as well as puzzle-solving – pay attention to your thoughts and ideas, even (especially) the ones that don’t work out!**

In the assignment below, you will be writing about your experience working on the MIU puzzle above. Ā Be sure to respond to EACH PART – detailed answers to part 2 and 3 below will gain the MOST credit.

- Write down what youĀ
**think**the answer is (is the statement TRUE or FALSE?). - Describe in as much detail as you can the process of working on the assignment. Ā What did you think in the beginning? What are the different things you tried in order to solve the problem? Ā Describe
**each one, in the order that you tried them.** - Give a detailed list of the resources you used, and how you used each one (these could include anything – your brain, pen & paper, a computer (what applications did you use), the internet, other people, and so on).
- What activity or resource do you feel was most effective for you in working on the assignment (what helped you the most in understanding the assignment and figuring out an answer)?
- If you believe the statement is TRUE, post an example of a goal word that youĀ encountered in your work that you think was most difficult to solve. Ā If you believe the statement is FALSE, post an example of a goal word that you believe is impossible to obtain by starting with the axiomatic word MI and following the rules.

**Extra Credit.** Ā Respond to one of your classmates’ submissions.

1) True

2) to work on the assignment I read all of the rules and made sure

I understood the many ways each could be used in different orders to get the same combination. in the beginning I thought that some combinations of U and I would be impossible to get but that was before I actually knew all of the rules perfectly. while i was solving a problem I would forget one of the rules which would have helped out the situation. one of the ways I thought would help me solve any problem was doubling any combination after the M. at first i would use the rule to put a U after the I then i would have MIU and double the IU but that did not help. my next choice was to just double the “I” many times and each time it would get 2 times larger than the original. after doing that I used the rule to choose any 3 “I’s” and make it a U. I did that where ever i needed i could use the rule to take 2 U’s close to each other and cancel them out. I think using that method you can solve any combination.

3) I used pen and paper to solve all of my combinations. the reason is because with the method i stated in number 2 eventually you have too many I’s to count in your head and even on paper that many similar lines close to each other may make you miss count.

4) the thing that helped me understand the rules better was in the beginning where there were a few easy combinations the professor asked us to solve also i understood even better after i tried creating my own combination.

5)MIUIUIIIUU

1. I believe the statement above is true in general BUT I believe it is false. A puzzle can be difficult to figure out but with time and patience it is possible to get to

the goal word started with MI. But I have no patience in figuring out ones patterns so it may not be the case for everyone. I think

it’s even possible to try for a whole day straight and still not get it. Using a different approach can possibly lead one to the

solution but other than failing multiple times I can’t come to grips with not being able to solve one.

2. Working on this assignment was a challenge. I would look at the other persons puzzle and then start from scratch on my own with MI

on a blank peice of paper. I apply certain rules that seem appropriate to get the letters exactly where I want them. The easier

part was the pairs that repeat becuase it was the easiest rule to follow. Even with trial and error it would still be difficult

there is only just many combination of steps you can try which I believe made it almost impossible. Once I couldn’t figure out the pair

I would work backwards. Redoing a combination of steps in order to go BACK to the beginning. That helped ever so slightly more.

3. In order to solve the problem I used scrap paper and a pen. I also looked at other solutions for different problems to see if it

was possible that some used the same steps in order; possibly changing one step in the end or something similar.

4. The most effective resource was the explanation and examples the Professor gave us in order to understand the game.

5. MUUIUIUUIUI

1) True

2) In the starting I thought this puzzle game was going to be easy but then as I went through the game it became quite complex. To be honest, I still don’t quite understand the rules of this games completely, but I do understand the basic guidelines. To me coming up with my goal word was a challenge. But I did it how ever by nibbling on my loosleaf and constantly racking my brain. I imagined different roads I can take when trying to establish my goal word. I kept getting lost, but eventually I realized that if I find this challenging, well that is the whole point for the assignment, to come up with a challenging word. So, I came up with a word in my head that I found quite confusing then evaluated on my own. That made me realize the complexity of a game that seems so simple. Hence, applying each rule gave me a better understanding of how the game was to prolong.

3) The tools I used for this puzzle game were as simple as my brain, sheet of loose leaf and a pencil. Going over the words, rules, crossing out ideas and coming up with a better understanding was the abuse my poor sheet of paper obtained.

4) What helped me the most in understanding the assignment was non other than the reference to the rules them self. I used each rule in my goal word I tried to keep it as simple as that, at first. But then I switched it up a bit to get a better hand at it.

5) The goal word I encountered that was most difficult was MIUUUIIIIUIII. It puzzles me how complex the word is since one thing is clear, there isn’t a repetitive pattern. So pretty much every rule is applied, atleast I think so.

1) True

2) Through repetition of “I”s in base 2 this can give us an infinite amount of “I”s. Then, any three consecutive “I”s can be transformed into a “U”. This can result in any combination of “U”s and “I”s. Furthermore, any two consecutive “U”s can be removed. So, any 3″I” may be transformed into a U, so we will let 3 “I”s equal one “U”. Any 6″I” may be removed.

Let x represent the amount of duplications, y represent all “III”s transformed to a “U”, z represent all “UU”s removed, and w represent the U added at the end. All numbers exist in the range of integers, with 0<=w<=1. To represent some MIU puzzles in terms of "I", this gives us:

F(w,x,y,z,)=2^x-(3y+6z)+3w

This equation represents the operations done only when duplication is done at the beginning. However, duplications can be done at any time, so to fully represent any MIU puzzle in terms of "I" this will yield the recursive function:

F(a,b,c,d)=(2^b)F(w,x,y,z)-(3c+6d)+3a.

This equation yields any integer ranging from negative infinity to positive infinity. With this I believe we can reach any combination of the MIU puzzle.

3) I just solved these puzzles in my head.

4) The puzzles were easy enough to understand. I did not need any additional resources to understand the puzzle.

5)MUIIIIUUIIIIU

Wow! May I ask if you read any special textbook about math? I feel strongly inside to learn from you. When you feel challenged, what do you usually do? Everything comes from your mind? I do not know if I asked the questions in a proper way, but I am just so curious. Please do not mind. š

1- True

2-To construct a goal word sounds easy, even reading the guidelines and the rules sounds easy to me . The hard part for me was to break the ice after the first MI I mean knowing where to go from them, then after read a couple examples I just played with the rules, I just double the I’s after each M and add U after I and when my puzzle reach a form where there is UU I get rid of it, each time I want to double the word after M; I always see if that can lead to a double UU because that actually will allow me to make change to the puzzle by eliminating UU. the puzzle is unstoppable in my opinion you can continue for as long as we want and that makes the statement true .

3- for this puzzle I used my hand and my phone.

4- I just log in open lab in my phone, read the rules , looked at the example we had, solved 2 of them until i got the concept in my head and started applying them one after another to construct my goal word. but if I the professor didn’t gave us those examples it would of been harder for me.

5-MUIIUUIIUI

1) I believe the statement is true

2) I’ve came up with many examples and I could solve them all

3) I used pen and paper. I wrote down some of the examples I came up with and try to solve them and I was able to easily.

4) I think the rules helped me a lot. By reading the rules, I can understand how to do it.

5) I don’t think there is any word that is difficult. It just takes some time to figure it out. No matter how long a word is, you can eventually solve it if you have patience.

I think the statement is false, but also I partially agree the statement is true since every person comes up with different puzzles by using the same set of rules. It is confusing because how it is that possible to check all the puzzles and some of them we havenāt seen yet ?As a puzzle was created by people by the MIU rules which means that there must be a solution corresponds to it. At the beginning, I felt very challenged and anxious about this series of questions followed by the puzzle. It seemed that I can do nothing with it, but I felt better when I followed the MIU puzzle rules. I found it interesting after I created my own puzzle and let classmates to solve.

There was one thing that I figured out when I made a mistake on my puzzle. I corrected it because I could not solve the one I made mistake on. However, Fuzail solved the one I made mistake. This example kind of proved the statement. One person cannot solve the puzzle which does not mean nobody can do it.

After I created my own puzzle, I also started working on a puzzle that no one worked out yet. I found Yuyu Chenās puzzle āMIUIIUIIUā that is kind of impossible for me. I used my pen, several scrap paper, and of course my āslow-wittedā brain to start my journal of solving it. It took me hours to think about the puzzle; I wondered how come a person can come up with a puzzle, but it is so hard to get it solved. I asked my friend from another school and wished he could help me out. Unfortunately, he could not get it. I thought about the Internet which is always my last option I could use for any problem I do not understand or solve.

I felt so happy when I saw the first sentence in Wikipedia which says āThe puzzle’s solution is no. It is impossible to change the stringĀ MIĀ intoĀ MUĀ by repeatedly applying the given rules.ā In other words, if a final puzzle starts with āMUā then there will be no solutions for it. However, this cannot prove that Yuyu Chenās puzzle cannot be solved because the first two letters are āMIā instead ofāMUā. I read more about the explanation from the website. It is such an amazing thing that a small puzzle contains so much knowledge. It talks about āinvariant propertyā which is mainly about the number of Is in the puzzle and see if it is divisible by 3. It makes much sense to me because when I solved the puzzle I always count the number of Is and to check which rule I should use.

This assignment made me think of the Division Algorithm from my Number Theory class. Right now, I could not say if there is any relationship because that is just my overall understanding while reading the rules. Actually, for my conclusion, I would like to say I just could not solve the puzzle. I prefer to use Yuyu Chenās puzzle āMIUIIUIIUā as my impossible puzzle. The following are the rules I used: MI (R1), MIIIIIIII(3*R3), MIUIIII(R4), MIUIIIIIUIIII(R3), MIUIIUUUI(R4), MIUIIUUUIIUIIUUUI(R3), MIUIIUIIUIIUI(R5). This is the result I have at the end which has āIIUIā as the extra compared to Yuyu Chenās goal word. Maybe I am wrong, and I hope anyone can help me with this puzzle and convince me that this original statement should be true instead of wrong.

I actually agree with you Mei, I attempted to solve the MU puzzle on my own but had no success on it since I couldn’t get the I out of the puzzle. Since I couldn’t solve it I concluded that it was unsolvable but am a little disappointed that there was no solution after spending time working on it.

1. True

2. At first, I looked at the rules chronologically which caused me to go through steps 1 – 4 in that order to try to create the goal word. Basically, I was looking at each individual letter and trying to figure out how to make it so. However, I wasn’t able to draw the connections between each rule. I was looking at them statically instead of dynamically. What was a major breakthrough was when I realized that any goal word can be converted into a the starting letter M and a number of I’s because U’s can be expressed as I’s. Then the goal became not to get U’s and I’s but just the correct number of I’s which can then be converted into U’s later. With this in mind, the goal was split into subgoals which made things a lot easier.

3. I only used a paper, pen and computer . I like mental math but sometimes my mind becomes disarray which is what this puzzle did to me. I wanted the rules to really settle into my mind and I wrote them down by hand even though they were available on my computer screen. I also used some space to experiment with word generation using the rules.

4. I think the paper and pen are really the most effective. I’m more of an old school learner and need to sort out my thoughts which helped me out the most. It was when I was writing out the word goal steps that I came to realize the relationship behind the rules.

5. MUIUIIUIIUII

1. I believe that is false, not all the goal word could be obtained.

2. I did read all the rules carefully and make sure I don’t make any mistake. At first, I think the statement should be true, because you can change I to u or double the strings. you have many ways to add I and U or get rids of it. it seems to be you could any words you like. However, I realize that if we continually double the Iās. we always get even number, so if our goal is odd number. it will be hard to reach it.

3. I don’t use internet to solve the problem, because I want to rely myself. if I use goolge, it is cheating and this game would be be interesting anymore. I only use pen, paper and my brain.

4. I think the most effective way to reach the goal word is using the pen and paper doing more practice, using difference rules to see if that works or not during the challenge.

5. The reason I’m thinking it is false because the word MU. I did try many ways to reach this word but it is unsuccessful

1. I believe that it is false for any word goals to be solved with the rules given. In the beginning of this assignment, there was some confusion as to knowing the rules. There were times I thought I could go from one step to another in my mind but then writing it out on a piece of paper made a difference of understanding the rules better. I also had the help of my handy calculator. At first I also believed that there are probably solutions to all the problems even if the word goal is tedious. The rules seem concrete enough to create a variety of different solutions but it didnāt seem impossible to have a solution to all word goals. I feel that simply a piece of paper and pencil and a calculator were the best instrument that could be used. It allowed for visualization and understanding. At first all I did was try and solve the problems mentally but then went to using paper and a pencil and noticed there were mistakes I had made mentally. There were times I would have to alter the steps in between working on the puzzle and go back to step four when I could have been on step eight. I attempted solving the Goal word of MU but noticed that there was always an āIā at the end that I couldnāt get rid of. In order to have no Iās means that you would have to replace them all with Uās or find a way to maneuver your way around it. Rule 4 states that the only way you can get rid of any Iās is by replacing three of them for one U. On a piece of paper I tried doubling the Iās and grouping the is in groups of three but couldnāt get a result. After using the paper and pencil, I went to my calculator and found the doubles that could occur. I notice a pattern in the doubles and that was that they all ended with 2,4,8,6 in this order and repeated as it continued. As a result, none of these numbers were divisible by three which leads me to conclude that it is not possible to get rid of all the Iās in this puzzle.

1. I believe the statement āAny goal word (any word with first letter M, followed by a combination of Uās and Iās) can be obtained from the starting word MI using the rules of the MIU puzzleā is false, because mathematics has certain situation where thing can work or not work.

2. At first I thought as true, because the rules of MIU puzzle it like circle that self-correct itself. There has be like this break point, because mathematics has certain situation where thing can work or not work. I grab a piece of paper. I wrote down all MI puzzles I can come up with that was outrageously long or just plain ending with U. An example MI puzzle I thought up was āMIIUIUā they limit what actions you take base on rules, which limit to double any string after the M. This force the puzzle to expand. Since rules āIII become U and UU can be remove is not present or after I can add U to endā this force This limit the rules you can take by great deal forcing person use rule for double string after M and rule for āremove UU and III become Uā cannot be performed. The idea try double string MIIUIU -> MIIUIUIIUIU, so retreat the last letter before will end in become start U then new string IIU and no way remove any letters.

3. I was unsure about the rules, because I thought at first you can go forward with rules and backward with them. The truth you can only go forward which limits a lot possible combination. Which simply the confuse I had, then I figure what rules has limits on itself that would be U ending and IIU begin after M. This limit the rules you can take by great deal forcing person use rule for double string after M and rule for āremove UU and III become U or after I a U ruleā cannot be performed.

4. I think was condition p -> g and negation was most helpful, because if one fall over then other fall over in sense.

5. MIIUIU

1. I think the statement is true because we can made infinite steps suing the five riles for the puzzle and in some point it can be solve.

2. At first, I thought the assignment would be fun but it was tricky. Specially the questions that was given by Prof. Reitz for the extra credit: MU and MIIIU. I tried to solve the problem, but finally gave up. But, I strongly believe that their is way to solve those puzzle.

3. I tried to solve the puzzles problem by myself. I read through the comment, followed the steps and tried to be independent worker.

4. Using pen and paper and also thinking deeply, was part of the assignment that I had to do. I had rethink and find my mistakes and redo the problem again.

5. MUUUIUI