OpenLab #9: Proof Journal

Your assignment for the next week is to try to prove the conjecture that your group created in class on Thursday, 11/6/14.  You must spend at least 90 minutes working on this (including 30 minutes in class on Tuesday, 11/11).  Trying to prove something can consist of many different activities, such as the following (you do NOT have to do all of these things – you can choose how to spend your time – they are provided for inspiration only).

  • coming up with ideas, and testing them out (for example, by creating puzzles and trying to solve them)
  • trying to understand what the conjecture says
  • trying to solve puzzles that other people created
  • trying to create puzzles (and solve them yourself)
  • communicating with other members of your group (talking, emailing, etc.)
  • trying to write down a proof
  • other stuff…

As you work, keep track of what you are doing, thinking, and feeling (this is metacognition – an idea that discussed way back in OpenLab #2).  What did you do during the time you spent?  Did you create any puzzles?  Did you solve puzzles?  Did you change your mind about whether the conjecture is true or false?   Did you have any new ideas about how to prove the conjecture?  Did you have any ideas that you gave up on?  How did you feel as you worked – were you frustrated/confused/happy/depressed? Why? Did your mood change along the way?

Assignment (Due Thursday, 11/13/14):  Submit a journal of your efforts in the comments below.  Your response should be at least 300 words.  Describe what you did during the 90 minutes you worked, and express in some way what you were thinking and feeling during the process.  Your response can include puzzles or other work you did along the way.

27 thoughts on “OpenLab #9: Proof Journal

  1. The conjecture that my group had is- if it is possible to fit domino pieces onto the checkerboard if two diagonal square pieces are removed. To figure this conjecture out, I did a series of test to prove whether this statement is true or false. As I was working on my conjecture I first thought about how many ways you can remove two squares diagonal. I saw that there are many ways to remove the squares diagonally. Then I thought to myself if removing the squares had to be diagonally together or separated.
    So I first tried out an example for when the two squares in the checkerboard are removed together diagonally. I removed squares (3,2) and (4,3), which are two white squares. As I was testing this out I saw a way that was easy to tell if the domino can fit into the checkerboard or not. I saw that instead of trying to solve for the whole 8×8 checkerboard to surround the pieces that was missing and solve from there. As I placed the dominoes I saw that there was one square piece left over, leaving me to conclude that it can not work for those squares removed. But I was still unsure because there are many other ways to test so, instead I did the long and hard way and tried making a 3×3, 4×4, 5×5, 6x, 7×7, and 8×8 checker board. I saw a pattern as I was checking with different size checkerboards for the even checkerboard such as 8×8, 6×6,4×4 those had two pieces left over, however the domino could not cover it because those pieces were diagonal to each other, and the rule states that you are only allowed to place the domino vertically and horizontally. Then for the 3×3, 5×5, and 7×7 I saw that there is one piece left over and the domino cannot cover this because a domino has two pieces to cover. So then I realized that the old checkerboard always leaves you having one square piece, and the even checkerboard leaves you having two pieces that are diagonal from each other left over. I was going do the same thing but removing two black pieces that are diagonal together but then I realized that there is going to be the same result as removing the two white pieces.
    After this I came to a conclusion that it is not possible for the domino to cover a checkerboard when two pieces that are diagonal together are removed. After establishing this result I felt a bit relief and happy because I solved part of a puzzle piece but then I realized that that was only half of my example to test. The other half to test was if the checkerboard removed two separated diagonal squares can be covered with dominos. This theory was a bit challenging because there was so many examples I can do. So I went and asked myself a set of possible questions. I saw that that checkerboard is an 8×8 square and I saw that there are odd and even number squares, and that there are black and white squares.
    So I asked my self if I take out two odd number squares that are separated and diagonal if it can be covered with dominos, I asked the same question for the even number squares. Then I asked the same question for an even and odd number square and for an odd and even number square. These were four theories that I had to test out using odd and even number squares. The second part was the color. I asked myself if two white separated diagonal squares can be covered with dominos, the same goes for two black squares, a white and black square and a black and white square. So out of two main topics which dealt with the color and the number. I had to test out 8 theories and that was not easy. My brain was about to explode testing these theories out.
    At first it was fun but as I had to go through more examples I saw that this was becoming annoying. I felt like I was going to be those type of mathematicians who stays in their attic trying to prove a problem where it takes them years to solve until they finally come to a conclusion. I almost thought I was about to grow gray hair.
    I wanted to be lazy but I decided to work on the problem until I had given up or until I went crazy. So for my first test I removed two odd squares which are (3,3) and (5,7) as I removed the squares I saw that the colors removed were both black. Then I realized that if I took any odd number squares they will be color black. I also tried this with even squares. I saw that with two even number squares removed, the pieces removed are black also. Then after realizing this I tested out if the squares (3,3) and (5,7) were removed if the domino would fit. To my conclusion the domino did not fit because the two pieces that were left over were diagonal and as I mentioned earlier, dominos are only suppose to be place horizontal and vertical. Finding this out was awesome because I knew that if I did that with the even numbers the domino wouldn’t fit into the checker board also because the even number squares were black as well.
    Making many puzzle and actually testing the puzzles was tedious work, I was getting frustrated at first but as I was slowly finding a connection and a pattern I knew I was close to finish. As I tested out removing an even number and odd number squares that are diagonal and separated, I saw that both colors removed was white and as I was experimenting I saw that the dominos was not able to fit into the checkerboard.
    Then I realized that I didn’t have to go through all this crazy hypothesis and test because I saw an easier way to check if dominos were able to fit into the checker board or not and that was by looking at the pieces removed then making a square from those pieces and you can easily find the solution.
    Another important discovery is that if you remove two of the same color whether it is two black or two white pieces then the domino will not cover because if you look at the checker board you will see that the checker board is black, white, black, white, black,…. When you have a domino you already know that domino can only cover two square pieces, and when you place one domino on the square pieces you will see that the domino is meant to cover a black and a white piece. If you remove two of the same color square pieces the domino will not fit because there will be an extra two piece of the same color that would not fit the domino, because the domino is only meant to cover a black and white piece.
    Finding this discovery and knowing the reason why was an awesome feeling. However I did feel like I was thinking too hard and did sooo much work for nothing.

    1. Julia, you did a great job of keeping track of your own thought process and method of working. I love that you proceeded from one idea, to many, and that you tried testing lots of different puzzles. I must disagree with your final comment ” I did feel like I was thinking too hard and did sooo much work for nothing.” — the insight that you came to in the previous paragraph (about black vs white squares, and the fact that each domino covers one of each) is just great, and well worth the time and effort you spent. This is how math goes!

    1. There is a 300 word minimun, but no maximum – you should use as much space as you need to respond to the assignment. Unlike (for example) a 5-paragraph essay, a prompt like this one will generate responses of widely varying length, no worries as long you meet the minimum.

  2. The conjecture that my group had was “suppose u = 3I and n the times rule 2 is applied, if a goal word has 2^n I’s then it can be solved in n + 1 turns.” For the first ten minutes I tried to understand what the conjecture is trying to say, I understood what 2^n I’s was trying to say but what got me confused of the u=3i and that goal word can be obtained using n + 1. After the 20 minutes or so I went back and saw the other goal words posted by groups members and tried to make a connection with the conjecture that I am trying to prove. I couldn’t seem to apply the rules when it came to u = 3i. I tried to create my own goal words and see if I could apply this conjecture to it but every time I tried to prove my goal words I would get stuck trying to prove as how u = 3i and if I couldn’t figure this part of the conjecture I couldn’t figure out the last part of the conjecture n + 1.
    Working on this conjecture caused me to get frustrated because I spent hours trying to understand where u = 3i came from. As I was working on the conjecture I came to a conclusion that I cannot solve this conjecture on my own that I needed to work with someone else to see if I were to together with someone else maybe it will all click. I called my brothers to see if they give a hand in trying to decided where can we get u = 3i. I explained the rules to them. We spent hours and hours trying to state whether or not this conjecture is true or false. After spending countless hours and having frustration we said that this conjecture that my group and I have come up with is true because we can get any number for n (which n equals to the number of I’s using rule 2) to have an even amount so that it can be divisible by 3 if we use n+1.

    1. In many great math problems, a lot of time is spent trying to clarify “what are we trying to prove?”, and you did just that, trying to understand exactly what your conjecture was trying to say. Great job, and appreciate your persistence. That’s half the battle (or more!). I love that you enlisted the help of your brothers.
      I’m also not entirely sure I understand the details of your group’s conjecture, though I do think you are on to something.

  3. The conjecture my group came up with was: If you remove a tile and any tile diagonal from the first tile removed than its impossible to complete the puzzle.
    So, the first thing I did was test this conjecture. I started by removing the tile (2,2) and than I removed the “closest diagonal tile,” (3,3). When I did this, the puzzle was not possible. After this, I decide to remove a center tile, (4,5), then I removed a “far diagonal tile,” (2,2). When I did this it sill wasn’t possible to complete the puzzle.
    After this, I was stuck I didn’t know what to do; so I reviewed the sample puzzles above and I noticed that I was just removing tiles of the colors. Next, I created three more puzzle. The first puzzle, I remove 2 same colored tiles in the same row (tiles removed: (3,2), (3,4)). The second puzzle , I remove two same colored tiles in the same column (tiles removed: (4,2), (4,8)). The final puzzle, I removed tiles from different rows and columns (tiles removed: (2,2), (8,6)). For these three puzzles it was unsolvable
    From there, I felt that I needed to checked if its impossible to solve the puzzle if I removed tiles of different colors. So again I created three puzzles following the same conditions for each corresponding puzzle. Puzzle 1, I removed different colored tiles: (2,5) and (7,5). Puzzle 2, I removed different colored tiles: (2,2) and (3,3). Puzzle 3, I removed different colored tiles: (2,2) and (3,6). I noticed that all three of these puzzles were solvable.
    Form there I thought about ways to prove or show if a conjecture is true so,
    Q= remove different colored tiles
    P= unsolvable
    If Q then P
    not Q= roomed same colored tiles
    not p= solvable
    If not Q the not P

    Since the contrapositive for my conjecture is true than my conjecture must be true.

    1. Thinking in terms of colors is a great move – gives an easy way to talk about the different kinds of tiles you are removing. I like the methodical way you looked at examples – very nice. Setting up your conjecture in terms of P and Q is a great way of taking advantage of our work in logic to help you in solving the puzzle (but do be careful – the contrapositive of “if Q then P” should be “if not P then not Q”).

  4. The conjecture that my group had was that if two squares of the same colour were removed then the dominoes weren’t able to fit in and another one was if the two squares removed are of one black and one white then the dominoes were able to cover the mutilated checkerboard. Even before I was hands-on experimenting with the checkerboard I was having the intuition that the dominoes are able to cover 62 squares no matter what square was removed because 2 can go into 62 evenly. I knew I was trying to be a genius when my guesswork turned out to be so wrong. Anyway my logic i.e. “2 can go into 62” seemed to work when the top right and bottom right corners were removed but didn’t work when the opposite corners were removed. I wasn’t amazed because it was a puzzle after all and I am not that clever to solve it in an instant. For a while I was only working with these two examples. The first example worked perfectly having the dominoes align vertically and sometimes horizontally. The second one i.e. when the two opposite ends were removed was the one I was having trouble with, somehow I tried to fit 60 squares with the dominoes but the remaining two squares were not directly next to each other which was impossible to fit because as for the conditions the dominoes could only either be aligned vertically or horizontally and not diagonally. If not for this condition the puzzle would be unchallenging.
    So far I think the conjecture is true because in the first example I was able to cover the checkerboard with the dominoes because the squares that were removed were black and white and the second example was impossible because the squares were both white. This conjecture to me was strong because it worked exactly as that. If I removed any two squares that were black and white it worked and I’m also not sure if this is the best conjecture because there was no math involved in it.

    1. What struck me in reading your proof journal was the way you used your intuition to generate ideas, and then focussed on specific examples to test them. Very much a mathematician’s process! I also like that you had two examples which you spent a lot of time thinking about – sometimes that can be more useful than looking at many examples briefly. I do disagree with your final comment “there was no math involved in it” – there’s a lot of math there, just no numbers or equations!

  5. Throughout working with my group, they have come up with a conjecture that was “suppose u=3I, N=the times rule 2 is applied. IF a goal word has 2^N I’S then it can be solved n+1 turns”. By looking at this conjecture in the first time, I felt that this conjecture will be challenging and a bit hard to prove. So, the first thing I did was trying to understand what the conjecture means, since it looked complicated for me. I spent most of my time trying to do examples following this formula to understand it if it’s true or false or even if I understood it the way it should be. After reading this conjecture many times, I understood some parts of it and the others I did not get at all, the most challenging part for me was if the word has 2^N I’S then it can be solved n+1 turns , which means the main, and most important part of the conjecture . Then I created many goal words and tried to solve them using this conjecture to see if I can make a connection between them. But still I did not understand the conjecture fully because there is part that I was not able to figure out, so it did not work well and I got stuck in proving it. Not understanding things fully could confuse a lot because it got me stuck in the middle of my proving. For me this conjecture is really challenging to prove. I have spent almost three hours trying to figure it out so I can come up with my prove but it did not work out with me. During working with this conjecture I felt really confused, sad, depressed that I was not able to prove it. I tried many times but it did not work out with me. At the end, I did not want to give up on it, because I really want to know how I can prove this conjecture. So I think I will have to ask my group mates if anyone was able to prove it to help me out or ask my partner that created it to help me understand it. But at the end, even though I got really depressed that I wasn’t able to do it. I just liked that this assignment made me think very hard for hours trying to come up with a prove.

    1. Marina, thank you for this post – your open self-reflection is awesome. Your description of what this experience was like reminded me of many instances in my own mathematical work – trying over and over again to understand something, and not making progress, is one of the most frustrating & depressing experiences, and I have been through it more than I’d like. Figuring out strategies to deal with this is essential – I like that you end with a comment about asking your group mates for help, as discussions with other can often lead to insight. And your final comment rings true – anytime you can think very hard for hours, I believe you are accomplishing something, even if you don’t find the solution that you seek. Great!

  6. My group’s conjecture for the MIU Game:
    Suppose U=3I and n= number of times rule 2 is applied. If a goal word has 2^n I’s, then it can be solved in n+1 turns.
    Our consensus was that it must be true because rule 3 states we can instantly replace any three consecutive I with U and because rule 2 doubles any string afterward.
    Since my group didn’t have a chance to convene on Tuesday, we were tasked with describing our thoughts independently from each other. As of this moment, I do not believe any members of our group have diverged from the consensus and I hope they don’t before this project is done.
    Upon first thought, I believed that this conjecture could be proved directly, but something was gnawing at me because it didn’t seem like it would completely prove the conjecture or there would be no simple method to set up a direct proof. Then we begun learning proofs by induction as of last week. The proverbial light bulb turns on in my head and I realize that if we can prove the first case, we can attempt to prove each next case. Hooray! A concept we can build on, I hope.
    In the first case however, it can be solved by applying Rule 1 in the end as opposed to Rule 3.
    So how will we go about proving this?
    I’m not really certain how to do this without algebra at this point. It just seems like it’s true.
    I don’t know what to feel at this point. I think I answered one question, but then more just reveal themselves. It’s both a feeling of achievement and trepidation. Where do we go from here? I guess I have to think some more.
    I hope one of my colleagues can make something out of this idea so we can really start.

    1. Great! I’m glad that you made a connection between our work on induction and your puzzle – I’ll be curious to see if something can be made of it. You should also consider (if you haven’t done so already) creating a few puzzles that fit your conjecture, and show that you can solve them. Very nice!
      (I’m not sure I entirely understand your conjecture, however – in particular, where does the “U” factor into the conclusion?)

  7. The conjecture from my group was suppose u=3i and n= number of times rule 2 is applied, if goal word has 2^n i’s then it can be solve n+1 turns”. At first I tried to figure out if the conjecture is true or false. During our group work in class, when we were making the lists of all our conjectures looking at this conjecture made me felt that this conjecture will be quite challenging to solve. So I first tried to understand the exact meaning of the conjecture chosen during our group work and for which I kept on reading the statement for few times. Even though I could not understand the last part of the conjecture which is “it can be solved n+1 turns” but I tried to solve the conjecture by starting with MI. Then I tried further with some examples by putting different numbers for n for an hour, but still felt like it is not working. It seemed confusing at that moment and I kept staring at the conjecture for couple of minutes. Then I thought I should solve the conjecture using direct or contrapositive proof but I gave up because I thought it will make this more complicated than it already is. I was hoping at that moment that during our group work in class if we would had more time to discuss about the conjecture then I might understand it properly. And then I would not be as frustrated as was at that moment. At that point I did not know what to do so I almost gave up and I even got a headache of thinking so hard about this conjecture, so I tried again after few hours later to solve this conjecture again but still the same problem I was getting stuck and had so many questions in mind. I feel like this conjecture is true even though I could not solve it and also anxious to see how my group members solve this conjecture.

      1. Great question – the instructions don’t specify (in fact, they don’t mention the number of turns at all), so it’s up to you to decide whether you want to make the Axiomatic Word MI be the “first turn” (n=1), or make the step from MI to whatever is next be the first turn. Just be sure to make your decision explicit (tell me what you decided).

    1. I like that you moved between looking at examples, to thinking about proof techniques – this is a very typical pattern when working on a mathematical problem. And your struggles to try to understand the conjecture also resonate for me. I hope you’re able to get together with your group and exchange ideas further!

  8. My groups puzzle was the checkerboard puzzle. The conjecture that we decided to work on as a group was if the pieces that are missing are diagonal from each other , can the puzzle be solved? First we needed to understand if the diagonal pieces had to be next to each other or they could be anywhere on board as long as they were diagonal. We decided that they could to anywhere. Just taking a guess I would think that it is possible. I tried it on my own first with the pieces next to each other. I took out (4,4) and (5,3) which were black squares. After completing the puzzle two squares left that I could not fill with the domino because the either diagonal from each other which doesn’t work , because the dominoes can only be horizontal or vertical or they were too far apart to cover . I then tried (1,6) and (2,5) which were white squares. After completing that one there were also two left that I could not fill. I also notice that the two parts that I couldn’t fill were of pieces opposite color, which I thought was interesting. Now to try if they are not next to each other but still diagonal. I took (1,2) and (5,6) which were white squares. I tried to cover the board with the dominoes and was left with one piece missing. Which was even more interesting because I was still expecting to have two missing pieces that I could not fill. I also notice that the part that I couldn’t fill was of the opposite color too. This is also true if the missing diagonal pieces are black. I was surprised to see that this conjecture did not work because if an even number of pieces is missing then you should be able to fill the checkerboard with the dominoes minus the missing pieces with no problem.

Leave a Reply

Your email address will not be published. Required fields are marked *