Your assignment for the coming week+ is to try to prove the conjecture that your group created in class on Tuesday, 10/21 (10/21 working space (Google Doc) with group conjectures is here).Â You may need to refine/expand your conjecture first (letâ€™s discuss this in class).Â Â **You must spend at least 90 minutesÂ working on this.**Â Â Trying to prove somethingÂ can consist of many different activities,Â such as the following (you do NOT have to do all of these things â€“ you can choose how to spend your time â€“ 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 stuffâ€¦

As you work, keep track of what you are doing, thinking, and feeling (this is **metacognition** – an important idea in education!). 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 â€“ were you frustrated/confused/happy/depressed? Why? Did your mood change along the way?

**Assignment (Due Tuesday, 11/9/21)**: Â Submit a journal of your efforts in the comments below. Â Your response should beÂ **at least 300 words**. Â DescribeÂ what you did during the 90 minutesÂ you worked, andÂ express in some way what you were thinking and feeling during the process. Â Your response can include puzzles (useÂ sketchtoy.com) or other work you did along the way.

**Extra Credit**. Respond to a fellow studentâ€™s comment. Did you do similar things? Different things? Do you have any suggestions for them? Be kind.

My groupâ€™s conjecture is, â€śYou have to start on the edges or the perimeter.â€ť So, in order to prove this, I started thinking about the various puzzles we were given, particularly from the assignment/post on 9/29. To begin with, I sketched out 3 of them from Example 4 and worked through them applying our conjecture, and also not – so by beginning at a node in the center. Something I realized early on is that our conjecture goes without saying if there is a walking tour that only has nodes on the edge/perimeter. For example, what if the provided graph only consists of edges – does that automatically mean itâ€™s solvable? While I didnâ€™t arrive at an answer, I did find that repeated attempts and a strategy of 1) start on the edge, then 2) donâ€™t start on the edge for each individual walking tour graph provided a clear system for trying to confirm our groupâ€™s conjecture. It led me from one question to another. From the question above concerning what if a walking tour graph only has edges, to does the number of line segments play a role, to, what is really the determining factor?

When I worked on graphs 6,7,8 from Example 4 on 9/29, I found I was unable to find solution when beginning from the center. Starting on the edge for all of them worked. Thus I started feeling confident that our groupâ€™s conjecture was accurate! I also began wondering what would a walking tour look like if it had a different number of line segments. So for graphs 6,7,8 – these three had an even number of line segments (assuming I counted correctly). When I attempted my own walking tours from the sketches I did on SketchToy, I found solutions when I started on the edge and ALSO when I started at the center! This finding completely discredited our conjecture then, as I originally tried it this way thinking it wouldnâ€™t work and thought in addition to starting on the edge, the number of line segments or â€śbridgesâ€ť crossed had to be an even number. Again, not true in the case of my sketches.

As a result, I can say our groupâ€™s conjecture is false since I was able to complete walking tours (not crossing a bridge more than once) by not starting on the edge. I revisited other conjectures we proposed and the one I originally suggested also does not work (Begin at a node where more than two line segments meet). While I can continue working through this, I donâ€™t feel particularly frustrated because I see this question as more a puzzle thatâ€™s fun to solve and might have more than one right answer. Perhaps if I were under a time limit Iâ€™d feel differently, but I didnâ€™t feel frustrated, only excited to determine if our conjecture could be shown to be true or not. I think with each new confirmation – that something did or did not work – it only encouraged me to continue working. The frustrating part would be finding out that there is no true conjecture to solving these walking tours after attempting to try to find something that works đź™‚

When I first started the bridges and walking tours project, the initial graphic of the bridges made me think back to many previous days of traveling internationally (as well as in NYC) to determine the best ways to get to various points of interest. In a way, as a chocolate lover, much of my days are spent figuring out how to get to the most of my favorite bakeries and chocolate shops as possible based on parameters such as the weather, my other activities, etc. I see the bridges and walking tours project, and puzzles more generally, as an exercise to visually explore the art of the possible through trial and error of various pathways to determine if valid connections exist. Iâ€™ve found that these puzzle exercises have challenged my mind in the way I think and explore problems â€“ in a more creative way than before. Even though puzzles â€“ or puzzling questions â€“ may or may not be solvable, I believe the exercise of challenging oneself to see things differently than the status quo is important to oneâ€™s growth and contribution to society. In thinking about this latest assignment, my first thought was the relationship between puzzles and building structures, in particular the importance of the architecture and engineering to meet perfectly and establish congruence to ensure the forces on the building are balanced so that the building will not collapse. Its made me think about the tragic collapse of the residential high-rise building in Miami several months ago â€“ the question of which connections developed faults and the puzzle of using the blueprints of the building, along with piecing together photographs, first-hand accounts and other clues, to hopefully solve that question. I believe the same puzzle approach is applied in many settings, even if we donâ€™t often use the puzzle visual, including investigations, to determine whether connections between evidence and crimes can be established. In this example, its interesting because in all cases of crimes, the crime is solvable, but in some cases of crimes, the crime is not solved. I did some digging and found an interesting section on the archives of the US DOJ website called The Crisis of Cold Cases: https://www.ojp.gov/files/archives/blogs/2019/crisis-cold-cases, which states â€śexperts estimate that, based on UCR data, our nation currently has 250,000 unsolved murders, a number that increases by about 6,000 each year.â€ť These are puzzles that havenâ€™t been solved. Unlike puzzles known publicly, these unsolved crimes, once no longer being investigated, are often never worked on again due to lack of resources and new cases to investigate. However, over the past several years technological advancements such as scientific capabilities and social media have helped solve a number of these cold cases through public participation in the puzzle solving efforts. One recent article details the now oldest cold case solved through genealogy: https://www.kiro7.com/news/trending/double-murder-1956-now-oldest-cold-case-solved-through-genetic-genealogy-cops-say/NNLY2M43SFCQ3MVL5LKHMHEA6M/. This 1956 case was picked up multiple times but never solved until new tools became available. In 2018, through the arrest of Californiaâ€™s Golden State Killer, forensic genealogy was introduced to the world as a groundbreaking tool to solve cold crimes. Through public, family history genealogy databases, investigators can mine DNA data for leads and identify gene trees and branches to investigate from suspect and victim DNA profiles. The detectives working on the 1956 case were able to determine three individuals genetically linked to the DNA profile and through further testing were able to definitively link the crimes to one person (deceased) who lived mere blocks away from one of the victims. Even though it took 65 years, this newly solved puzzle brings much deserved closure as well as the promise for solving so many other cold cases through this technological advancement. Throughout my thought process on this assignment, I kept open graph theory images that Iâ€™ve recently started reading about which, besides looking like our puzzle problems, has helped me visualize spatial relations through the networks of interconnected points. I just clicked on the article attached to the image that most resonated with me, and scrolled down, and sure enough found the image of our bridges and walking tours project, so its good to know my thought process wasnâ€™t too far off

My group’s conjecture was the most optimal tour would be going along the perimeter. During the ninety minutes, we tried different variations to find the most optimal route, even if we didn’t hit every bridge.

Looking more into it on my own, I realized there’s no specific spot you can begin on the tour (whether it’s the corner, center, along any of the perimeters, etc) I realized that depending on the examples of the graph, and when drawing the graphs out separately, it depends on the route that’s taken. So in some cases going along the perimeter makes sense, whereas in others it doesn’t. That’s why the route that is used matters. I think there are many ways to approach the conjecture, but for the most part, it solely depends on which route to take. This means the conclusion to a concrete conjecture is ambiguous as there are many solutions to the puzzle.

Repost: *Forgot to add some information

On my own time, I also started to think about different parameters that can be applied to the puzzle. For example, if we were not allowed to cross the same line again, the most optimal way to have solved the puzzle is by at a corner (if we base it off the picture, that would be the bridge at the bend) and then going around the perimeter Similarly, if we start with the bridge on the middle right side, going around the perimeter would still work. But these are my observations from the image themselves.

If we look at the other graphs given, the geometry matters as to how we follow the paths, and once again the parameters for the puzzle. I noticed (especially when drawing our own) that if the path is rectangular, starting in the center and forming an x, weâ€™d hit the center each time, but wonâ€™t cross any of the lines, however, weâ€™ll hit each node (assuming the edges are the nodes). If we are allowed to cross lines and donâ€™t necessarily have to hit each node once, then there are many ways of solving the puzzle as there arenâ€™t as many parameters to follow, but the conjectures as to how the puzzle will be solved primarily depend on the geometry of the puzzle, as well as the rules necessary to follow in order to solve the puzzle.

I was not a part of any group, although at the very beginning I had been put in the breakup room with two ladies; that time I had had nothing to contribute. I have been convinced that solving puzzles does enable multiple choices to solve but under the condition that a hand will not be raised in course of the solving. In other words, the puzzle should be solved with on uninterrupted line. I did my own puzzle via http://sketchtoy.com/70247046 (a simple one). Unfortunately, I have been missing the sense, meaning, and purpose of this activity (I do not doubt that there is one but unless I see where it is going, I still will be confused). Yes, I liked it, but the purpose was far away. Yes, it gave me joyful and funny moments, as all the Proof class has been giving me, but I just do not have any grip on the outcome. So far! I had been solving the puzzles that Professor Reitz had posted when he had introduced the conjecture as well as the puzzles of my smart colleagues, too. Let me be a bit personal now: I am scheduled to go to the doctor located in Manhattan. I will be riding a bike. I have never been in Manhattan on bike, and never crossed the East River over the bridge. Since I have no bike phone holder mounted on my bike, I expect Iâ€™ will be zig-zagging to the destination in the same manner as my own puzzle has been designed on the sketchtoy.

My group’s conjecture is â€śIt’s best to start in a corner to allow flexibility in movementâ€ť. I had no idea how I was supposed to prove this. But then I started brainstorming and thought to myself as to why I believe this was true. I began to look back at the bridges and walking tours conjectures, I noticed that in order to solve it , I always chose to begin from the sides. For example, https://sketchtoy.com/70196434, I began from point E, and had three options, to go left, up or right( proves mobility). I decided to go up, and then make a right, which led me all around the route till I got to point K, then went in a route of an hourglass figure. Then I started experimenting and made another conjecture, ex: http://sketchtoy.com/70251093 . In this conjecture, I began from point G, and went left to point A then went back down B, and went by a route of triangle BFD, then went back to point B and went through other triangle routes, BGF and then FED and DCB. After this, I began thinking to myself that what if starting from the corner/ sides only works best when dealing with triangles?

Repost: *Forgot to add some information

My groupâ€™s conjecture was the most optimal tour would be going along the perimeter. During the ninety minutes, we tried different variations to find the most optimal route, even if we didnâ€™t hit every bridge.

Looking more into it on my own, I realized thereâ€™s no specific spot you can begin on the tour (whether itâ€™s the corner, center, along any of the perimeters, etc) I realized that depending on the examples of the graph, and when drawing the graphs out separately, it depends on the route thatâ€™s taken. So in some cases going along the perimeter makes sense, whereas in others it doesnâ€™t. Thatâ€™s why the route that is used matters. I think there are many ways to approach the conjecture, but for the most part, it solely depends on which route to take. This means the conclusion to a concrete conjecture is ambiguous as there are many solutions to the puzzle.

On my own time, I also started to think about different parameters that can be applied to the puzzle. For example, if we were not allowed to cross the same line again, the most optimal way to have solved the puzzle is by at a corner (if we base it off the picture, that would be the bridge at the bend) and then going around the perimeter Similarly, if we start with the bridge on the middle right side, going around the perimeter would still work. But these are my observations from the image themselves.

If we look at the other graphs given, the geometry matters as to how we follow the paths, and once again the parameters for the puzzle. I noticed (especially when drawing our own) that if the path is rectangular, starting in the center and forming an x, weâ€™d hit the center each time, but wonâ€™t cross any of the lines, however, weâ€™ll hit each node (assuming the edges are the nodes). If we are allowed to cross lines and donâ€™t necessarily have to hit each node once, then there are many ways of solving the puzzle as there arenâ€™t as many parameters to follow, but the conjectures as to how the puzzle will be solved primarily depend on the geometry of the puzzle, as well as the rules necessary to follow in order to solve the puzzle.

Repost: *Forgot to add some information

On my own time, I also started to think about different parameters that can be applied to the puzzle. For example, if we were not allowed to cross the same line again, the most optimal way to have solved the puzzle is by at a corner (if we base it off the picture, that would be the bridge at the bend) and then going around the perimeter Similarly, if we start with the bridge on the middle right side, going around the perimeter would still work. But these are my observations from the image themselves.

If we look at the other graphs given, the geometry matters as to how we follow the paths, and once again the parameters for the puzzle. I noticed (especially when drawing our own) that if the path is rectangular, starting in the center and forming an x, weâ€™d hit the center each time, but wonâ€™t cross any of the lines, however, weâ€™ll hit each node (assuming the edges are the nodes). If we are allowed to cross lines and donâ€™t necessarily have to hit each node once, then there are many ways of solving the puzzle as there arenâ€™t as many parameters to follow, but the conjectures as to how the puzzle will be solved primarily depend on the geometry of the puzzle, as well as the rules necessary to follow in order to solve the puzzle.

In trying to verify the conjecture that my group came up with which was; You have to start on the edges or the perimeter./ It is best to move horizontally/vertically first, proposed by Rachel. I decided to focus on primarily; â€ścoming up with ideas and testing them out, trying to understand what the conjecture says, & trying to solve puzzles others created.â€ť Keeping in mind the conjecture I decided to look at one of the sketches we had discussed in our group and find a solution by any means first. I looked at https://sketchtoy.com/70193302 I was able to determine a solution by going from D-G-B-E-F-C-A-B-D. In this solution we are not starting at the edges or perimeter. I am taking a moment to clarify that starting at the perimeter means that we will run the perimeter before moving to the center. Going back to the sketch toy I noticed that there was no way of running the perimeter and then the middle without any overlap; I did notice that some of the intersections were not labeled. This observation motivated me to speculate that in working the entire perimeter 1st our conjecture might work if the added that every angle/vertex and intersection should be labeled to be considered a stopping point. I labeled the intersections that are unlabeled 1,2,3. This allowed me to go back to the sketch today and apply our conjecture making the solution C-F-3-G-B-E-2-A-1-D-C-1-3-2-B-3-C.

With this new refinement I decided to try another sketch we discussed as a team to check if my theory of how our conjecture could possibly evolve was true. I looked at https://sketchtoy.com/70193330, and after multiple tries I decided the sketch as it currently is does not have a solution that could prove our conjecture as it was originally stated. But taking into consideration labeling all vertices and intersections We notice that in the middle of this sketch there is an unlabeled intersection which I decided to label G. When Every intersection and vertex was labeled our group conjecture works making the solution to this F-E-G-D-C-B-G-A-F-C.

Now my question is, could we refine our conjecture to include that to travel the edges of the perimeter first you must consider all vertices and intersections a stopping point. To check if this could be true I looked at a third sketch. I looked specifically at one I had created that had a solution that did not prove our conjecture. The sketch I looked at was the following https://sketchtoy.com/70194491 since this was one of the sketches I had posted I looked at the solutions one of the members from my team(Sue) commented on the post. She had stated that a possible solution could be B-A-F-B-A-D-E-C-D-F-C-B-E-D-E (I added on the last DE). I then tested if the sketch, as is, had a solution with our postulated conjecture which it did not. I then applied my theory about labeling all intersections and vertices. I tested it multiple times and didnâ€™t seem to find a solution. This being said, I think a second pair of eyes testing the evolution of our conjecture could help clarify if I am just not testing every possible scenario or if the conjecture in this case did not work.

As we know group work is all about good communication and sharing ideas with each other in order to make a conclusion with solid evidence to support this. Similarly, in the beginning, I have shared my idea about this connecting bridges problem. We all explained what approaches we have used to solve this problem, unsurprisingly we all used some kind of different strategies to solve this problem. For example one of our groupmates believe that it’s not best to start from a corner and Itâ€™s best to not start where only two line segments meet, My other groupmate thinks You have to start on the edges or the perimeter./ It is best to move horizontally/vertically first, and according to our other groupmate, It is best to start where you can move diagonally. and from their perspective they are right I believe. However, I think It is best to start from the middle. Eventually, we all agreed that even though our method was different but all of them are still effective to find the solution to this bridge’s problem. However some of our methods might be more efficient than others, consequently, we have used trial and error and analyses our connecting dot sketches.