Talk:Seven Bridges of Königsberg
From Math Images
Contents |
Abram 7/10
Just about there. One more thing.
"At vertex 2, we can see that this solution fails because we must choose to go to either vertex 1 or 4, but with no possible way out of those vertexes. Note that 2 is neither a starting point nor an end point, and thus with an odd number of lines, scraps any possible solution."
This description doesn't really match your picture. First, in this picture the solution fails at vertex 1, not vertex 2. Also, your second sentence is true, but again, it's the oddness of vertex 1's edges, not vertex 2's edges, that are the problem in this particular example.
Can you change your text to reflect this? Something about how even though you weren't finished with the walk yet, you got stuck at vertex 1. You entered but weren't able to exit because it has an odd number of lines.
Abram 7/8
Nice job with the interim picture. Two things that will help this section:
- Could you label each purple line as "step 1, step 2", etc.
- You say that 2 isn't a start-point or an end-point, which ruins this possible solution, but the problem is not with vertex 2 -- the problem is that you are about to be stuck at vertex 1 or vertex 4. You could just scrap this sentence about vertex 2 scrapping any possible solution. You could replace this with something about how you get stuck at vertex 1 or vertex 4 because they have an odd number of edges.
I still have a slight objection to:
- Now we make the key observation that the walker must enter and exit every landmass.
It would be more precise if you rewrote this as:
- Now we make the key observation that, except for the starting and finishing landmass, the walker must exit each landmass exactly as many times as she enters it, and each entry and exit must be through a different line.
The page otherwise looks great. I retract my previous statement about the plural of "vertex" being "vertices". Wikipedia tells me that "vertexes" is acceptable, and far be it from me to complain about language evolving in ways that make things simpler.
Anna 7/4
I'd still really like to see a diagram indicating a possible path, and why the process fails.
I had forwarded the idea to the drexel people, but I can make an interim picture.
- much appreciated :)
Abram 7/2
Really nice job laying out the solution to this problem. The way you reduce the context of the problem to its graph theoretical essence is especially elegant.
I want to pick up on the conversation from last week, and make two suggestions. One is to edit the following sentence:
- Now we make the key observation that the walker must enter and exit every landmass.
It would be more precise if you rewrote the first sentence as:
- Now we make the key observation that, except for the starting and finishing landmass, the walker must exit each landmass exactly as many times as she enters it, and each entry and exit must be through a different line.
The other suggestion would be to further flesh this out by choosing one of the vertices and walking the reader through an explanation of why that vertex couldn't be a non-starting-or-ending landmass. Vertex 2 works really well because it has 5 edges, so people can really see the process unfold.
The one other thing is that the plural of "vertex" is "vertices", not "vertexes." I know this is really small. It's just so easy to fix and seems like one of those things that should be right for anything that's "in print".
Anna 6/26
I don't think you need a real illustration, but more of a direct explanation. My other thought would be to create a path, noting which bridges are crossed when. This can simply be a numbering, and you can see that you'll get stuck on a vertex. Maybe, just take your very simplified picture (the lines and vertices) and add arrows indicating a path? that seems easiest...
Alan, 6/25
It would be great to have something illustrating this paragraph: "Now we make the key observation that the walker must enter and exit every landmass. In order for this to be possible there must be an even number of lines at every vertex with the exceptions of the starting and finishing vertexes of the walk. For those two vertexes, an odd number of vertexes is permitted since the walker only exits the starting vertex and only enters the final vertex."
Would a picture of a walker walking through an island suffice?
Gene, 6/21
I think you got good comments in your presentation, so I'll be brief, Alan. I like very much the idea of having an interactive image upon which people could try various solutions.
While we could literally test out every possible case by hand, this would be extremely tedious and prone to error. But possible!
"Instead we will analyze the problem abstractly. By abstract, we mean to essentialize" ["reduce the problem to its essentials"? You shouldn't "verb" when you don't need to] Something about eliminating all inessential feature helps to get a better grip on the problem?
"Keeping these observations in mind, we resize the landmasses to points, and the bridges to lines." Maybe impose letters on the previous image and your graph showing the landmasses?
It would be great to have something illustrating this paragraph: "Now we make the key observation that the walker must enter and exit every landmass. In order for this to be possible there must be an even number of lines at every vertex with the exceptions of the starting and finishing vertexes of the walk. For those two vertexes, an odd number of vertexes is permitted since the walker only exits the starting vertex and only enters the final vertex."
Good job, Alan. Isn't it Euler who solved the problem? Also, you're a bit close to the Wikipedia treatment. I think you should make sure there are some different illustrations, etc.