**The assignment below is due BEFORE CLASS on Thursday, October 10th (it is essential that you complete it before class, as we will be doing a class activity building on the assignment).**

We are going to play a game creating walking tours of cities with bridges. Â We begin in the city of Kingâ€™s Mountain, which is built on four land masses â€“ both shores of a river and two islands in midstream â€“ connected by a total of seven bridges (shown in green).

EXAMPLE 1:Â Â Can you create a walking tour of the city that crosses every bridge exactly once? Â *You can begin anywhere you like, and end anywhere you like, as long as you cross each bridge just once.*

### Background –Â Graph Theory

We can simplify the picture of Kingâ€™s Mountain to make it easier to deal with:

The key elements of the map are the four land masses (letâ€™s label them A, B, C, and D) and the seven bridges (p,q,r,s,t,u and v) (*thanks to mathisfun.com for the images)*:

For the purposes of our problem, we can simply think about each land mass as a point (A, B, C, and D), and the bridges as lines connecting the points (p,q,r,s,t,u and v) – like this:

We call this kind of picture a **graph** – the points are called **vertices** and the the lines are called **edges**. Â Our goal of finding â€śa walking tour that crosses each bridge onceâ€ť is now matter of tracing out all the edges without lifting our pencil (and without repeating any edge).

### Assignment, Due Thursday 10/10 (beginning of class)

**Warm upÂ ***(This Warm Up is**Â just for practice – you do NOT need to submit your answers – see below for the three-part Assignment to be submitted).* Â The following examples build on the example above.

EXAMPLEÂ 2: If you are given the freedom to build one new bridge in King’s Mountain (“make one new edge in the graph”), can you do it in such a way the walking tour becomes possible? Â Do it!

EXAMPLE 3: If you are given the freedom to destroy one bridge (“erase one edge”), can you do it in such a way that the walking tour becomes possible? Do it!

EXAMPLEÂ 4: Construct walking tours for each of the following graphs (or decide if it is impossible).

**Assignment.**Â Your assignment has 4Â parts.

PART 1.Â For each of the four graphs below (G1 – G4), decide whether it is possible to create a walking tour crossing each bridge exactly once.Â Post your solutions here (TO POST A SOLUTION, JUST LIST THE POINTS OF YOUR WALKING TOUR IN ORDER).Â If it is not possible to create a create a solution, say so!

PART 2. Â Challenge your friends: Â Now itâ€™s up to you to build your own graph, and challenge your classmates to construct a walking tour (or to determine if it is impossible). Â It can consist of as many points as you wish, and as many bridges (edges) connecting them. Â You MUST label your pointsÂ “A, B, C…” etc. Â When youâ€™re finished, decide for yourself if a walking tour crossing each bridge exactly once is possible. Â *Â Remember, the most challenging puzzles are the ones where the answer is difficult to determine.* Post **two** puzzles in the comments. Â See the note Â “POSTING YOUR PUZZLE ONLINE” below for instructions on how to draw and share graphs online.

PART 3. Â Solve a friend’s puzzle. Â Leave a response to a friend’s posted puzzle, giving a solution. Â TO POST A SOLUTION, JUST LIST THE POINTS OF YOUR WALKING TOUR IN ORDER.

Example:

Here is a puzzle:Â http://sketchtoy.com/67467551

Here is a solution: (start at A) –Â A, B, D, A, E, B, C, E

PART 4. Â The third part of your assignment is to write a short paragraph (at least 3 sentences) responding to the following prompt. Â Be sure to respond to each part:

*Writing Prompt**: Â **Did you enjoy this assignment? Why or why not? Â Describe a connection between this assignment and our work in the class. Â (If you don’t believe there is a connection, try to *imagine* why we are doing this). Â Leave your response in the comments.*

POSTING YOUR PUZZLE ONLINE. Â I recommend the site sketchtoy.comÂ – it allows you to draw something, then click “SAVE” and get a link to your drawing. Â You can post the link in a comment, and we’ll be able to click on it and view your drawing. Â Â Don’t worry if it’s not pretty! Â For example, here is a graph that I drew (can you find a walking tour that crosses all edges?):Â http://sketchtoy.com/67467556

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