Construct a loop with a 11" by 1" strip of paper by connecting end to end and taping. Notice that the loop has two sides and two edges. Draw a line down the middle of the outside of the loop until you end where you started. Cut along this line. What is the result? Place the results in the table below.
Construct a twisted loop with a 11" by 1" strip of paper by connecting one end to other end that has been given a half-twist, and then tape. This loop is called a Möbius Strip. Notice that the loop has one side and one edge. Draw a line down the middle of the outside of the loop until you end where you started. Cut along this line. What is the result? Place the results in the table below.
Repeat the actions above with loops with two, three, four and five twists. Record the results in the table below. Generalize to the case with 100 twists and then n twists.
| Twists | Sides | Edges | Result After Cutting |
| 0 | |||
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 100 | |||
| n |
A network consists of a series of connected vertices (points) and arcs (paths). Networks are either traversable or not traversable. If a network is traversable, it can be completely traced without lifting your pencil from the paper. Each point is a vertex of the network. Networks that are traversable can be either Type 1, which can be traced by starting and ending at the same vertex, or Type 2, which can be traced by starting at one vertex and ending at another. Networks can be analyzed by examining the vertices and arcs leading to those vertices. Examine each of the networks below, find the number of even vertices, the number of odd vertices, and determine whether they are traversable or not. Place your results in the table below the networks.
| Network Name |
Number of Even Vertices |
Number of Odd Vertices |
Traversable Type 1 |
Traversable Type 2 |
| A | ||||
| B | ||||
| C | ||||
| D | ||||
| E | ||||
| F |
What do you notice about the Type 1 networks?
Notice anything about the Type 2 networks?
Test your guesses by drawing networks of your own.
In the city of Königsberg, now called Kalingrad, and located in the GUS, there were seven bridges connecting parts of the city that are situated in and around the Pregel River. A common pastime in Königsberg was to take a leisurely walk through the city, passing across each bridge exactly once. Is this possible? If so, draw the path that crosses all seven bridges. If not, why not? Construct a network to represent the Königsberg Bridge Problem, using arcs for bridges and points for land areas.
Construct the corresponding network below.
Is this network traversable? Why or why not?
How would a change in the map affect a change in the network?
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