## Exploring Discrete Mathematics Summary

### Monday - Friday, July 3 - 7, 2006

We spent this week working on our projects. During the first two days, we completed the two activities we started last week. On day four and five we worked on two other activities. Both of these activities use puzzles to motivate students to explain why certain graphs cannot be planar, meaning that no edges can cross.

The first activity investigates why K5, which is a complete graph of five vertices, can't be planar. We motivate this activity with the following question: "Imagine there are four cities that all must be connected to each other by railroad tracks. To minimize accidents, no tracks are allowed to cross. Draw a sketch on a piece of paper showing how this can be done. Now try to do the same thing with five cities." The students then investigate patterns and develop an inequality that shows that K5 cannot be done.

In the second activity, we examine whether K3,3 is planar. K3,3 is a graph that connects three vertices in one set to each of three vertices in another set, but the vertices within each set are not connected to each other. Students are presented with the following puzzle: "Imagine there are two houses and three separate utilities (gas, water, electricity). Can we connect each house to each of the utilities without crossing any lines? How about two houses to four utilities? How about three houses to three utilities?" Again, students investigate patterns and develop an inequality that shows that K3,3 cannot be done. We also spent some time investigating how Euler's formula can be used to show that there are only five Platonic solids.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.