# 3.1 Educational Tools

In an attempt to make the subtleties to CopyCat more approachable, I have made a couple of handouts. One is a page of cutouts: it has all both the cube and octahedron shapes faces so that the shapes can be constructed, and also all the fixed positions of the shapes so that the fixed positions can be arranged to make the graphs or trees mentioned on the page called Educational Value . Preferably, the cutout page should be photocopied on to bristle board for rigidity--the 3D models will hold together better.

Another handout is two grids. These grids can be used to roll the shapes on. For the cube, the grid is the standard square grid that everyone is familiar with. For the octahedron, the grid is triangular. The triangular gird is very important in helping the students to see why the octahedron only has 24 fixed positions rather than 48. Both grids exactly fit the two 3D models from the first handout.

The last set of handouts I have my doubts if they are useable since they are a bit involved. The first is a chart of the minimum distances between the nodes in the cube graph. It is like the distances chart in the corner of road maps telling you that Vancouver and Hope is 60 km. It is interesting since in this chart Hope to Vancouver could be 100 km because of the non commutativity of the graph. My goal with the chart is to possibly have players try to come up with algorithms that use the chart to compute the shortest path and be sure that it really is the shortest path. For simple patterns it is do-able but it would get pretty tedious with patterns that use a large number of different faces. To compliment the chart, there is a set of 4 transparencies that can be overlaid on the cube graphs that show the shortest path from a particular node to any other node. Also another version of the cube graph is needed to keep the number of transparencies needed to 4 rather than 8. So, with the chart the player can compute the shortest distances and with the graphs and transparencies the player can find an exact path to achieve the shortest path. Once an algorithm is perfected for the task it becomes clean why people want computers to run the algoritms.

These handouts are written directly as postscript files so to print them out with high quality you may have to take them to some copy place. If you try these with a class let me know how it goes. I'm open to suggestions.
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