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Above is a graph that could be used to represent the vertices of a dodecahedron (below). Each circle represents a vertex, and each line represents an edge between two vertices.
Challenge #1 Try to color the faces of the dodecahedron, that is, the left-over spaces formed between the lines and circles, in such a way that no two spaces of the same color share the same edge. What is the least number of colors you can use to accomplish this?
Challenge #2 Try to color the circles in such a way that no two circles of the same color are connected by a line. What is the least number of colors you can use to accomplish this?
Challenge #3 Try to color the edges in such a way that no two edges of the same color are connected to the same circle. As before, what is the least number colors you can use to accomplish this?
Click on the image above to download a black and white BMP file of the graph that you can try coloring, for example, in MSPAINT.exe using the fill tool.
For a 3 dimensional version of challenge #3, try constructing the dodecahedron from origami modules of different colors. This can be done, for example, using 30 PHiZZ units.
If you're really willing to spend some time on this problem, try to do the same for all the other polyhedra. There are 80 of them. You'll find a beautiful catalog of polyhedra here.
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Think you got the answer? Send it to me and earn eternal (ephemeral?) fame --> 
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