Minimum Colors for an Icosahedron

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The discussion centers on determining the minimum number of colors needed to color the faces of an icosahedron so that no two adjacent faces share the same color. It references the four-color theorem, noting that extending this problem to three-dimensional shapes is not straightforward. The consensus is that solving this problem often involves guesswork, making it challenging to prove the minimality of a solution. The chromatic number for the dodecahedral graph, which is relevant to this discussion, is identified as three. This highlights the complexity and intrigue of coloring problems in three-dimensional geometry.
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How could you mathematically solve for the minimum amount of colors needed for each face of an icosahedron or any regular polyhedron to not touch? (E.g. a tetrahedron pyramid would need four unique colors.)
 
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"There is no obvious extension of the coloring problem to three-dimensional solid regions."
-taken from http://en.wikipedia.org/wiki/Four_color_theorem.

So it looks like it's guess and check mostly, and even then, you'd have a pretty tough time proving your solution is indeed minimal. It is an interesting problem though.
 
If I'm reading this right, what you want is the chromatic number of the dodecahedral (!) graph, which is 3.
 

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