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Number of colored hexagons up to cyclic symmetry

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If C6 acts on a regular hexagon by rotation and each of the vertices is colored red, blue or green, use the Burnsideââ?¬â?¢s formula to determine how many possible colorings there are up to cyclic symmetry.

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Solution Summary

We show in detail how Burnside's theorem can be used to obtain the number of colored hexagons up to the cyclic symmetry.

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The number of colorings can be written as:

1/|G| Sum over all g in G of Fix(g)

Here G is the symmetry group, |G| is the number of elements in G (which is 6 in this case), and Fix(g) is the number of colorings that g will keep fixed. Let's consider all the rotations that are in G:

1) Identity. If we do nothing, then all possible colorings will be left fixed. There are 3^6 colorings in total, because there are 3 choices for each of the 6 vertices. We thus have:

Fix(Identity) = 3^6

2) Clockwise rotation over 60 degrees. Under this group element, a coloring changes such that the new color of a vertex becomes the old color of the vertex located 60 degrees counterclockwise. ...

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