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Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible. Please do the 4 problems as soon as possible.
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Explanation & Answer

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Problem 1:
The set X on which G D 6 acts is the set of functions from the set of sides of the hexagon to the set of
3 colors {color 1, color 2, color 3}. Hence, we have the following table of fixed point sets
By Burnside’s lemma, the number ( N ) of nonequivalent 3-colorings of the sides of a hexagon is
calculated by
1 6
3 3 34 4 33 2 32 2 31
1...
