57.8 Repeat Problem 57.7 with a regular hexagon in place of a regular pentagon.

Question

57.8 Repeat Problem 57.7 with a regular hexagon in place of a regular pentagon. Problem 57.7 is--- State and solve the problem that results from replacing a square by a regular pentagon in Example 57.2. Compare Problem 57.4. Example 57.2 is In how many distinguishable ways can the four edges of a square be painted with four different colors if there is no restriction on the number of times each color can be used, and two ways are considered indistinguishable if one can be obtained from the other by an isometry in the group of symmetries of the square? (This would be the case for a square that could be either rotated in the plane or turned over; the latter corresponds to reflection through a line in the plane.) The appropriate set S in this case is the set of 44 = 256 ways of painting the edges without regard to equivalence. The group is the one described at the beginning of Section 56. If ? is a group element, then ?(?) = 4k, where k is the number of independent choices to be made in painting the edges so as to have invariance under ?. ?(?0) = 44 (always if is the identity

Explanation / Answer

COMPLETE ANNSWER IS AT http://mathforum.org/library/drmath/sets/high_perms_combs.html?start_at=121

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