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Arrangements of vertices of nonagon

Source:

March 18, 2011
countingdistinguishabilitygeometrygeometric transformationrotationAMCAIME

Problem Statement

The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 11 through 99 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 33. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.