Problem:
The vertices of a regular nonagon (-sided polygon) are to be labeled with the digits through in such a way that the sum of the numbers on every three consecutive vertices is a multiple of . 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.
Solution:
Let the nonagon be , and let the digits on the vertices be , and , respectively. It may be assumed that , so . If , it is impossible for to be a multiple of . Therefore one of and belongs to the set , and the other belongs to . It follows that the only possible sequences of digits are and . In each of the two sequences, there are possible choices for the ordered pair and possible choices for each of the ordered triples and . Thus the total number of distinguishable acceptable arrangements is . .
The problems on this page are the property of the MAA's American Mathematics Competitions