Last year, the \mathrm{U.S.} House of Representatives passed a bill which would make Washington, \mathrm{D.C.} into the $51$st state. Naturally, the mathematicians are upset that Congress won’t prioritize mathematical interest of flag design in choosing how many \mathrm{U.S.} states there should be. Suppose the \mathrm{U.S.} flag must contain, as it does now, stars arranged in rows alternating between n and n - 1 stars, starting and ending with rows of n stars, where n \geq 2 is some integer and the flag has more than one row. What is the minimum number of states that the \mathrm{U.S.} would need to contain so that there are at least three different ways, excluding rotations, to arrange the stars on the flag?