The change-making problem for six coin values and beyond

The change-making problem asks: given a positive integer $v$ and a collection $C$ of integer coin values $c_1=1<c_2<c_3<\cdots <c_n$, what is the minimum number of coins needed to represent $v$ with coin values from $C$? For some coin systems $C$, the greedy algorithm finds a representation with a minimum number of coins for all $v$. We call such coin systems orderly. However, there are coin systems where the greedy algorithm fails to always produce a minimal representation. Over the past fifty years, progress has been made on the change-making problem, including finding a characterization of all orderly coin systems with 3, 4, and 5 coin values. We characterize orderly coin systems with 6 coin values, and we make generalizations to orderly coin systems with $n$ coin values.